Survival Analysis Using SAS: A Practical Guide, Second Edition Paul D Allison SAS Publishing (2010)

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Survival Analysis
Using SAS
®

A Practical Guide Second Edition

Paul D. Allison

The correct bibliographic citation for this manual is as follows: Allison, Paul D. 2010. Survival Analysis Using SAS :
A Practical Guide, Second Edition. Cary, NC: SAS Institute Inc.
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Survival Analysis Using SAS : A Practical Guide, Second Edition
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CONTENTS
PREFACE vii

Chapter 1 Introduction 1
What Is Survival Analysis? 1
What Is Survival Data?

Why Use Survival Analysis? 4
Approaches to Survival Analysis 5
What You Need to Know 6
Computing Notes 7

Chapter 2 Basic Concepts of Survival Analysis 9
Introduction 9
Censoring 9
Describing Survival Distributions 15
Interpretations of the Hazard Function 18
Some Simple Hazard Models 20
The Origin of Time 23
Data Structure 26

Chapter 3 Estimating and Comparing Survival Curves with
PROC LIFETEST 29
Introduction

The Kaplan-Meier Method

Testing for Differences in Survivor Functions 38
The Life-Table Method 49
Life Tables from Grouped Data 55
Testing for Effects of Covariates 59
Log Survival and Smoothed Hazard Plots 64
Conclusion 69

CONTENTS

Chapter 4 Estimating Parametric Regression Models with
PROC LIFEREG 71
Introduction 71
The Accelerated Failure Time Model 72
Alternative Distributions 77
Categorical Variables and the CLASS Statement 87
Maximum Likelihood Estimation 89
Hypothesis Tests 95
Goodness-of-Fit Tests with the Likelihood-Ratio Statistic 98
Graphical Methods for Evaluating Model Fit 100
Left Censoring and Interval Censoring 103
Generating Predictions and Hazard Functions 108
The Piecewise Exponential Model 112
Bayesian Estimation and Testing 117
Conclusion 124

Chapter 5 Estimating Cox Regression Models with
PROC PHREG 125
Introduction 125
The Proportional Hazards Model 126
Partial Likelihood 128
Tied Data 142
Time-Dependent Covariates 153
Cox Models with Nonproportional Hazards 172
Interactions with Time as Time-Dependent Covariates 177
Nonproportionality via Stratification 179
Left Truncation and Late Entry into the Risk Set 183
Estimating Survivor Functions 186
Testing Linear Hypotheses with CONTRAST or TEST Statements 192
Customized Hazard Ratios 195
Bayesian Estimation and Testing 197
Conclusion 200

Chapter 6 Competing Risks

203

Introduction 203
Type-Specific Hazards 204
Time in Power for Leaders of Countries: Example 207
Estimates and Tests without Covariates 208

CONTENTS

Covariate Effects via Cox Models 213
Accelerated Failure Time Models 220
Alternative Approaches to Multiple Event Types 227
Conclusion 232

Chapter 7 Analysis of Tied or Discrete Data with
PROC LOGISTIC 235
Introduction 235
The Logit Model for Discrete Time 236
The Complementary Log-Log Model for Continuous-Time Processes 240
Data with Time-Dependent Covariates 243
Issues and Extensions 246
Conclusion 255

Chapter 8 Heterogeneity, Repeated Events, and Other
Topics 257
Introduction 257
Unobserved Heterogeneity 257
Repeated Events 260
2

Generalized R

282

Sensitivity Analysis for Informative Censoring 283

Chapter 9 A Guide for the Perplexed

289

How to Choose a Method 289
Conclusion 292

Appendix 1 Macro Programs

293

Introduction 293
The LIFEHAZ Macro 293
The PREDICT Macro 296

Appendix 2 Data Sets

299

Introduction 299
The MYEL Data Set: Myelomatosis Patients 299
The RECID Data Set: Arrest Times for Released Prisoners 300
The STAN Data Set: Stanford Heart Transplant Patients 301
The BREAST Data Set: Survival Data for Breast Cancer Patients 302
The JOBDUR Data Set: Durations of Jobs 302

CONTENTS

The ALCO Data Set: Survival of Cirrhosis Patients 302
The LEADERS Data Set: Time in Power for Leaders of Countries 303
The RANK Data Set: Promotions in Rank for Biochemists 304
The JOBMULT Data Set: Repeated Job Changes 305

References
Index

313

307

PREFACE
When the first edition of Survival Analysis Using SAS was published
in 1995, my goal was to provide an accessible, data-based introduction to
methods of survival analysis, one that focused on methods available in SAS
and that also used SAS for the examples. The success of that book
confirmed my belief that statistical methods are most effectively taught by
showing researchers how to implement them with familiar software using
real data.
Of course, the downside of a software-based statistics text is that the
software often changes more rapidly than the statistical methodology. In the
15 years that the first edition of the book has been in print, there have been
so many changes to the features and syntax of SAS procedures for survival
analysis that a new edition has been long overdue. Indeed, I have been
working on this second edition for several years, but got partially
sidetracked by a four-year term as department chair. So, it’s a great relief
that I no longer have to warn potential readers about out-of-date SAS code.
Although the basic structure and content of the book remain the same,
there are numerous small changes and several large changes. One global
change is that all the figures use ODS Graphics. Here are the other major
changes and additions:




Chapter 3, “Estimating and Comparing Survival Curves
with PROC LIFETEST.” This chapter documents some
major enhancements to the STRATA statement, which
now offers several alternative tests for comparing survivor
functions. It also allows for pairwise comparisons and for
adjustment of p-values for multiple comparisons. In the
first edition, I demonstrated the use of a macro called
SMOOTH, which I had written to produce smoothed
graphs of hazard function. That macro is no longer
necessary, however, because the PLOTS option (combined
with ODS Graphics) can now produce smoothed hazard
functions using a variety of methods.
Chapter 4, “Estimating Parametric Regression Models with
PROC LIFEREG.” This chapter now includes a section on
the PROBPLOT command, which offers graphical
methods to evaluate the fit of each model. The last section

PREFACE

VIII



introduces the new BAYES statement, which (as the name
suggests) makes it possible to do a Bayesian analysis of
any of the parametric models using MCMC methods.
Chapter 5, “Estimating Cox Regression Models with PROC
LIFEREG.” The big change here is the use of the counting
process syntax as an alternative method for handling timedependent covariates. When I wrote the first edition, the
counting process syntax had just been introduced, and I
did not fully appreciate its usefulness for handling
predictor variables that vary over time. Another new topic
is the use of the ASSESS statement to evaluate the
proportional hazards assumption. Finally, there is a
section that describes the BAYES statement for estimating
Cox models and piecewise exponential models.
Chapter 6, “Competing Risks.” This chapter now contains
a section on cumulative incidence functions, which is a
popular alternative approach to competing risks.
Chapter 7, “Analysis of Tied or Discrete Data with the
LOGISTIC Procedure.” The first edition also used the
PROBIT and GENMOD procedures to do discrete time
analysis. But, PROC LOGISTIC has been enhanced to the
point where the other procedures are no longer needed for
this application.
Chapter 8, “Heterogeneity, Repeated Events, and Other
Topics.” For repeated events and other kinds of clustered
data, the WLW macro that I described in the first edition
has been superseded by the built-in option
COVSANDWICH. In this chapter, I also describe the use of
the new GLIMMIX procedure to estimate random-effects
models for discrete time data.

Please note that I use the following convention for presenting SAS
programs. All words that are part of the SAS language are shown in
uppercase. All user-specified variable names and data set names are in
lowercase. In the main text itself, both SAS keywords and user-specified
variables are in uppercase.
I am most grateful to my editor, George McDaniel, for his patient
persistence in getting me to finish this new edition.

CHAPTER 1
Introduction
What Is Survival Analysis? 1
What Is Survival Data?

Why Use Survival Analysis? 4
Approaches to Survival Analysis 5
What You Need to Know 6
Computing Notes 7

WHAT IS SURVIVAL ANALYSIS?
Survival analysis is a class of statistical methods for studying the
occurrence and timing of events. These methods are most often applied to
the study of deaths. In fact, they were originally designed for that purpose,
which explains the name. That name is somewhat unfortunate, however,
because it encourages a highly restricted view of the potential applications
of these methods. Survival analysis is extremely useful for studying many
different kinds of events in both the social and natural sciences, including
disease onset, equipment failures, earthquakes, automobile accidents,
stock market crashes, revolutions, job terminations, births, marriages,
divorces, promotions, retirements, and arrests. Because these methods
have been adapted—and sometimes independently discovered—by
researchers in several different fields, they also go by several different
names: event history analysis (sociology), reliability analysis (engineering),
failure time analysis (engineering), duration analysis (economics), and
transition analysis (economics). These different names don’t imply any
real difference in techniques, although different disciplines may
emphasize slightly different approaches. Because survival analysis is the
name that is most widely used and recognized, it is the name I use here.
This book is about doing survival analysis with SAS. I have also
written an introduction to survival analysis that is not oriented toward a
specific statistical package (Allison, 1984), but I prefer the approach taken
here. To learn any kind of statistical analysis, you need to see how it’s
actually performed in some detail. And to do that, you must use a
particular computer program. But which one? Although I have performed
survival analysis with many different statistical packages, I am convinced
that SAS currently has the most comprehensive set of full-featured

CHAPTER 1

Introduction

procedures for doing survival analysis. When I compare SAS with any of
its competitors in this area, I invariably find some crucial capability that
SAS has but that the other package does not. When you factor in the
extremely powerful tools that SAS provides for data management and
manipulation, the choice is clear. On the other hand, no statistical package
can do everything, and some methods of survival analysis are not available
in SAS. I occasionally mention such methods, but the predominant
emphasis in this book is on those things that SAS can actually do.
I don’t intend to explain every feature of the SAS procedures
discussed in this book. Instead, I focus on those features that are most
widely used, most potentially useful, or most likely to cause problems and
confusion. It’s always a good idea to check the official SAS documentation
or online help file.

WHAT IS SURVIVAL DATA?
Survival analysis was designed for longitudinal data on the
occurrence of events. But what is an event? Biostatisticians haven’t written
much about this question because they have been overwhelmingly
concerned with deaths. When you consider other kinds of events,
however, it’s important to clarify what is an event and what is not. I define
an event as a qualitative change that can be situated in time. By a
qualitative change, I mean a transition from one discrete state to another. A
marriage, for example, is a transition from the state of being unmarried to
the state of being married. A promotion consists of the transition from a job
at one level to a job at a higher level. An arrest can be thought of as a
transition from, say, two previous arrests to three previous arrests.
To apply survival analysis, you need to know more than just who is
married and who is not married. You need to know when the change
occurred. That is, you should be able to situate the event in time. Ideally,
the transitions occur virtually instantaneously, and you know the exact
times at which they occur. Some transitions may take a little time,
however, and the exact time of onset may be unknown or ambiguous. If the
event of interest is a political revolution, for example, you may know only
the year in which it began. That’s all right so long as the interval in which
the event occurs is short relative to the overall duration of the observation.
You can even treat changes in quantitative variables as events if the
change is large and sudden compared to the usual variation over time. A
fever, for example, is a sudden, sustained elevation in body temperature. A
stock market crash could be defined as any single-day loss of more than 20
percent in some market index. Some researchers also define events as

CHAPTER 1

Introduction

occurring when a quantitative variable crosses a threshold. For example, a
person is said to have fallen into poverty when income goes below some
designated level. This practice may not be unreasonable when the
threshold is an intrinsic feature of the phenomenon itself or when the
threshold is legally mandated. But I have reservations about the
application of survival methods when the threshold is arbitrarily set by the
researcher. Ideally, statistical models should reflect the process generating
the observations. It’s hard to see how such arbitrary thresholds can
accurately represent the phenomenon under investigation.
For survival analysis, the best observation plan is prospective. You
begin observing a set of individuals at some well-defined point in time,
and you follow them for some substantial period of time, recording the
times at which the events of interest occur. It’s not necessary that every
individual experience the event. For some applications, you may also want
to distinguish between different kinds of events. If the events are deaths,
for example, you might record the cause of death. Unlike deaths, events
like arrests, accidents, or promotions are repeatable; that is, they may
occur two or more times to the same individual. While it is definitely
desirable to observe and record multiple occurrences of the same event,
you need specialized methods of survival analysis to handle these data
appropriately.
You can perform survival analysis when the data consist only of the
times of events, but a common aim of survival analysis is to estimate
causal or predictive models in which the risk of an event depends on
covariates. If this is the goal, the data set must obviously contain
measurements of the covariates. Some of these covariates, like race and
sex, may be constant over time. Others, like income, marital status, or
blood pressure, may vary with time. For time-varying covariates, the data
set should include as much detail as possible on their temporal variation.
Survival analysis is frequently used with retrospective data in which
people are asked to recall the dates of events like marriages, child births,
and promotions. There is nothing intrinsically wrong with this approach
as long as you recognize the potential limitations. For one thing, people
may make substantial errors in recalling the times of events, and they may
forget some events entirely. They may also have difficulty providing
accurate information on time-dependent covariates. A more subtle
problem is that the sample of people who are actually interviewed may be
a biased subsample of those who may have been at risk of the event. For
example, people who have died or moved away will not be included.
Nevertheless, although prospective data are certainly preferable, much can
be learned from retrospective data.

CHAPTER 1

Introduction

WHY USE SURVIVAL ANALYSIS?
Survival data have two common features that are difficult to handle
with conventional statistical methods: censoring and time-dependent
covariates (sometimes called time-varying explanatory variables).
Consider the following example, which illustrates both these problems. A
sample of 432 inmates released from Maryland state prisons was followed
for one year after release (Rossi et al., 1980). The event of interest was the
first arrest. The aim was to determine how the occurrence and timing of
arrests depended on several covariates (predictor variables). Some of these
covariates (like race, age at release, and number of previous convictions)
remained constant over the one-year interval. Others (like marital and
employment status) could change at any time during the follow-up period.
How do you analyze such data using conventional methods? One
possibility is to perform a logistic regression analysis with a dichotomous
dependent variable: arrested or not arrested. But this analysis ignores
information on the timing of arrests. It’s natural to suspect that people who
are arrested one week after release have, on average, a higher propensity to
be arrested than those who are not arrested until the 52nd week. At the
least, ignoring that information should reduce the precision of the
estimates.
One solution to this problem is to make the dependent variable the
length of time between release and first arrest and then estimate a
conventional linear regression model. But what do you do with the
persons who were not arrested during the one-year follow-up? Such cases
are referred to as censored. There are a couple of obvious ad-hoc methods
for dealing with censored cases, but neither method works well. One
method is to discard the censored cases. That method might work
reasonably well if the proportion of censored cases is small. In our
recidivism example, however, fully 75 percent of the cases were not
arrested during the first year after release. That’s a lot of data to discard,
and it has been shown that large biases may result. Alternatively, you
could set the time of arrest at one year for all those who were not arrested.
That’s clearly an underestimate, however, and some of those ex-convicts
may never be arrested. Again, large biases may occur.
Whichever method you use, it’s not at all clear how a timedependent variable like employment status can be appropriately
incorporated into either the logistic model for the occurrence of arrests or
the linear model for the timing of arrests. The data set contains information
on whether each person was working full time during each of the 52 weeks
of follow-up. You could, I suppose, estimate a model with 52 indicator
(dummy) variables for employment status. Aside from the computational

CHAPTER 1

Introduction

awkwardness and statistical inefficiency of such a procedure, there is a
more fundamental problem: all the employment indicators for weeks after
an arrest might be consequences of the arrest rather than causes. In
particular, someone who is jailed after an arrest is not likely to be working
full time in subsequent weeks. In short, conventional methods don’t offer
much hope for dealing with either censoring or time-dependent covariates.
By contrast, all methods of survival analysis allow for censoring, and
many also allow for time-dependent covariates. In the case of censoring,
the trick is to devise a procedure that combines the information in the
censored and uncensored cases in a way that produces consistent
estimates of the parameters of interest. You can easily accomplish this by
the method of maximum likelihood or its close cousin, partial likelihood.
Time-dependent covariates can also be incorporated with these likelihoodbased methods. Later chapters explain how you can usefully apply these
methods to the recidivism data.

APPROACHES TO SURVIVAL ANALYSIS
One of the confusing things about survival analysis is that there are
so many different methods: life tables, Kaplan-Meier estimators,
exponential regression, log-normal regression, proportional hazards
regression, competing risks models, and discrete-time methods, to name
only a few. Sometimes these methods are complementary. Life tables have
a very different purpose than regression models, for example, and discretetime methods are designed for a different kind of data than continuoustime methods. On the other hand, it frequently happens that two or more
methods may seem attractive for a given application, and the researcher
may be hard-pressed to find a good reason for choosing one over another.
How do you choose between a log-normal regression model (estimated
with the LIFEREG procedure) and a proportional hazards model (estimated
with the PHREG procedure)? Even in the case of discrete-time versus
continuous-time methods, there is often considerable uncertainty about
whether time is best treated as continuous or discrete. One of the aims of
this book is to help you make intelligent decisions about which method is
most suitable for your particular application. SAS/STAT software contains
six procedures that can be used for survival analysis. Here’s an overview of
what they do:
LIFETEST

is primarily designed for univariate analysis of the timing
of events. It produces life tables and graphs of survival
curves (also called survivor functions). Using several

CHAPTER 1

Introduction

methods, this procedure tests whether survival curves are
the same in two or more groups. PROC LIFETEST also tests
for associations between event times and time-constant
covariates, but it does not produce estimates of parameters.
LIFEREG

estimates regression models with censored, continuoustime data under several alternative distributional
assumptions. PROC LIFEREG allows for several varieties of
censoring, but it does not allow for time-dependent
covariates.

PHREG

uses Cox’s partial likelihood method to estimate regression
models with censored data. The model is less restrictive
than the models in PROC LIFEREG, and the estimation
method allows for time-dependent covariates. PROC
PHREG handles both continuous-time and discrete-time
data.

LOGISTIC

is designed for general problems in categorical data
analysis, but it is effective and flexible in estimating
survival models for discrete-time data with time-dependent
covariates.

GENMOD

estimates the same discrete-time survival models as PROC
LOGISTIC, but it can handle repeated events using
generalized estimating equation (GEE) methods.

NLMIXED

estimates random-effects (mixed) parametric models for
repeated events.

All of these procedures can be used to estimate competing risks
models that allow for multiple kinds of events, as described in Chapter 6,
“Competing Risks.”

WHAT YOU NEED TO KNOW
I have written this book for the person who wants to analyze survival
data using SAS but who knows little or nothing about survival analysis.
The book should also be useful if you are already knowledgeable about
survival analysis and simply want to know how to do it with SAS. I
assume that you have a good deal of practical experience with ordinary
least squares regression analysis and that you are reasonably familiar with

CHAPTER 1

Introduction

the assumptions of the linear model. There is little point in trying to
estimate and interpret regression models for survival data if you don’t
understand ordinary linear regression analysis.
You do not need to know matrix algebra, although I sometimes use
the vector notation βx = β1 x1 + β 2 x2 + ...β k xk to simplify the presentation of
regression models. A basic knowledge of limits, derivatives, and definite
integrals is helpful in following the discussion of hazard functions, but
you can get by without those tools. Familiarity with standard properties of
logarithms and exponentials is essential, however. Chapters 4 and 5 each
contain a more technical section that you can skip without loss of
continuity. (I note this at the beginning of the section.) Naturally, the more
experience you have with SAS/STAT and the SAS DATA step, the easier it
will be to follow the discussion of SAS statements. On the other hand, the
syntax for most of the models considered here is rather simple and
intuitive, so don’t be intimidated if you are a SAS neophyte.

COMPUTING NOTES
Most of the examples in this book were executed on a Dell laptop
running Windows Vista at 1.80 GHz with 1 GB of memory, using SAS 9.2.
Occasionally, I report comparisons of computing times for different
procedures, options, sample sizes, and program code. These comparisons
should only be taken as rough illustrations, however, because computing
time depends heavily on both the software and hardware configuration.

CHAPTER 1

Introduction

CHAPTER 2
Basic Concepts of Survival Analysis
Introduction 9
Censoring 9
Describing Survival Distributions 15
Interpretations of the Hazard Function 18
Some Simple Hazard Models 20
The Origin of Time 23
Data Structure 26

INTRODUCTION
In this chapter, I discuss several topics that are common to many
different methods of survival analysis:
 censoring, a nearly universal feature of survival data
 common ways of representing the probability distribution of
event times, especially the survivor function and the hazard
function
 choice of origin in the measurement of time, a tricky but
important issue that is rarely discussed in the literature on
survival analysis
 the basic data structure required for most computer programs
that perform survival analysis.

CENSORING
Not all survival data contain censored observations, and censoring
may occur in applications other than survival analysis. Nevertheless,
because censored survival data are so common and because censoring
requires special treatment, it is this topic more than anything else that
unifies the many approaches to survival analysis.
Censoring comes in many forms and occurs for many different
reasons. The most basic distinction is between left censoring and right
censoring. An observation on a variable T is right censored if all you know
about T is that it is greater than some value c. In survival analysis, T is

CHAPTER 2

Basic Concepts of Survival Analysis

typically the time of occurrence for some event, and cases are right
censored because observation is terminated before the event occurs. Thus,
if T is a person’s age at death (in years), you may know only that T > 50, in
which case the person’s death time is right censored at age 50. This notion
of censoring is not restricted to event times. If you know only that a
person’s income is greater than $75,000 per year, then income is right
censored at $75,000.
Symmetrically, left censoring occurs when all you know about an
observation on a variable T is that it is less than some value. Again, you
can apply this notion to any sort of variable, not just an event time. In the
context of survival data, left censoring is most likely to occur when you
begin observing a sample at a time when some of the individuals may have
already experienced the event. If you are studying menarche (the onset of
menstruation), for example, and you begin following girls at age 12, you
may find that some of them have already begun menstruating. Unless you
can obtain information on the starting date for those girls, the age of
menarche is said to be left censored at age 12. (In the social sciences, left
censoring often means something quite different. Observations are said to
be left censored if the origin time, not the event time, is known only to be
less than some value. According to the definitions used here, such
observations are actually right censored.)
In both the natural and social sciences, right censoring is far more
common than left censoring, and most computer programs for survival
analysis do not allow for left censored data. Note, however, that the
LIFEREG procedure will handle left censoring as well as interval
censoring. Interval censoring combines both right and left censoring. An
observation on a variable T is interval censored if all you know about T is
that a < T < b, for some values of a and b. For survival data, this sort of
censoring is likely to occur when observations are made at infrequent
intervals, and there is no way to get retrospective information on the exact
timing of events. Suppose, for example, that a sample of people is tested
annually for HIV infection. If a person who was not infected at the end of
year 2 is then found to be infected at the end of year 3, the time of
infection is interval censored between 2 and 3. When all observations are
interval censored and the intervals are equally spaced, it is often
convenient to treat such data as discrete-time data, which is the subject of
Chapter 7, “Analysis of Tied or Discrete Data with PROC LOGISTIC.”
I won’t say anything more about left censoring and interval
censoring until Chapter 4, “Estimating Parametric Regression Models with
PROC LIFEREG.” The rest of this section is about the various patterns of

CHAPTER 2

Basic Concepts of Survival Analysis

right-censored data and the possible mechanisms generating such data.
The distinctions are important because some kinds of censoring are
unproblematic, while other kinds require some potentially dubious
assumptions.
The simplest and most common situation is depicted in Figure 2.1.
For concreteness, suppose that this figure depicts some of the data from a
study in which all persons receive heart surgery at time 0 and are followed
for 3 years thereafter. The horizontal axis represents time. Each of the
horizontal lines labeled A through E represents a different person. An X
indicates that a death occurred at that point in time. The vertical line at 3
is the point at which we stop following the patients. Any deaths occurring
at time 3 or earlier are observed and, hence, those death times are
uncensored. Any deaths occurring after time 3 are not observed, and those
death times are censored at time 3. Therefore, persons A, C, and D have
uncensored death times, while persons B and E have right-censored death
times. Observations that are censored in this way are referred to as singly
Type I censored.

Figure 2.1

Singly Right-Censored Data

CHAPTER 2

Basic Concepts of Survival Analysis

Type I means that the censoring time is fixed (that is, under the
control of the investigator), and singly refers to the fact that all the
observations have the same censoring time. Even observations that are not
censored are said to have a censoring time, in this case 3 years. It’s just that
their death times did not exceed their censoring time. Of course, censoring
times can also vary across individuals. For example, you might want to
combine data from two experiments, one with observation terminating
after 3 years and another with observation terminating after 5 years. This is
still Type I censoring, provided the censoring time is fixed by the design of
the experiment.
Type II censoring occurs when observation is terminated after a
prespecified number of events have occurred. Thus, a researcher running
an experiment with 100 laboratory rats may decide that the experiment
will stop when 50 of them have died. This sort of censoring is uncommon
in the social sciences.
Random censoring occurs when observations are terminated for
reasons that are not under the control of the investigator. There are many
possible reasons why this might happen. Suppose you are interested in
divorces, so you follow a sample of couples for 10 years beginning with
the marriage, and you record the timing of all divorces. Clearly, couples
that are still married after 10 years are censored by a Type I mechanism.
But for some couples, either the husband or the wife may die before the 10
years are up. Some couples may move out of state or to another country,
and it may be impossible to contact them. Still other couples may refuse to
participate in the study after, say, 5 years. These kinds of censoring are
depicted in Figure 2.2, where the O for couples B and C indicates that
observation is censored at that point in time. Regardless of the subject
matter, nearly all prospective studies end up with some cases that didn’t
make it to the maximum observation time for one reason or another.

CHAPTER 2

Basic Concepts of Survival Analysis

Figure 2.2

Randomly Censored Data

Random censoring can also be produced when there is a single
termination time, but entry times vary randomly across individuals.
Consider again the example in which people are followed from heart
surgery until death. A more likely scenario is one in which people receive
heart surgery at various points in time, but the study has to be terminated
on a single date (say, December 31, 2010). All persons still alive on that
date are considered censored, but their survival times from surgery will
vary. This censoring is considered random because the entry times are
typically not under the control of the investigator.
Standard methods of survival analysis do not distinguish among
Type I, Type II, and random censoring. They are all treated as generic rightcensored observations. Why make the distinctions, then? Well, if you have
only Type I or Type II censoring, you’re in good shape. The maximum
likelihood and partial likelihood methods discussed in this book handle
these types of censoring with no appreciable bias. Things are not so simple
with random censoring, however. Standard methods require that random

CHAPTER 2

Basic Concepts of Survival Analysis

censoring be noninformative. Here’s how Cox and Oakes (1984) describe
this condition:
A crucial condition is that, conditionally on the
values of any explanatory variables, the prognosis for
any individual who has survived to ci should not be
affected if the individual is censored at ci. That is, an
individual who is censored at c should be
representative of all those subjects with the same
values of the explanatory variables who survive to c
(p. 5).
The best way to understand this condition is to think about possible
violations. Suppose you follow a cohort of new graduate students to see
what factors affect how long it takes them to get a Ph.D. Many students
drop out before completing the degree, and these observations are
randomly censored. Unfortunately, there is good reason to suspect that
those who drop out are among those who would take a long time to finish
if they stayed until completion. This is called informative censoring. In the
divorce example mentioned earlier, it is plausible that those couples who
refuse to continue participating in the study are more likely to be
experiencing marital difficulties and, hence, at greater risk of divorce.
Again, the censoring is informative (assuming that measured covariates do
not fully account for the association between dropout and marital
difficulty).
Informative censoring can, at least in principle, lead to severe biases,
but it is difficult in most situations to gauge the magnitude or direction of
those biases. In the case of graduate student drop out, where the censored
cases are likely to be those who would have had long times to the event,
one consequence of censoring is to underestimate the median survival
time.
An easy solution to censoring that is random because of random
entry times is to include entry time as a covariate in a regression model.
This solution should work well in many situations, but it can lead to
computational difficulties if a large proportion of the observations is
censored.
Unfortunately, there is no statistical test for informative censoring
versus noninformative censoring. The best you can do is a kind of
sensitivity analysis that is described in Chapter 8, “Heterogeneity,

CHAPTER 2

Basic Concepts of Survival Analysis

Repeated Events, and Other Topics.” Aside from that, there are three
important lessons here:
1. First, in designing and conducting studies, you should do
everything possible to reduce the amount of random censoring.
You can’t rely on statistical methods to completely adjust for
such censoring.
2. Second, you should make an effort to measure and include in the
model any covariates that are likely to affect the rate of
censoring.
3. Third, in studies with high levels of random censoring, you may
want to place less confidence in your results than calculated
confidence intervals indicate.

DESCRIBING SURVIVAL DISTRIBUTIONS
All of the standard approaches to survival analysis are probabilistic
or stochastic. That is, the times at which events occur are assumed to be
realizations of some random process. It follows that T, the event time for
some particular individual, is a random variable having a probability
distribution. There are many different models for survival data, and what
often distinguishes one model from another is the probability distribution
for T. Before looking at these different models, you need to understand
three different ways of describing probability distributions.
Cumulative Distribution Function

One way that works for all random variables is the cumulative
distribution function, or c.d.f. The c.d.f. of a variable T, denoted by F(t), is
a function that tells us the probability that the variable will be less than or
equal to any value t that we choose. Thus, F(t) = Pr{T ≤ t}. If we know the
value of F for every value of t, then we know all there is to know about the
univariate distribution of T. In survival analysis, it is more common to
work with a closely related function called the survivor function, defined
as S(t) = Pr{T > t} = 1 – F(t). If the event of interest is a death, the survivor
function gives the probability of surviving beyond t. Because S is a
probability, we know that it is bounded by 0 and 1. And because T cannot
be negative, we know that S(0) = 1. Finally, as t gets larger, S never
increases (and usually decreases). Within these restrictions, S can have a
wide variety of shapes.

CHAPTER 2

Basic Concepts of Survival Analysis

Chapter 3, “Estimating and Comparing Survival Curves with PROC
LIFETEST,” explains how to estimate survivor functions using life-table
and Kaplan-Meier methods. Often, the objective is to compare survivor
functions for different subgroups in a sample. If the survivor function for
one group is always higher than the survivor function for another group,
then the first group clearly lives longer than the second group. If survivor
functions cross, however, the situation is more ambiguous.
Probability Density Function

When variables are continuous, another common way of describing
their probability distributions is the probability density function, or p.d.f.
This function is defined as

f (t ) =

dF (t )
dS (t )
=−
dt
dt

(2.1)

That is, the p.d.f. is just the derivative or slope of the c.d.f. Although this
definition is considerably less intuitive than that for the c.d.f., it is the
p.d.f. that most directly corresponds to our intuitive notions of
distributional shape. For example, the familiar bell-shaped curve that is
associated with the normal distribution is given by its p.d.f., not its c.d.f.
Hazard Function

For continuous survival data, the hazard function is actually more
popular than the p.d.f. as a way of describing distributions. The hazard
function is defined as

Pr (t ≤ T < t + ∆t | T ≥ t )
∆t →0
∆t

h(t ) = lim

(2.2)

Instead of h(t), some authors denote the hazard by λ(t) or r (t).
Because the hazard function is so central to survival analysis, it is worth
taking some time to explain this definition. The aim of the definition is to
quantify the instantaneous risk that an event will occur at time t. Because
time is continuous, the probability that an event will occur at exactly time
t is necessarily 0. But we can talk about the probability that an event
occurs in the small interval between t and t + ∆t. We also want to make this
probability conditional on the individual surviving to time t. Why?
Because if individuals have already died (that is, experienced the event),
they are clearly no longer at risk of the event. Thus, we want to consider
only those individuals who have made it to the beginning of the interval
[t, t + ∆t). These considerations point to the numerator in equation (2.2):
Pr(t ≤ T c, Sˆ (t ) is undefined.
Things get more complicated when some censoring times are smaller
than some event times. In that instance, the observed proportion of cases
with event times greater than t can be biased downward because cases that
are censored before t may, in fact, have “died” before t without our
knowledge. The solution is as follows. Suppose there are k distinct event
times, t1 < t2 < … tk. When there are
censored times greater than tk, Sˆ (t ) is undefined for t greater than the

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

largest censoring time. For an explanation of the rationale for equation
(3.1), see The Life-Table Method later in this chapter.
Here’s an example of how to get the KM estimator using PROC
LIFETEST with the myelomatosis data shown in Output 2.1:
DATA myel;
INPUT dur status treat renal;
DATALINES;
8
1
1
1
180
1
2
0
632
1
2
0
852
0
1
0
52
1
1
1
2240
0
2
0
220
1
1
0
63
1
1
1
195
1
2
0
76
1
2
0
70
1
2
0
8
1
1
0
13
1
2
1
1990
0
2
0
1976
0
1
0
18
1
2
1
700
1
2
0
1296
0
1
0
1460
0
1
0
210
1
2
0
63
1
1
1
1328
0
1
0
1296
1
2
0
365
0
1
0
23
1
2
1
;
PROC LIFETEST DATA=myel;
TIME dur*status(0);
RUN;

The KM estimator is the default, so you do not need to request it. If
you want to be explicit, you can put METHOD=KM on the PROC
LIFETEST statement. The syntax DUR*STATUS(0) is common to PROC
LIFETEST, PROC LIFEREG, and PROC PHREG. The first variable is the
time of the event or censoring, the second variable contains information on
whether or not the observation was censored, and the numbers in
parentheses (there can be more than one) are values of the second variable
that correspond to censored observations. These statements produce the
results shown in Output 3.1.

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

32
Output 3.1 Kaplan-Meier Estimates for Myelomatosis Data

Product-Limit Survival Estimates

dur

Survival

0.00
8.00
8.00
13.00
18.00
23.00
52.00
63.00
63.00
70.00
76.00
180.00
195.00
210.00
220.00
365.00*
632.00
700.00
852.00*
1296.00
1296.00*
1328.00*
1460.00*
1976.00*
1990.00*
2240.00*

Failure

1.0000
.
0.9200
0.8800
0.8400
0.8000
0.7600
.
0.6800
0.6400
0.6000
0.5600
0.5200
0.4800
0.4400
.
0.3960
0.3520
.
0.3017
.
.
.
.
.
.

0
.
0.0800
0.1200
0.1600
0.2000
0.2400
.
0.3200
0.3600
0.4000
0.4400
0.4800
0.5200
0.5600
.
0.6040
0.6480
.
0.6983
.
.
.
.
.
.

Survival
Standard
Error

Number
Failed

Number
Left

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
14
15
16
16
17
17
17
17
17
17
17

25
24
23
22
21
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0

0
.
0.0543
0.0650
0.0733
0.0800
0.0854
.
0.0933
0.0960
0.0980
0.0993
0.0999
0.0999
0.0993
.
0.0986
0.0970
.
0.0953
.
.
.
.
.
.

NOTE: The marked survival times are censored observations.
Summary Statistics for Time Variable dur
Quartile Estimates

Percent

Point
Estimate

75
50
25

.
210.00
63.00

95% Confidence Interval
Transform
[Lower
Upper)
LOGLOG
LOGLOG
LOGLOG

220.00
63.00
8.00

.
1296.00
180.00

(continued)

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

33
Output 3.1 (continued)

Mean

Standard Error

562.76

117.32

NOTE: The mean survival time and its standard error were underestimated
because the largest observation was censored and the estimation was
restricted to the largest event time.

Summary of the Number of Censored and Uncensored Values

Total

Failed

Censored

Percent
Censored

32.00

Each line of numbers in Output 3.1 corresponds to one of the 25
cases, arranged in ascending order (except for the first line, which is for
time 0). Censored observations are starred. The crucial column is the
second—labeled Survival—which gives the KM estimates. At 180 days, for
example, the KM estimate is .56. We say, then, that the estimated
probability that a patient will survive for 180 days or more is .56. When
there are tied values (two or more cases that die at the same reported time),
as we have at 8 days and 63 days, the KM estimate is reported only for the
last of the tied cases. No KM estimates are reported for the censored times.
In fact, however, the KM estimator is defined for any time between 0
and the largest event or censoring time. It’s just that it changes only at an
observed event time. Thus, the estimated survival probability for any time
from 70 days up to (but not including) 76 days is .64. The 1-year (365 days)
survival probability is .44, the same as it was at 220 days. After 2,240 days
(the largest censoring time), the KM estimate is undefined.
The third column, labeled Failure, is just 1 minus the KM estimate,
which is the estimated probability of a death prior to the specified time.
Thus, it is an estimate of the cumulative distribution function. The fourth
column, labeled Survival Standard Error, is an estimate of the standard
error of the KM estimate, obtained by the well-known Greenwood’s
formula (Collett, 2003). Although this standard error could be used
directly to construct confidence intervals around the survival probabilities,
PROC LIFETEST has better methods that we shall examine later.
The fifth column, labeled Number Failed, is just the cumulative
number of cases that experienced events prior to and including each point

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

in time. The column labeled Number Left is the number of cases that have
neither experienced events nor been censored prior to each point in time.
This is the size of the risk set for each time point. Below the main table,
you find the estimated 75th, 50th, and 25th percentiles (labeled Quartile
Estimates). If these were not given, you could easily determine them from
the Failure column. Thus, the 25th percentile (63 in this case) is the
smallest event time such that the probability of dying earlier is greater than
.25. No value is reported for the 75th percentile because the KM estimator
for these data never reaches a failure probability greater than .70.
Of greatest interest is the 50th percentile, which is, of course, the
median death time. Here, the median is 210 days, with a 95% confidence
interval of 63 to 1296. As noted in the table, the confidence intervals are
calculated using a log-log transform that preserves the upper bound of 1
and the lower bound of 0 on the survival probabilities. Although other
transforms are optionally available, there is rarely any need to use them.
An estimated mean time of death is also reported. This value is
calculated directly from the estimated survival function. It is not simply
the mean time of death for those who died. As noted in the output, the
mean is biased downward when there are censoring times greater than the
largest event time. Even when this is not the case, the upper tail of the
distribution will be poorly estimated when a substantial number of the
cases are censored, and this can greatly affect estimates of the mean.
Consequently, the median is typically a better measure of central tendency
for censored survival data.
Usually you will want to see a plot of the estimated survival
function. PROC LIFETEST can produce plots in three different modes: line
printer, traditional graphics, and the output delivery system (ODS). Here I
use ODS because it has a much richer array of features. Here’s how to get
the survivor function graph using ODS:
ODS GRAPHICS ON;
PROC LIFETEST DATA=myel PLOTS=S;
TIME dur*status(0);
RUN;
ODS GRAPHICS OFF;

The resulting graph (shown in Output 3.2) can be accessed by doubleclicking on the graph icon in the Results window.

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

Output 3.2

Plot of the Survivor Function for Myelomatosis Data

Notice that the graph contains plus signs (that look like tick marks)
wherever there are censored observations. These marks can be very
distracting when the data set is large with lots of censored observations.
They can be suppressed by using the NOCENSOR option. Another useful
option is ATRISK, which adds the number of individuals still at risk (not
yet dead or censored) to the graph. To get a graph with 95% confidence
limits around the survivor function, use the CL option. All three options
can be specified with the statement
PROC LIFETEST DATA=myel PLOTS=S(NOCENSOR ATRISK CL);

which produces the graph shown in Output 3.3.

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

Output 3.3

Plot of the Survivor Function for Myelomatosis Data with Pointwise Confidence Limits

The confidence limits shown in Output 3.3 are pointwise limits,
meaning that for each specified survival time, we are 95% confident that
the probability of surviving to that time is within those limits. Note that
the confidence limits only extend to the largest event time.
Suppose we want confidence bands that can be interpreted by
saying that we are 95% confident that the entire survivor function falls
within the upper curve and the lower curve. More complex methods are
needed to produce such bands, and PROC LIFETEST offers two: the HallWellner method and the equal precision (EP) method. I prefer the EP
method because it tends to produce confidence bands that are more stable
in the tails. To implement this method, the option becomes
PLOTS=S(CB=EP). To get both pointwise and EP bands, use the option
PLOTS=S(CL CB=EP), which produces Output 3.4. As is evident in that
graph, the confidence bands are always wider than the pointwise
confidence limits.

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

Output 3.4

Survivor Function for Myelomatosis Data with Confidence Bands and Pointwise Limits

You can also write the pointwise limits to a SAS data set using the
OUTSURV option:
PROC LIFETEST DATA=myel OUTSURV=a;
TIME dur*status(0);
PROC PRINT DATA=a;
RUN;

Output 3.5 shows the printed data set. Note that the data set contains
one record for each unique event or censoring time, plus a record for time
0. The two right-hand columns give the upper and lower limits for a 95%
confidence interval around each survival probability. (To get 90%
intervals, use ALPHA=.10 as an option in the PROC LIFETEST statement.)

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

Output 3.5 Data Set Produced by the OUTSURV= Option
Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24

dur
0
8
13
18
23
52
63
70
76
180
195
210
220
365
632
700
852
1296
1296
1328
1460
1976
1990
2240

_CENSOR_
.
0
0
0
0
0
0
0
0
0
0
0
0
1
0
0
1
0
1
1
1
1
1
1

SURVIVAL
1.00000
0.92000
0.88000
0.84000
0.80000
0.76000
0.68000
0.64000
0.60000
0.56000
0.52000
0.48000
0.44000
0.44000
0.39600
0.35200
0.35200
0.30171
.
.
.
.
.
.

SDF_LCL
1.00000
0.71639
0.67256
0.62806
0.58445
0.54205
0.46093
0.42215
0.38449
0.34794
0.31249
0.27813
0.24490
.
0.20826
0.17355
.
0.13419
.
.
.
.
.
.

SDF_UCL
1.00000
0.97937
0.95964
0.93673
0.91146
0.88428
0.82527
0.79378
0.76109
0.72728
0.69238
0.65640
0.61936
.
0.57872
0.53660
.
0.48925
.
.
.
.
.
.

By default, these limits are calculated by adding and subtracting 1.96
standard errors to the log-log transformation of the survivor function,
log − log Sˆ (t ) , and then reversing the transformation to get back to the
original metric. This method ensures that the confidence limits will not be
greater than 1 or less than 0. Other transformations are available, the most
attractive being the logit: log[Sˆ (t ) /(1 − Sˆ (t ))] . To switch to this transform,
include the CONFTYPE=LOGIT option in the PROC statement. You can
also write the equal precision confidence bands to the OUTSURV data set
by including the CONFBAND=EP option in the PROC statement.

(

TESTING

FOR

)

DIFFERENCES

SURVIVOR FUNCTIONS

If an experimental treatment has been applied to one group but not
another, the obvious question to ask is “Did the treatment make a
difference in the survival experience of the two groups?” Because the
survivor function gives a complete accounting of the survival experience

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

of each group, a natural approach to answering this question is to test the
null hypothesis that the survivor functions are the same in the two groups
(that is, S1(t) = S2(t) for all t, where the subscripts distinguish the two
groups). PROC LIFETEST can calculate several different statistics for
testing this null hypothesis.
For the myelomatosis data, it is clearly desirable to test whether the
treatment variable (TREAT) has any effect on the survival experience of the
two groups. To do that with PROC LIFETEST, simply add a STRATA
statement after the TIME statement:
ODS GRAPHICS ON;
PROC LIFETEST DATA=myel PLOTS=S(TEST);
TIME dur*status(0);
STRATA treat;
RUN;
ODS GRAPHICS OFF;

The STRATA statement has three consequences:
1.

First, instead of a single table with KM estimates, separate tables
(not shown here) are produced for each of the two treatment
groups.
Second, corresponding to the two tables are two graphs of the
survivor function, superimposed on the same axes for easy
comparison.
Third, PROC LIFETEST reports several statistics related to
testing for differences between the two groups. Also, the TEST
option (after PLOTS=S) includes the log-rank test in the survivor
plot.

Look at the graphs in Output 3.6. Before 220 days, the two survival
curves are virtually indistinguishable, with little visual evidence of a
treatment effect. The gap that develops after 220 days reflects the fact that
no additional deaths occur in treatment group 1 after that time.

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

Output 3.6

Survival Curves for Two Treatment Groups

Next, PROC LIFETEST prints log-rank and Wilcoxon statistics for
each treatment group, followed by an estimate of their covariance matrix.
These are used to compute the chi-square statistics shown near the bottom
of Output 3.7. In most cases, you can ignore the preliminaries and look
only at the p-values. Here they are far from significant, which is hardly
surprising given the graphical results and the small sample size. Thus,
there is no evidence that would support the rejection of the null
hypothesis that the two groups have exactly the same survivor function
(that is, exactly the same probability distribution for the event times). The
p-value for a likelihood-ratio test (-2log(LR)) is also reported; this test is
usually inferior to the other two because it requires the unnecessary
assumption that the hazard function is constant in each group, implying
an exponential distribution for event times.

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

Statistics for Testing for Differences between Two Groups

Output 3.7

Testing Homogeneity of Survival Curves for dur over Strata
Rank Statistics
treat
1
2

Log-Rank

Wilcoxon

-2.3376
2.3376

-18.000
18.000

Covariance Matrix for the Log-Rank Statistics
treat
1
2

4.16301
-4.16301

-4.16301
4.16301

Covariance Matrix for the Wilcoxon Statistics
treat
1
2

1301.00
-1301.00

-1301.00
1301.00

Test of Equality over Strata

Test
Log-Rank
Wilcoxon
-2Log(LR)

Chi-Square

Pr >
Chi-Square

1.3126
0.2490
1.5240

1
1
1

0.2519
0.6178
0.2170

The log-rank test is the best known and most widely used test for
differences in the survivor function, but the Wilcoxon test is also popular.
Are there any reasons for choosing one over the other? Each statistic can be
written as a function of deviations of observed numbers of events from
expected numbers. For group 1, the log-rank statistic can be written as
r

∑ (d
j =1

− e1 j )

where the summation is over all unique event times (in both groups), and
there are a total of r such times. d1j is the number of deaths that occur in
group 1 at time j, and e1j is the expected number of events in group 1 at
time j. The expected number is given by n1jdj/nj, where nj is the total

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

number of cases that are at risk just prior to time j, n1j is the number at risk
just prior to time j in group 1, and dj is the total number of deaths at time j
in both groups. (This is just the usual formula for computing expected cell
counts in a 2 × 2 table, under the hypothesis of independence.) As shown
in Output 3.7, the log-rank statistic in group 1 is –2.3376. The chi-square
statistic is calculated by squaring this number and dividing by the
estimated variance, which is 4.16301 in this case.
The Wilcoxon statistic, given by
r

∑ n (d
j =1

− e1 j )

differs from the log-rank statistic only by the presence of nj, the total
number at risk at each time point. Thus, it is a weighted sum of the
deviations of observed numbers of events from expected numbers of
events. As with the log-rank statistic, the chi-square test is calculated by
squaring the Wilcoxon statistic for either group (–18 for group 1 in this
example) and dividing by the estimated variance (1301).
Because the Wilcoxon test gives more weight to early times than to
late times (nj never increases with time), it is less sensitive than the logrank test to differences between groups that occur at later points in time.
To put it another way, although both statistics test the same null
hypothesis, they differ in their sensitivity to various kinds of departures
from that hypothesis. In particular, the log-rank test is more powerful for
detecting differences of the form

S1 (t ) = [ S 2 (t )]γ
where γ is some positive number other than 1.0. This equation defines a
proportional hazards model, which is discussed in detail in Chapter 5. (As
we will see in that chapter, the log-rank test is closely related to tests for
differences between two groups that are performed within the framework
of Cox’s proportional hazards model.) In contrast, the Wilcoxon test is
more powerful than the log-rank test in situations where event times have
log-normal distributions (discussed in the next chapter) with a common
variance but with different means in the two groups. Neither test is
particularly good at detecting differences when survival curves cross.
Now let’s see an example where the survival distributions clearly
differ across two groups. Using the same data set, we can stratify on
RENAL, the variable that has a value of 1 if renal functioning was impaired
at the time of randomization; otherwise, the variable has a value of 0.
Output 3.8 shows that the survival curve for those with impaired renal
functioning drops precipitously, while the curve for those with normal

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

functioning declines much more gradually. The three hypothesis tests are
unanimous in rejecting the null hypothesis of no difference between the
two groups.

Output 3.8

Graphs and Tests for Stratifying by Renal Functioning

Test of Equality over Strata
Test
Log-Rank
Wilcoxon
-2Log(LR)

Chi-Square

Pr >
Chi-Square

24.0188
20.8739
35.4813

1
1
1

0.0001
0.0001
0.0001

In addition to the log-rank and Wilcoxon tests, which are produced
by default, the STRATA statement also has options for four other
nonparametric tests: Tarone-Ware, Peto-Peto, modified Peto-Peto, and
Fleming-Harrington. Like the Wilcoxon and log-rank tests, all of these can
be represented as a weighted sum of observed and expected numbers of
events:

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

44
r

∑ w (d
j =1

− e1 j )

For the Tarone-Ware test, wj is the square root of nj, so this test behaves
much like the Wilcoxon test in being more sensitive to differences at earlier
rather than later times. That’s also true of the two Peto tests, for which wj is
a function of the survivor function itself. Fleming-Harrington is actually a
family of tests in which the weights depend on two parameters, p and q,
which can be chosen by the user:

w j = Sˆ (t j ) p [1 − Sˆ (t j )]q
Although p and q can be any nonnegative numbers, here are a few special
cases. When both p and q are 0, you get the log-rank test. When p is 1 and q
is 0, you get something very close to the Peto-Peto test. When q is 1 and p is
0, wi increases with time, unlike any of the other tests. When both q and p
are 1, you get a weight function that is maximized at the median and gets
small for large or small times.
To get all these tests, the STRATA statement can be changed to
STRATA treat / TESTS=ALL;

which produces the results in Output 3.9. By default, the FlemingHarrington test is performed with p=1 and q=0.

Output 3.9

Optional Tests for Differences between Survival Functions
Test of Equality over Strata
Test

Chi-Square

Log-Rank
Wilcoxon
Tarone
Peto
Modified Peto
Fleming(1)

1.3126
0.2490
0.6514
0.3766
0.3321
0.3045

Pr >
Chi-Square

1
1
1
1
1
1

0.2519
0.6178
0.4196
0.5394
0.5644
0.5811

So far, we have only tested for differences between two groups. The
tests readily generalize to three or more groups, with the null hypothesis
that all groups have the same survivor function. If the null hypothesis is
true, the test statistics all have chi-square distributions with degrees of

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

freedom equal to the number of groups minus 1. The STRATA statement
provides great flexibility in defining those groups. PROC LIFETEST
constructs a stratum corresponding to each unique value of the variable in
the STRATA statement. That variable may be either numeric or character.
If the STRATA statement contains two or more variable names, PROC
LIFETEST constructs one group for every unique combination of values of
the variables.
To illustrate this and other options, we need a different data set.
Output 3.10 displays the first 20 observations from the recidivism study
that was briefly described in Chapter 1, “Introduction,” in the section Why
Use Survival Analysis? The sample consisted of 432 male inmates who
were released from Maryland state prisons in the early 1970s (Rossi et al.,
1980). These men were followed for 1 year after their release, and the dates
of any arrests were recorded. We’ll look only at the first arrest here. (In
fact, there weren’t enough men with two or more arrests to use the
techniques for repeated events discussed in Chapter 8, “Heterogeneity,
Repeated Events, and Other Topics.”) The WEEK variable contains the
week of the first arrest after release. The variable ARREST has a value of 1
for those who were arrested during the 1-year follow-up, and it has a value
of 0 for those who were not. Only 26 percent of the men were arrested. The
data are singly right censored so that all the censored cases have a value of
52 for WEEK.

Output 3.10

Recidivism Data (First 20 Cases Out of 432)

OBS WEEK ARREST FIN
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

52
52
52
52
52
12
52
19
52
15
8
52
52
36
52
4
45
52
52
52

0
0
0
0
0
1
0
1
0
1
1
0
0
1
1
1
1
1
0
0

1
0
1
0
1
1
0
0
0
1
1
1
1
1
0
0
1
0
1
0

AGE

RACE

WEXP

MAR

PARO

PRIO

24
29
20
20
31
22
24
18
18
22
21
21
21
19
33
18
18
21
20
22

1
1
1
1
0
1
1
1
1
1
1
1
0
1
1
1
1
1
1
1

1
1
1
0
1
1
1
0
0
0
1
0
1
0
1
1
0
0
0
1

0
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
1
0

1
1
1
1
1
1
1
0
1
1
1
1
1
1
1
0
0
0
0
0

1
3
1
1
3
2
2
2
3
3
4
1
1
2
2
1
5
0
1
1

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

The covariates shown in Output 3.10 are defined as follows:
FIN

AGE
RACE
WEXP

MAR
PARO
PRIO

has a value of 1 if the inmate received financial aid after
release; otherwise, FIN has a value of 0. This variable
was randomly assigned with equal numbers in each
category.
specifies age in years at the time of release.
has a value of 1 if the person was black; otherwise,
RACE has a value of 0.
has a value of 1 if the inmate had full-time work
experience before incarceration; otherwise, WEXP has a
value of 0.
has a value of 1 if the inmate was married at the time of
release; otherwise, MAR has a value of 0.
has a value of 1 if the inmate was released on parole;
otherwise, PARO has a value of 0.
specifies the number of convictions an inmate had prior to
incarceration.

The following program stratifies by the four combinations of WEXP
and PARO. The ADJUST option (new in SAS 9.2) tells PROC LIFETEST to
produce p-values for all six pairwise comparisons of the four strata and
then to report p-values that have been adjusted for multiple comparisons
using Tukey’s method (other methods are also available):
PROC LIFETEST DAta=my.recid;
TIME week*arrest(0);
STRATA wexp paro / ADJUST=TUKEY;
RUN;

Results are shown in Output 3.11. The first table shows how the four
strata correspond to the possible combinations of WEXP and PARO. Next
we get overall chi-square tests of the null hypothesis that the survivor
functions are identical across the four strata. All three tests are significant
at the .05 level. Last, we see the log-rank tests comparing each possible
pair of strata. Three of the tests are significant using the “raw” p-value, but
only one of these comparisons is significant after the Tukey adjustment.
(Not shown is a similar table for the Wilcoxon test.)

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

Output 3.11

Tests Comparing More Than Two Groups, with Multiple Comparison Adjustment
Stratum
1
2
3
4

wexp

paro

0
0
1
1

0
1
0
1

Test of Equality over Strata

Test
Log-Rank
Wilcoxon
-2Log(LR)

Chi-Square

10.2074
11.2596
9.2725

3
3
3

Pr >
Chi-Square
0.0169
0.0104
0.0259

Adjustment for Multiple Comparisons for the Logrank Test
p-Values
Tukey_Stratum_
_Stratum_
Chi-Square
Raw
Kramer
1
1
1
2
2
3

2
3
4
3
4
4

0.00542
4.5853
6.6370
3.5427
5.2322
0.4506

0.9413
0.0322
0.0100
0.0598
0.0222
0.5021

0.9999
0.1401
0.0491
0.2357
0.1009
0.9080

For numeric variables, you can use the STRATA statement to define
groups by intervals rather than by unique values. For the recidivism data,
the AGE variable can be stratified as
STRATA age(21 24 28);

This statement produces four strata, corresponding to the intervals:
age < 21
21 ≤ age < 24
24 ≤ age < 28
28 ≤ age

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

Output 3.12 displays the results. Here I have also requested multiple
comparisons but now using the Bonferroni adjustment (ADJUST=BON).
Note that except for the two extreme strata, each stratum is identified by
the midpoint of the interval. Clearly one must reject the null hypothesis
that all age strata have the same survivor function. Three of the six
pairwise comparisons are also significant, either with or without the
adjustment.

Output 3.12

Tests for Multiple Age Groups
Summary of the Number of Censored and Uncensored Values

Stratum

age

Total

Failed

Censored

Percent
Censored

1
<21
127
52
75
59.06
2
22.5
114
29
85
74.56
3
26
91
17
74
81.32
4
>=28
100
16
84
84.00
--------------------------------------------------------------Total
432
114
318
73.61

Test of Equality over Strata

Test

Chi-Square

Pr >
Chi-Square

22.2316
20.9222
19.8330

3
3
3

<.0001
0.0001
0.0002

Log-Rank
Wilcoxon
-2Log(LR)

Adjustment for Multiple Comparisons for the Logrank Test
Strata Comparison
p-Values
age
age
Chi-Square
Raw
Bonferroni
21.0000
21.0000
21.0000
22.5000
22.5000
26

22.5000
26
28.0000
26
28.0000
28.0000

8.4518
15.6465
18.3765
0.8272
1.6912
0.1836

0.0036
<.0001
<.0001
0.3631
0.1934
0.6683

0.0219
0.0005
0.0001
1.0000
1.0000
1.0000

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

THE LIFE-TABLE METHOD
If the number of observations is large and if event times are precisely
measured, there will be many unique event times. The KM method then
produces long tables that may be unwieldy for presentation and
interpretation. One way to solve this problem is to use the TIMELIST
option (in the PROC LIFEREG statement), which reports the KM estimates
only at specified points in time. An alternative solution is to switch to the
life-table method, in which event times are grouped into intervals that can
be as long or short as you please. In addition, the life-table method (also
known as the actuarial method) can produce estimates and plots of the
hazard function. The downside to the life-table method is that the choice
of intervals is usually somewhat arbitrary, leading to arbitrariness in the
results and possible uncertainty about how to choose the intervals. There
is inevitably some loss of information as well. Note, however, that PROC
LIFETEST computes the log-rank and Wilcoxon statistics (as well as other
optional test statistics) from the ungrouped data (if available) so they are
unaffected by the choice of intervals for the life-table method.
We now request a life table for the recidivism data, using the default
specification for interval lengths:
PROC LIFETEST DATA=recid METHOD=LIFE;
TIME week*arrest(0);
RUN;

Output 3.13 shows the results. PROC LIFETEST constructs six
intervals, starting at 0 and incrementing by periods of 10 weeks. The
algorithm for the default choice of intervals is fairly complex (see the
SAS/STAT User’s Guide for details). You can override the default by
specifying WIDTH=w in the PROC LIFETEST statement. The intervals will
then begin with [0, w) and will increment by w. Alternatively, you can get
even more control over the intervals by specifying INTERVALS=a b c … in
the PROC LIFETEST statement, where a, b, and c are cut points. For
example, INTERVALS= 15 20 30 50 produces the intervals [0, 15), [15, 20),
[20, 30), [30, 50), [50, ∞). See the SAS/STAT User’s Guide for other options.
Note that intervals do not have to be the same length. It’s often desirable to
make later intervals longer so that they include enough events to get
reliable estimates of the hazard and other statistics.

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

Output 3.13

Applying the Life-Table Method to the Recidivism Data
Life Table Survival Estimates

Interval
[Lower,
Upper)
0
10
20
30
40
50

10
20
30
40
50
60

Interval
[Lower,
Upper)
0
10
20
30
40
50

10
20
30
40
50
60

Number
Failed

Number
Censored

Effective
Sample
Size

14
21
23
23
26
7

0
0
0
0
0
318

432.0
418.0
397.0
374.0
351.0
166.0

Conditional
Probability
of Failure

Conditional
Probability
Standard
Error

0.0324
0.0502
0.0579
0.0615
0.0741
0.0422

0.00852
0.0107
0.0117
0.0124
0.0140
0.0156

Survival

Failure

Survival
Standard
Error

1.0000
0.9676
0.9190
0.8657
0.8125
0.7523

0
0.0324
0.0810
0.1343
0.1875
0.2477

0
0.00852
0.0131
0.0164
0.0188
0.0208

Median
Residual
Lifetime

Median
Standard
Error

.
.
.
.
.
.

Evaluated at the Midpoint of the Interval

Interval
[Lower,
Upper)
0
10
20
30
40
50

10
20
30
40
50
60

PDF

PDF
Standard
Error

Hazard

Hazard
Standard
Error

0.00324
0.00486
0.00532
0.00532
0.00602
0.00317

0.000852
0.00103
0.00108
0.00108
0.00114
0.00118

0.003294
0.005153
0.005966
0.006345
0.007692
0.004308

0.00088
0.001124
0.001244
0.001322
0.001507
0.001628

Summary of the Number of Censored and Uncensored Values

Total

Failed

432

114

Censored
318

Percent
Censored
73.61

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

For each interval, 14 different statistics are reported. The four
statistics displayed in the first panel, while not of major interest in
themselves, are necessary for calculating the later statistics. Number Failed
and Number Censored should be self-explanatory. Effective Sample Size is
straightforward for the first five intervals because they contain no censored
cases. The effective sample size for these intervals is just the number of
persons who had not yet been arrested at the start of the interval. For the
last interval, however, the effective sample size is only 166, even though
351 persons made it to the 50th week without an arrest. Why? The answer
is a fundamental property of the life-table method. The method treats any
cases censored within an interval as if they were censored at the midpoint
of the interval. This treatment is equivalent to assuming that the
distribution of censoring times is uniform within the interval. Because
censored cases are only at risk for half of the interval, they only count for
half in figuring the effective sample size. Thus, the effective sample size
for the last interval is 7+(318/2)=166. The 7 corresponds to the seven men
who were arrested in the interval; they are treated as though they were at
risk for the whole interval.
The Conditional Probability of Failure is an estimate of the
probability that a person will be arrested in the interval, given that he
made it to the start of the interval. This estimate is calculated as (number
failed)/(effective sample size). An estimate of its standard error is given in
the next panel. The Survival column is the life-table estimate of the
survivor function (that is, the probability that the event occurs at a time
greater than or equal to the start time of each interval). For example, the
estimated probability that an inmate will not be arrested until week 30 or
later is .8657.
The survival estimate is calculated from the conditional probabilities
of failure in the following way. For interval i, let ti be the start time and qi
be the estimated conditional probability of failure. The probability of
surviving to ti or beyond is then
i −1

Sˆ (t ) = ∏ (1 − q j ) .
j =1

For i = 1 and, hence, ti = 0, the survival probability is set to 1.0.
The rationale for equation (3.2) is a fairly simple application of
conditional probability theory. Suppose we want an expression for the
probability of surviving to t4 or beyond. To obtain this, let’s define the
following events:
A = survival to t2 or beyond.
B = survival to t3 or beyond.
C = survival to t4 or beyond.

(3.2)

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

We want the probability of C. But since you can’t get past t4 without
getting past t2 and t3, we can write Pr(C) = Pr(A, B, C). By the definition of
conditional probability, we can rewrite this as
Pr(A, B, C)=Pr(C|A, B)Pr(B|A)Pr(A)=(1 – q3)(1 – q2)(1 – q1).

Extending this argument to other intervals yields the formula in equation
(3.2). Note the similarity between this formula and equation (3.1) for the
KM estimator. In equation (3.1), dj/nj is equivalent to qj in equation (3.2);
both are estimates of the probability of failure in an interval given survival
to the start of the interval. The major differences between the two formulas
are as follows:
 The number of censored observations in an interval is not halved in
the KM estimator.
 The interval boundaries for the KM estimator are determined by the
event times themselves.
Thus, each interval for KM estimation extends from one unique event time
up to, but not including, the next unique event time.
Continuing with the second panel of Output 3.13, the Failure
column is just 1 minus the Survival column. We are also given the
standard errors of the Survival probabilities. The Median Residual
Lifetime column is, in principle, an estimate of the remaining time until an
event for an individual who survived to the start of the interval. For this
example, however, the estimates are all missing. To calculate this statistic
for a given interval, there must be a later interval whose survival
probability is less than half the survival probability associated with the
interval of interest. It is apparent from Output 3.13 that no interval satisfies
this criterion. For many data sets, there will be at least some later intervals
for which this statistic cannot be calculated.
The PDF Standard Error column gives the estimated value of the
probability density function at the midpoint of the interval. Of greater
interest is the Hazard column, which gives estimates of the hazard
function evaluated at the midpoint of each interval. This is calculated as

di
h(tim ) =
w d 

bi  ni − i − i 
2 2


(3.3)

where for the ith interval, tim is the midpoint, di is the number of events, bi
is the width of the interval, ni is the number still at risk at the beginning of
the interval, and wi is the number of cases withdrawn (censored) within
the interval. A better estimate of the hazard could be obtained by di/Ti,
where Ti is the total exposure time within interval i. For each individual,

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

exposure time is the amount of time actually observed within the interval.
For an individual who is known to have survived the whole interval,
exposure time is just the interval width bi. For individuals who had events
or who withdrew in the interval, exposure time is the time from the
beginning of the interval until the event or withdrawal. Total exposure
time is the sum of all the individual exposure times. The denominator in
equation (3.3) is an approximation to total exposure time, such that all
events and all withdrawals are presumed to occur at the midpoint of the
interval (thus, the division by 2). Why use an inferior estimator? Well,
exact exposure times are not always available (see the next section), so the
estimator in equation (3.3) has become the standard for life tables.
You can get plots of the survival and hazard estimates by using
PLOTS=(S,H) in the PROC LIFETEST statement, as in the following
program:
ODS GRAPHICS ON;
PROC LIFETEST DATA=recid METHOD=LIFE PLOTS=(S,H);
TIME week*arrest(0);
RUN;
ODS GRAPHICS OFF;

Output 3.14 displays the graph of the hazard function. Apparently,
the hazard of arrest increases steadily until the 50-60 week interval, when
it drops precipitously from .077 to .043. This drop is an artifact of the way
that the last interval is constructed, however. Although the interval runs
from 50 to 60, in fact, no one was at risk of an arrest after week 52 when
the study was terminated. As a result, the denominator in equation (3.3) is
a gross overestimate of the exposure time in the interval.

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

Output 3.14

Hazard Estimates for Recidivism Data

We can fix this by explicitly setting the last interval to end at 53. (If
we set it at 52, the interval will not include arrests that occurred in week
52 because the interval is open on the right). At the same time, it is better
to recode the censored times from 52 to 53 because they are not censored
within the interval but rather at the end. Recoding effectively credits the
full interval (rather than only half) as exposure time for the censored cases.
Here’s the revised code:
DATA newrecid;
SET recid;
IF arrest=0 THEN week=53;
PROC LIFETEST DATA=newrecid METHOD=LIFE PLOTS=(S,H)
INTERVALS=10 20 30 40 50 53;
TIME week*arrest(0);
RUN;

The resulting hazard estimates and plot in Output 3.15 show only a
slight tendency for the hazard to decline in the last interval.

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

55
Output 3.15

Corrected Hazard Estimates and Plot for Recidivism Data
Evaluated at the Midpoint of the Interval

Median
Interval
Standard
[Lower, Upper)
Error
0
10
20
30
40
50
53

LIFE TABLES

FROM

10
20
30
40
50
53
.

.
.
.
.
.
.
.

PDF
0.00324
0.00486
0.00532
0.00532
0.00602
0.00540
.

PDF
Standard
Error

Hazard

Hazard
Standard
Error

0.000852
0.00103
0.00108
0.00108
0.00114
0.00202
.

0.003294
0.005153
0.005966
0.006345
0.007692
0.007258
.

0.00088
0.001124
0.001244
0.001322
0.001507
0.002743
.

GROUPED DATA

Although PROC LIFETEST estimates survival probabilities from
individual-level data with exact event times, you can also easily construct
life tables from published data that provide only
 the boundaries of the intervals
 the number of events in each interval
 the number of censored cases in each interval.

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

Consider the following survival data for 68 patients from the Stanford
Heart Transplantation Program, as reported by Elandt-Johnson and
Johnson (1980):
Number of Days
0-50

Number of Deaths
16

Number Censored
3

50-100

100-200

200-400

400-700

700-1000

1000-1300

1300-1600

1600+

Time is measured from the date of the transplant. Although this sample is
rather small for constructing a life table, it will do fine for illustration. The
trick is to create a separate observation for each of the frequency counts in
the table. For each observation, the value of the time variable (TIME) can
be anywhere within the interval—we’ll use the midpoint. A second
variable (STATUS) is created with a value of 1 for the death counts and a
value of 0 for the censored counts. The frequency count (NUMBER) is used
as a weight variable with the FREQ statement in PROC LIFETEST. Here’s
the SAS code:
DATA;
INPUT time status number;
DATALINES;
25 1 16
25 0 3
75 1 11
75 0 0
150 1 4
150 0 2
300 1 5
300 0 4
550 1 2
550 0 6
850 1 4
850 0 3
1150 1 1
1150 0 2
1450 1 1

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

1450 0 3
1700 1 0
1700 0 1
;
PROC LIFETEST METHOD=LIFE INTERVALS=50 100 200 400 700
1000 1300 1600 PLOTS=(S,H);
TIME time*status(0);
FREQ number;
RUN;

Two of the data lines (75 0 0, 1700 1 0) are unnecessary because the
frequency is 0, but they are included here for completeness. Output 3.16
shows the tabular results. The hazard plot is in Output 3.17.
The most striking fact about this example is the rapid decline in the
hazard of death from the origin to about 200 days after surgery. After that, the
hazard is fairly stable. This decline is reflected in the Median Residual
Lifetime column. At time 0, the median residual lifetime of 257.7 days is an
estimate of the median survival time for the entire sample. However, of those
patients still alive at 50 days, the median residual lifetime rises to 686.6 days.
The median remaining life continues to rise until it reaches a peak of 982.9
days for those who were still alive 400 days after surgery.
Output 3.16

Life-Table Estimates from Grouped Data

Life Table Survival Estimates

Interval
[Lower, Upper)
0
50
100
200
400
700
1000
1300
1600

50
100
200
400
700
1000
1300
1600
.

Number
Failed

Number
Censored

Effective
Sample
Size

16
11
4
5
2
4
1
1
0

3
0
2
4
6
3
2
3
1

66.5
49.0
37.0
30.0
20.0
13.5
7.0
3.5
0.5

Conditional
Probability
of Failure
0.2406
0.2245
0.1081
0.1667
0.1000
0.2963
0.1429
0.2857
0

(continued)

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

58
Output 3.16 (continued)

Interval
[Lower, Upper)
0
50
50
100
100
200
200
400
400
700
700
1000
1000
1300
1300
1600
1600
.

Conditional
Probability
Standard
Error
0.0524
0.0596
0.0510
0.0680
0.0671
0.1243
0.1323
0.2415
0

Survival
1.0000
0.7594
0.5889
0.5253
0.4377
0.3939
0.2772
0.2376
0.1697

Failure
0
0.2406
0.4111
0.4747
0.5623
0.6061
0.7228
0.7624
0.8303

Survival
Standard
Error
0
0.0524
0.0608
0.0620
0.0628
0.0637
0.0664
0.0677
0.0750

Median
Residual
Lifetime
257.7
686.6
855.7
910.5
982.9
779.6
.
.
.

Evaluated at the Midpoint of the Interval

Interval
[Lower, Upper)
0
50
100
200
400
700
1000
1300
1600

50
100
200
400
700
1000
1300
1600
.

Median
Standard
Error
140.1
139.4
124.4
363.2
216.3
236.9
.
.
.

PDF

PDF
Standard
Error

Hazard

Hazard
Standard
Error

0.00481
0.00341
0.000637
0.000438
0.000146
0.000389
0.000132
0.000226
.

0.00105
0.000935
0.000308
0.000186
0.0001
0.000175
0.000126
0.000202
.

0.00547
0.005057
0.001143
0.000909
0.000351
0.001159
0.000513
0.001111
.

0.001355
0.001513
0.00057
0.000405
0.000248
0.000571
0.000511
0.001096
.

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

Output 3.17

TESTING

FOR

Hazard Plot for Grouped Data

EFFECTS

COVARIATES

Besides tests of differences between groups, PROC LIFETEST can
test whether quantitative covariates are associated with survival time.
Given a list of covariates, PROC LIFETEST produces a test statistic for each
one and ignores the others. It also treats them as a set, testing the null
hypothesis that they are jointly unrelated to survival time and also testing
for certain incremental effects of adding variables to the set. The statistics
are generalizations of the log-rank and Wilcoxon tests discussed earlier in
this chapter. They can also be interpreted as nonparametric tests of the
coefficients of the accelerated failure time model discussed in Chapter 4,
“Estimating Parametric Regression Models with PROC LIFEREG.”

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

You can test the same sorts of hypotheses with PROC LIFEREG or
PROC PHREG. In fact, the log-rank chi-square reported by PROC
LIFETEST is identical to the score statistic given by PROC PHREG for the
null hypothesis that all coefficients are 0 (when the data contain no tied
event times). However, in most cases, you are better off switching to the
regression procedures, for two reasons. First, PROC LIFETEST doesn’t give
coefficient estimates, so there is no way to quantify the effect of a covariate
on survival time. Second, the incremental tests do not really test the effect
of each variable controlling for all the others. Instead, you get a test of the
effect of each variable controlling for those variables that have already
been included. Because you have no control over the order of inclusion,
these tests can be misleading. Nevertheless, PROC LIFETEST can be useful
for screening a large number of covariates before proceeding to estimate
regression models. Because the log-rank and Wilcoxon tests do not require
iterative calculations, they use relatively little computer time. (This is also
true for the SELECTION=SCORE option in PROC PHREG.)
Let’s look at the recidivism data as an example. The covariate tests
are invoked by listing the variable names in a TEST statement:
PROC LIFETEST DATA=recid;
TIME week*arrest(0);
TEST fin age race wexp mar paro prio;
RUN;

Output 3.18 shows selections from the output. I have omitted the
Wilcoxon statistics because they are nearly identical to the log-rank
statistics for this example. I also omitted the variance-covariance matrix for
the statistics because it is primarily useful as input to other analyses.

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

Output 3.18

Covariate Tests for the Recidivism Data
Univariate Chi-Squares for the LOG RANK Test
Variable

Test
Statistic

Variance

Chi-Square

Pr >
Chi-Square

10.4256
233.2
-2.7093
16.4141
7.1773
2.9471
-108.8

28.4744
4305.3
12.8100
27.3305
13.1535
26.7927
812.3

3.8172
12.6318
0.5730
9.8580
3.9164
0.3242
14.5602

0.0507
0.0004
0.4491
0.0017
0.0478
0.5691
0.0001

FIN
AGE
RACE
WEXP
MAR
PARO
PRIO

Forward Stepwise Sequence of Chi-Squares for the LOG RANK Test
Variable
PRIO
AGE
FIN
MAR
RACE
WEXP
PARO

DF
1
2
3
4
5
6
7

Chi-Square
14.5602
25.4905
28.8871
31.0920
32.4214
33.2800
33.3828

Pr >
Chi-Square

Chi-Square
Increment

0.0001
0.0001
0.0001
0.0001
0.0001
0.0001
0.0001

14.5602
10.9303
3.3966
2.2050
1.3294
0.8585
0.1029

Pr >
Increment
0.0001
0.0009
0.0653
0.1376
0.2489
0.3541
0.7484

The top panel shows that age at release (AGE), work experience
(WEXP), and number of prior convictions (PRIO) have highly significant
associations with time to arrest. The effects of marital status (MAR) and
financial aid (FIN) are more marginal, while race and parole status (PARO)
are apparently unrelated to survival time. The signs of the log-rank test
statistics tell you the direction of the relationship. The negative sign for
PRIO indicates that inmates with more prior convictions tend to have
shorter times to arrest. On the other hand, the positive coefficient for AGE
indicates that older inmates have longer times to arrest. As already noted,
none of these tests controls or adjusts for any of the other covariates.
The lower panel displays results from a forward inclusion
procedure. PROC LIFETEST first finds the variable with the highest chisquare statistic in the top panel—in this case, PRIO—and puts it in the set
to be tested. Because PRIO is the only variable in the set, the results for
PRIO are the same in both panels. Then PROC LIFETEST finds the variable
that produces the largest increment in the joint chi-square for the set of
two variables—in this case, AGE. The joint chi-square of 25.49 in line 2
tests the null hypothesis that the coefficients of AGE and PRIO in an

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

accelerated-failure time model are both 0. The chi-square increment of
10.93 is merely the difference between the joint chi-square in lines 1 and
2. It is a test of the null hypothesis that the coefficient for AGE is 0 when
PRIO is controlled. On the other hand, there is no test for the effect of PRIO
controlling for AGE.
This process is repeated until all the variables are added. For each
variable, we get a test of the hypothesis that the variable is unrelated to
survival time controlling for all the variables above it (but none of the
variables below it). For variables near the end of the sequence, the
incremental chi-square values are likely to be similar to what you might
find with PROC LIFEREG or PROC PHREG. For variables near the
beginning of the sequence, however, the results can be quite different.
For this example, the forward inclusion procedure leads to some
substantially different conclusions from the univariate procedure. While
WEXP has a highly significant effect on survival time when considered by
itself, there is no evidence of such an effect when other variables are
controlled. The reason is that work experience is moderately correlated
with age and the number of prior convictions, both of which have
substantial effects on survival time. Marital status also loses its statistical
significance in the forward inclusion test.
What is the relationship between the STRATA statement and the
TEST statement? For a dichotomous variable like FIN, the statement TEST
FIN is a possible alternative to STRATA FIN. Both produce a test of the
null hypothesis that the survivor functions are the same for the two
categories of FIN. In fact, if there are no ties in the data (no cases with
exactly the same event time), the two statements will produce identical
chi-square statistics and p-values. In the presence of ties, however,
STRATA and TEST use somewhat different formulas, which may result in
slight differences in the p-values. (If you’re interested in the details, see
Collett, 2003.) In the recidivism data, for example, the 114 arrests occurred
at only 49 unique arrest times, so the number of ties was substantial. The
STRATA statement produces a log-rank chi-square of 3.8376 for a p-value
of .0501 and a Wilcoxon chi-square of 3.7495 for a p-value of .0528. The
TEST statement produces a log-rank chi-square of 3.8172 for a p-value of
.0507 and a Wilcoxon chi-square of 3.7485 for a p-value of .0529.
Obviously the differences are minuscule in this case.
Other considerations should govern the choice between the STRATA
and TEST statements. While the STRATA statement produces separate
tables and graphs of the survivor function for the two groups, the TEST
statement produces only the single table and graph for the entire sample.
With the TEST statement, you can test for the effects of many dichotomous
variables with a single statement, but the STRATA statement requires a

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

new PROC LIFETEST step for each variable tested. Of course, if a variable
has more than two values, the STRATA statement treats each value as a
separate group while the TEST statement treats the variable as a
quantitative measure.
What happens when you include both a STRATA statement and a
TEST statement? Adding a TEST statement has no effect whatever on the
results from the STRATA statement. This fact implies that the hypothesis
test produced by the STRATA statement in no way controls for the
variables listed in the TEST statement. On the other hand, the TEST
statement can produce quite different results, depending on whether you
also have a STRATA statement. When you have a STRATA statement, the
log-rank and Wilcoxon statistics produced by the TEST statement are first
calculated within strata and then averaged across strata. In other words,
they are stratified statistics that control for whatever variable or variables
are listed in the STRATA statement. Suppose, for example, that for the
myelomatosis data, we want to test the effect of the treatment while
controlling for renal functioning. We can submit these statements:
PROC LIFETEST DATA=myel;
TIME dur*status(0);
STRATA renal;
TEST treat;
RUN;

The resulting log-rank chi-square for TREAT is 5.791 with a p-value
of .016. This result is in sharp contrast with the unstratified chi-square of
only 1.3126 that we saw earlier in this chapter (Output 3.7).
An alternative way to get a stratified test comparing survivor
functions for different groups is to use the GROUP option in the STRATA
statement, as follows:
PROC LIFETEST DATA=myel;
TIME dur*status(0);
STRATA renal / GROUP=treat;
RUN;

This produces the same log-rank chi-square for TREAT of 5.791 with a
p-value of .016. But the advantage of this approach is that the GROUP
variable (unlike a variable in the TEST statement) can have more than two
categories. As we’ll see in Chapter 5, “Estimating Cox Regression Models
with PROC PHREG,” you can also get nearly identical results using PROC
PHREG with stratification and the score test.

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

LOG SURVIVAL AND SMOOTHED HAZARD PLOTS
PROC LIFETEST produces two other plots that give useful
information about the shape of the hazard function, the log-survival (LS)
plot and the log-log survival (LLS) plot. In this section, we’ll see how these
plots help determine whether the hazard function can be accurately
described by certain parametric models discussed in Chapter 2. We’ll also
see how to get smoothed estimates of the hazard function with ungrouped
data.
Suppose we specify PLOTS=(S, LS, LLS) in the PROC LIFETEST
statement. The S gives us the now-familiar survival curve. LS produces a
plot of − log Sˆ (t ) versus t. To explain how this plot is useful, we need a
little background. From equation (2.6), one can readily see that
t

− log S (t ) = ∫ h(u )du
0

Because of this relationship, the log survivor function is commonly
referred to as the cumulative hazard function, frequently denoted by Λ(t).
Now, if h(t) is a constant with a value of λ (which implies an exponential
distribution for event times), then the cumulative hazard function is just
Λ(t) = λt. This result implies that a plot of − log Sˆ (t ) versus t should yield a
straight line with an origin at 0. Moreover, an examination of the logsurvival plot can tell us whether the hazard is constant, increasing, or
decreasing with time.
Output 3.19 displays the LS plot for the myelomatosis data. Instead
of a straight line, the graph appears to increase at a decreasing rate. This
fact suggests that the hazard is not constant but rather declines with time.
If the plot had curved upward rather than downward, it would suggest that
the hazard was increasing with time. Of course, because the sample size is
quite small, caution is advisable in drawing any conclusions. A formal test,
such as the one described in the next chapter, might not show a significant
decrease in the hazard.

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

Output 3.19

Log-Survival Plot for Myelomatosis Data

The LLS keyword produces a plot of log[− log Sˆ (t )] versus log t. If
survival times follow a Weibull distribution, which has a hazard given by
log h(t ) = α + β log t , then the log-log survival plot (log cumulative hazard
plot) should be a straight line with a slope of β. Examining Output 3.20,
we see a rather rough plot with a slight tendency to turn downward at later
times. Again, however, the data are so sparse that this is probably not
sufficient evidence for rejecting the Weibull distribution. In Chapter 4,
we’ll see how to construct similar plots for other distributions such as the
log-normal and log-logistic.

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

66
Output 3.20

Log-Log Survival Plot for Myelomatosis Data

Although graphs based on transformations of the survivor function
are certainly useful, they are ultimately rather frustrating. What we really
want to see is a graph of the hazard function. We got that from the life-table
method, but at the cost of grouping the event times into arbitrarily chosen
intervals. While it’s possible to estimate hazards from ungrouped data, the
estimates usually vary so wildly from one time to the next as to be almost
useless. There are several ways to smooth these estimates by calculating
some sort of moving average. One method, known as kernel smoothing,
has been shown to have good properties for hazard functions (RamlauHansen, 1983). Using this method, PROC LIFETEST (SAS 9.2 and later)
can produce smoothed graphs of hazard functions with the ODS version of
the PLOT option. Here’s how to do it for the recidivism data:
ODS GRAPHICS ON;
PROC LIFETEST DATA=recid PLOTS=H;
TIME week*arrest(0);
RUN;
ODS GRAPHICS OFF;

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

Output 3.21 shows the smoothed hazard function. The graph bears
some resemblance to the grouped hazard plot in Output 3.14, although
here we see a rather pronounced plateau between about 20 and 30 weeks
and no downturn at the end.

Output 3.21 Smoothed Hazard Function Estimate for Recidivism Data

Notice the Bandwidth=11.4839 in the lower left corner. This means
that, when calculating the hazard for any specified point in time, the
smoothing function only uses data within 11.5 weeks on either side of the
time point. The bandwidth has a big impact on the appearance of the
graph: the larger the bandwidth, the smoother the graph. By default, PROC
LIFETEST selects the bandwidth using a method that has certain
optimality properties. But sometimes you may want to experiment with
different bandwidths. Here’s how to force the bandwidth to be 5 weeks:
PROC LIFETEST DATA=recid PLOTS=H(BW=5);

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

As Output 3.22 shows, the resulting hazard function is much choppier.

Output 3.22 Smoothed Hazard Function Estimate with Smaller Bandwidth

If you want confidence limits around the hazard function, you can
get them using the CL option:
PROC LIFETEST DATA=recid PLOTS=H(CL);

However, as we see in Output 3.23, the hazard curve is greatly
compressed because the confidence limits are so wide at the earliest and
latest times.

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

Output 3.23 Smoothed Hazard Function with Confidence Limits

CONCLUSION
PROC LIFETEST is a useful procedure for preliminary analysis of
survival data and for testing simple hypotheses about differences in
survival across groups. For experimental studies, PROC LIFETEST (with
the STRATA statement) gives tests that are analogous to a one-way analysis
of variance. But the procedure is not adequate for two-factor designs
because there is no way to test for interactions. Similarly, while the TEST
statement in PROC LIFETEST may be useful for screening large numbers of
quantitative covariates, it is not adequate for examining the effects of
variables controlling for other covariates. In most cases, therefore, you will
need to move to the estimation of regression models with PROC LIFEREG
or PROC PHREG.

CHAPTER 3

Estimating and Comparing Survival Curves with PROC LIFETEST

It is also important to recognize that survival curves and their
associated hazard functions can be misleading when the sample is
heterogeneous. As explained in Chapter 8, uncontrolled heterogeneity
tends to make hazard functions look as though they are declining, even
when there is no real decline for any individual in the sample. As we will
see in Chapter 5, one way to reduce the effect of heterogeneity is to
estimate and plot baseline survivor functions after fitting Cox regression
models.

CHAPTER 4
Estimating Parametric Regression Models with
PROC LIFEREG
Introduction 71
The Accelerated Failure Time Model 72
Alternative Distributions 77
Categorical Variables and the CLASS Statement 87
Maximum Likelihood Estimation 89
Hypothesis Tests 95
Goodness-of-Fit Tests with the Likelihood-Ratio Statistic 98
Graphical Methods for Evaluating Model Fit 100
Left Censoring and Interval Censoring 103
Generating Predictions and Hazard Functions 108
The Piecewise Exponential Model 112
Bayesian Estimation and Testing 117
Conclusion 124

INTRODUCTION
The LIFEREG procedure produces estimates of parametric regression
models with censored survival data using the method of maximum
likelihood. To some degree, PROC LIFEREG has been eclipsed by the
PHREG procedure, which does semiparametric regression analysis using a
method known as partial likelihood. The reasons for PROC PHREG’s
popularity will become apparent in the next chapter. PROC LIFEREG is by
no means obsolete, however. It can do some things better than PROC
PHREG, and it can do other things that PROC PHREG can’t do at all:
 PROC LIFEREG accommodates left censoring and interval
censoring. PROC PHREG allows only right censoring.
 With PROC LIFEREG, you can test certain hypotheses about the
shape of the hazard function. PROC PHREG gives you only
nonparametric estimates of the survivor function, which can be
difficult to interpret.

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

If the shape of the survival distribution is known, PROC
LIFEREG produces more efficient estimates (with smaller
standard errors) than PROC PHREG.
 PROC LIFEREG can easily generate predicted event times for any
specified set of covariate values. This is more difficult with
PHREG, and often impossible.
PROC LIFEREG’s greatest limitation is that it does not handle
time-dependent covariates, something at which PROC PHREG excels.


THE ACCELERATED FAILURE TIME MODEL
The class of regression models estimated by PROC LIFEREG is
known as the accelerated failure time (AFT) model. In its most general
form, the AFT model describes a relationship between the survivor
functions of any two individuals. If Si(t) is the survivor function for
individual i, then for any other individual j, the AFT model holds that

S j (t ) = Si (φij t ) for all t

(4.1)

where φij is a constant that is specific to the pair (i, j). This model says, in
effect, that what makes one individual different from another is the rate at
which they age. A good example is the conventional wisdom that a year
for a dog is equivalent to 7 years for a human. This relationship can be
represented by equation (4.1), with Si being the survival probability for a
dog, Sj the survival probability for a human, and φij = 7.
What PROC LIFEREG actually estimates is a special case of this
model that is quite similar in form to an ordinary linear regression model.
Let Ti be a random variable denoting the event time for the ith individual
in the sample, and let xi1, …, xik be the values of k covariates for that same
individual. The model is then

log Ti = β 0 + β1 xi1 + ... + β k xik + σε i

(4.2)

where εi is a random disturbance term, and β0,…,βk, and σ are parameters
to be estimated. Exponentiating both sides of equation (4.2) gives an
alternative way of expressing the model:

Ti = exp( β 0 + β1 xi1 + ... + β k xik + σε i )
So far, the only differences between the model in equation (4.2) and
the usual linear regression model are that there is a σ before the ε and the
dependent variable is logged. The σ can be omitted, which requires that
the variance of ε be allowed to vary from one data set to another. But it is

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

simpler to fix the variance of ε at some standard value (for example, 1.0)
and let σ change in value to accommodate changes in the disturbance
variance. (This notational strategy could also be used for linear models.)
As for the log transformation of T, its main purpose is to ensure that
predicted values of T are positive, regardless of the values of the x’s and
the β’s.
In a linear regression model, it is typical to assume that εi has a
normal distribution with a mean and variance that is constant over i, and
that the ε’s are independent across observations. One member of the AFT
class, the log-normal model, has exactly these assumptions. It is called the
log-normal model because if log T has a normal distribution, then T has a
log-normal distribution. Other AFT models allow distributions for ε
besides the normal distribution but retain the assumptions of constant
mean and variance, as well as independence across observations. We will
consider these alternative models in some detail, but for the moment let’s
stick with the log-normal.
If there are no censored data, we can readily estimate this model by
ordinary least squares (OLS). Simply create a new variable in a DATA step
that is equal to the log of the event time and use the REG procedure with
the transformed variable as the dependent variable. This process yields
best linear unbiased estimates of the β coefficients, regardless of the shape
of the distribution of ε. If ε is normal, the OLS estimates will also be
maximum likelihood estimates and will have minimum variance among
all estimators, both linear and nonlinear. In fact, the coefficients and
standard errors produced by PROC REG will be identical to those
produced by PROC LIFEREG with a log-normal specification.
But survival data typically have at least some censored observations,
and these are difficult to handle with OLS. Instead, we can use maximum
likelihood estimation. Later, this chapter examines the mathematics of
maximum likelihood (ML) for censored regression models in some detail.
First, let’s look at an example of how a regression application is set up
with PROC LIFEREG and what results it produces. In the section The Life
Table Method in Chapter 3, “Estimating and Comparing Survival Curves
with PROC LIFETEST,” I described the recidivism data set in which 432
inmates were followed for 1 year after release. That data set is used
throughout this chapter, so you may want to reread the earlier description
(or see Appendix 2, “Data Sets”).
In this recidivism example, the variable WEEK contains the week of
the first arrest or censoring. The variable ARREST is equal to 1 if WEEK is
uncensored or 0 if right censored. There are seven covariates. To estimate
the log-normal model, we specify

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

PROC LIFEREG DATA=recid;
MODEL week*arrest(0)=fin age race wexp mar paro prio
/ DISTRIBUTION=LNORMAL;
RUN;

Note that WEEK*ARREST(0) in the MODEL statement follows the
same syntax as the TIME statement in PROC LIFETEST. However, it is now
followed by an equal sign and a list of covariates. The slash (/) separates
the variable list from the specification of options, of which there are
several possibilities. Here, we have merely indicated our choice of the lognormal distribution.
Output 4.1 displays the results. The output first provides some
preliminary information: the names of the time and censoring variables,
the values that correspond to censoring, and the numbers of cases with
each type of censoring. Then the output shows that the log-likelihood for
the model is –322.6946. This is an important statistic that we will use later
to test various hypotheses.
The table labeled “Fit Statistics” gives four modifications of the loglikelihood that may be used in assessing model fit. The first is simply the
log-likelihood multiplied by –2. The second, Akaike’s information
criterion (AIC), is a modification of the –2 log-likelihood that penalizes
models for having more covariates. Specifically,

AIC = −2 log L + 2k
where k is the number of covariates. The next statistic (AICC) is a
“corrected” version of the AIC that may have better behavior in small
samples:

AICC = AIC +

2k (k + 1)
n − k −1

Finally, the Bayesian information criterion (also known as Schwarz’s
criterion) gives a more severe penalization for additional covariates, at
least for most applications:

BIC = −2 log L + k log n
All three of these penalized statistics can be used to compare models with
different sets of covariates. The models being compared do not have to be
nested, in the sense of one model being a special case of another. However,
these statistics cannot be used to construct a formal hypothesis test, so the
comparison is only informal. For all three statistics, smaller values mean a

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

better fit. But keep in mind that the overall magnitude of these statistics
depends heavily on sample size.
For this example, the “Type III Analysis of Effects” table is
completely redundant with the table that follows. But the Type III table
will be important later when we consider CLASS variables.
Finally, we get a table of estimated coefficients, their standard errors,
chi-square statistics for the null hypothesis that each coefficient is 0, and
p-values associated with those statistics. The chi-squares are calculated by
dividing each coefficient by its estimated standard error and squaring the
result.

Output 4.1 Results from Fitting a Log-Normal Model to the Recidivism Data

The LIFEREG Procedure
Model Information
Data Set
Dependent Variable
Censoring Variable
Censoring Value(s)
Number of Observations
Noncensored Values
Right Censored Values
Left Censored Values
Interval Censored Values
Name of Distribution
Log Likelihood

RECID
Log(week)
arrest
0
432
114
318
0
0
Lognormal
-322.6945851

Number of Observations Read
Number of Observations Used

432
432

Fit Statistics
-2 Log Likelihood
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)

645.389
663.389
663.816
700.005

Algorithm converged.

(continued)

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

76
Output 4.1 (continued)

Type III Analysis of Effects

Effect
fin
age
race
wexp
mar
paro
prio

Wald
Chi-Square

Pr > ChiSq

1
1
1
1
1
1
1

4.3657
2.9806
1.8824
2.2466
2.4328
0.1092
5.8489

0.0367
0.0843
0.1701
0.1339
0.1188
0.7411
0.0156

Analysis of Maximum Likelihood Parameter Estimates

Parameter
Intercept
fin
age
race
wexp
mar
paro
prio
Scale

DF Estimate
1
1
1
1
1
1
1
1
1

4.2677
0.3428
0.0272
-0.3632
0.2681
0.4604
0.0559
-0.0655
1.2946

Standard
Error
0.4617
0.1641
0.0158
0.2647
0.1789
0.2951
0.1691
0.0271
0.0990

95% Confidence
Limits
3.3628
0.0212
-0.0037
-0.8819
-0.0825
-0.1181
-0.2756
-0.1186
1.1145

5.1726
0.6645
0.0581
0.1556
0.6187
1.0388
0.3873
-0.0124
1.5038

ChiSquare Pr > ChiSq
85.44
4.37
2.98
1.88
2.25
2.43
0.11
5.85

<.0001
0.0367
0.0843
0.1701
0.1339
0.1188
0.7411
0.0156

Two variables meet the .05 criterion for statistical significance: FIN
(whether the inmate received financial aid) and PRIO (number of prior
convictions). The signs of the coefficients tell us the direction of the
relationship. The positive coefficient for FIN indicates that those who
received financial aid had longer times to arrest than those who did not.
The negative coefficient for PRIO indicates that additional convictions
were associated with shorter times to arrest. As in any regression
procedure, these coefficients adjust or control for the other covariates in
the model.
The numerical magnitudes of the coefficients are not very
informative in the reported metrics, but a simple transformation leads to a
more intuitive interpretation. For a 1-0 variable like FIN, if we simply take
eβ, we get the estimated ratio of the expected (mean) survival times for the
two groups. Thus, e.3428= 1.41. Therefore, controlling for the other
covariates, the expected time to arrest for those who received financial aid

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

is 41 percent greater than for those who did not receive financial aid. (This
statement also applies to the median time to arrest, or any other percentile
for that matter.) For a quantitative variable like PRIO, we can use the
transformation 100(eβ – 1), which gives the percent change in the expected
survival time for each one-unit increase in the variable. Thus,
100(exp(–.0655)– 1) = –6.34. According to the model, then, each additional
prior conviction is associated with a 6.34 percent decrease in expected
time to arrest, holding other covariates constant. We can also interpret the
coefficients for any of the other AFT models discussed in this chapter in
this way.
The output line labeled SCALE is an estimate of the σ parameter in
equation (4.2), along with its estimated standard error. For some
distributions, changes in the value of this parameter can produce
qualitative differences in the shape of the hazard function. For the lognormal model, however, changes in σ merely compress or stretch the
hazard function.

ALTERNATIVE DISTRIBUTIONS
In ordinary linear regression, the assumption of a normal
distribution for the disturbance term is routinely invoked for a wide range
of applications. Yet PROC LIFEREG allows for four additional distributions
for ε: extreme value (2 parameter), extreme value (1 parameter), loggamma, and logistic. For each of these distributions, there is a
corresponding distribution for T:
Distribution of 

Distribution of T

extreme value (2 par.)

Weibull

extreme value (1 par.)

exponential

log-gamma

gamma

logistic

log-logistic

normal

log-normal

Incidentally, all AFT models are named for the distribution of T
rather than for the distribution of ε or log T. Because the logistic and
normal lead to the log-logistic and log-normal, you might expect that the
gamma will lead to the log-gamma. But it is just the reverse. This is one of
those unfortunate inconsistencies in terminology that we just have to live
with.

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

What is it about survival analysis that makes these alternatives worth
considering? The main reason for allowing other distributions is that they
have different implications for hazard functions that may, in turn, lead to
different substantive interpretations. The remainder of this section
explores each of these alternatives in some detail.
The Exponential Model

The simplest model that PROC LIFEREG estimates is the exponential
model, invoked by DISTRIBUTION=EXPONENTIAL in the MODEL
statement. This model specifies that ε has a standard extreme-value
distribution and constrains σ = 1. If ε has an extreme-value distribution,
then log T also has an extreme-value distribution, conditional on the
covariates. This implies that T itself has an exponential distribution,
which is why we call it the exponential model. The standard extreme
value distribution is also known as a Gumbel distribution or a double
exponential distribution. It has a p.d.f. of f(ε) = exp[ε – exp(ε)]. Like the
normal distribution, this is a unimodal distribution defined on the entire
real line. Unlike the normal, however, it is not symmetrical, being slightly
skewed to the left.
As we saw in Chapter 2, “Basic Concepts of Survival Analysis,” an
exponential distribution for T corresponds to a constant hazard function,
which is the most characteristic feature of this model. However, equation
(2.12) expresses the exponential regression model as

log h(t ) = β 0• + β1• x1 +  + β k• xk

(4.3)

where the •’s have been added to distinguish these coefficients from those
in equation (4.2). Although the dependent variable in equation (4.2) is the
log of time, in equation (4.3) it is the log of the hazard. It turns out that the
two models are completely equivalent. Furthermore, there is a simple
relationship between the coefficients in equation (4.2) and equation (4.3):
namely, that βj = –βj• for all j.
The change in signs makes intuitive sense. If the hazard is high, then
events occur quickly and survival times are short. On the other hand,
when the hazard is low, events are unlikely to occur and survival times are
long. It is important to be able to shift back and forth between these two
ways of expressing the model so that you can compare results across
different computer programs. In particular, because PROC PHREG reports
coefficients in log-hazard form, we need to make the conversion in order to
compare PROC LIFEREG output with PROC PHREG output.
You may wonder why there is no disturbance term in equation (4.3)
(a characteristic it shares with the more familiar logistic regression model).
No disturbance term is needed because there is implicit random variation

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

in the relationship between h(t), the unobserved hazard, and the observed
event time T. Even if two individuals have exactly the same covariate
values (and, therefore, the same hazard), they will not have the same event
time. Nevertheless, in Chapter 8, “Heterogeneity, Repeated Events, and
Other Topics,” we will see that there have been some attempts to add a
disturbance term to models like this to represent unobserved heterogeneity.
Output 4.2 shows the results of fitting the exponential model to the
recidivism data. Comparing this with the log-normal results in Output 4.1,
we see some noteworthy differences. The coefficient for AGE is about
twice as large in the exponential model, and its p-value declines from .08
to .01. Similarly, the coefficient for PRIO increases somewhat in
magnitude, and its p-value also goes down substantially. On the other
hand, the p-value for FIN increases to slightly above the .05 level.

Exponential Model Applied to Recidivism Data

Output 4.2

Analysis of Maximum Likelihood Parameter Estimates

Parameter
Intercept
fin
age
race
wexp
mar
paro
prio
Scale
Weibull Shape

DF Estimate
1
1
1
1
1
1
1
1
0
0

Standard
Error

4.0507
0.3663
0.0556
-0.3049
0.1467
0.4270
0.0826
-0.0857
1.0000
1.0000

0.5860
0.1911
0.0218
0.3079
0.2117
0.3814
0.1956
0.0283
0.0000
0.0000

95% Confidence
Limits
2.9021
-0.0083
0.0128
-0.9085
-0.2682
-0.3205
-0.3007
-0.1412
1.0000
1.0000

5.1993
0.7408
0.0984
0.2986
0.5617
1.1745
0.4660
-0.0302
1.0000
1.0000

ChiSquare Pr > ChiSq
47.78
3.67
6.48
0.98
0.48
1.25
0.18
9.15

<.0001
0.0553
0.0109
0.3220
0.4882
0.2629
0.6726
0.0025

Lagrange Multiplier Statistics
Parameter
Scale

Chi-Square
24.9302

Pr > ChiSq
<.0001

Clearly, the choice of model can make a substantive difference. Later,
this chapter considers some criteria for choosing among these and other
models. Notice that the SCALE parameter σ is forced equal to 1.0 which, as
noted above, is what distinguishes the exponential model from the Weibull
model. The line labeled “Weibull Shape” will be explained when we
discuss the Weibull model.

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

The “Lagrange Multiplier Statistics” table reports a chi-square of
24.9302 with a p-value less than .0001. This is a 1 degree-of-freedom test
for the null hypothesis that σ = 1. Here the null hypothesis is soundly
rejected, indicating that the hazard function is not constant over time.
While this might suggest that the log-normal model is superior, things are
not quite that simple. There are other models to consider as well.
The Weibull Model

The Weibull model is a slight modification of the exponential model,
with big consequences. By specifying DISTRIBUTION=WEIBULL in the
MODEL statement, we retain the assumption that ε has a standard extremevalue distribution, but we relax the assumption that σ = 1. When σ >1, the
hazard decreases with time. When .5<σ<1, the hazard is increasing at a
decreasing rate. When 0<σ<.5, the hazard is increasing at an increasing
rate. And when σ=.5, the hazard function is an increasing straight line
with an origin at 0. Graphs of these hazard functions appear in Figure 2.3
(with the α in the figure equal to 1/σ – 1).
We call this the Weibull model because T has a Weibull distribution,
conditional on the covariates. The Weibull distribution has long been the
most popular parametric model in the biostatistical literature, for two
reasons. First, it has a relatively simple survivor function that is easy to
manipulate mathematically:

[


Si (t ) = exp− ti e − βx i


] 
1



where xi is a vector of the covariate values and β is a vector of coefficients.
Second, in addition to being an AFT model, the Weibull model is also a
proportional hazards model. This means that its coefficients (when
suitably transformed) can be interpreted as relative hazard ratios. In fact,
the Weibull model (and its special case, the exponential model) is the only
model that is simultaneously a member of both these classes.
As with the exponential model, there is an exact equivalence
between the log-hazard form of the model

log h(t ) = α log t + β 0• + β1• x1 +  + β k• xk
and the log-survival time model

log Ti = β 0 + β1 x1 + ... + β k xk + σε
The relationship between the parameters is slightly more
complicated, however. Specifically, for the Weibull model

β •j =

− βj

for j = 1,…, k

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

and α = (1/σ )– 1. Because βj = 0 if and only if βj• = 0, a test of the null
hypothesis that a coefficient is 0 will be the same regardless of which form
you use. On the other hand, standard errors and confidence intervals for
coefficients in the log-survival time format are not so easily converted to
the log-hazard format. Collett (2003) gives formulas for accomplishing this.
Output 4.3 shows the results from fitting the Weibull model to the
recidivism data. Compared with the exponential model in Output 4.2, the
coefficients are all somewhat attenuated. But the standard errors are also
smaller, so the chi-square statistics and p-values are hardly affected at all.
Furthermore, if we convert the coefficients to the log-hazard format by
changing the sign and dividing by σˆ (the Scale estimate of .7124 in the
output), we get
FIN
AGE
RACE
WEXP
MAR
PARO
PRIO

-0.382
-0.057
0.316
-0.150
-0.437
-0.083
0.092.

These coefficients are much closer to the log-hazard coefficients for the
exponential model (which differ only in sign from the log-survival time
coefficients).

Output 4.3

Weibull Model Applied to Recidivism Data

Parameter
Intercept
fin
age
race
wexp
mar
paro
prio
Scale
Weibull Shape

Standard
DF Estimate
Error
1
1
1
1
1
1
1
1
1
1

3.9901
0.2722
0.0407
-0.2248
0.1066
0.3113
0.0588
-0.0658
0.7124
1.4037

0.4191
0.1380
0.0160
0.2202
0.1515
0.2733
0.1396
0.0209
0.0634
0.1250

95% Confidence
Limits
3.1687
0.0018
0.0093
-0.6563
-0.1905
-0.2244
-0.2149
-0.1069
0.5983
1.1789

4.8115
0.5426
0.0721
0.2067
0.4036
0.8469
0.3325
-0.0248
0.8482
1.6713

ChiSquare Pr > ChiSq
90.65
3.89
6.47
1.04
0.49
1.30
0.18
9.88

<.0001
0.0485
0.0110
0.3072
0.4820
0.2547
0.6735
0.0017

Because σˆ (labeled “Scale” in Output 4.3) is between 0 and 1, we
conclude that the hazard is increasing at a decreasing rate. We can also
calculate α̂ = (1/.7124)–1=0.4037, which is the coefficient for log t in the
log-hazard model. Because both the dependent and independent variables

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

are logged, this coefficient can be interpreted as follows: a 1 percent
increase in time since release is associated with a 0.40 percent increase in
the hazard for arrest.
The last line is labeled “Weibull Shape.” This is not an independent
parameter but merely the reciprocal of the Shape parameter (that is, 1/ σˆ ).
It is included in the output because some statisticians prefer this way of
parameterizing the model.
The Log-Normal Model

Although we have already discussed the log-normal model and
applied it to the recidivism data, we have not yet considered the shape of
its hazard function. Unlike the Weibull model, the log-normal model has a
nonmonotonic hazard function. The hazard is 0 when t=0. It rises to a peak
and then declines toward 0 as t goes to infinity. The log-normal is not a
proportional hazards model, and its hazard function cannot be expressed
in closed form (it involves the c.d.f. of a standard normal variable). It can,
however, be expressed as a regression model in which the dependent
variable is the logarithm of the hazard. Specifically,

log h(t ) = log h0 (te − βx ) − βx
where h0(.) can be interpreted as the hazard function for an individual
with x = 0. This equation also applies to the log-logistic and gamma
models to be discussed shortly, except that h0(.) is different in each case.
Some typical log-normal hazard functions are shown in Figure 4.1.
All three functions correspond to distributions with a median of 1.0. When
σ is large, the hazard peaks so rapidly that the function is almost
indistinguishable from those like the Weibull and log-logistic that may
have an infinite hazard when t = 0.
The inverted U-shape of the log-normal hazard is often appropriate
for repeatable events. Suppose, for example, that the event of interest is a
residential move. Immediately after a move, the hazard of another move is
likely to be extremely low. People need to rest and recoup the substantial
costs involved in moving. The hazard will certainly rise with time, but
much empirical evidence indicates that it eventually begins to decline.
One explanation is that, as time goes by, people become increasingly
invested in a particular location or community. However, Chapter 8 shows
how the declining portion of the hazard function may also be a
consequence of unobserved heterogeneity.

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

83
Figure 4.1 Typical Hazard Functions for a Log-Normal Model

The Log-Logistic Model

Another model that allows for an inverted U-shaped hazard is the
log-logistic model, which assumes that ε has a logistic distribution with
p.d.f.

f (ε ) =

eε
.
(1 + eε ) 2

The logistic distribution is symmetric with a mean of 0, and is quite
similar in shape to the normal distribution. It is well known to students of
the logistic (logit) regression model, which can be derived by assuming a
linear model with a logistically distributed error term and a
dichotomization of the dependent variable.
If ε has a logistic distribution, then so does log T (although with a
nonzero mean). It follows that T has a log-logistic distribution. The loglogistic hazard function is

h(t ) =

λγ (λt )γ −1
1 + (λt )γ

where γ = 1/σ and λ = exp{–[β0 + β1x1 + … + βkxk]}. This produces the
characteristic shapes shown in Figure 4.2, all of which correspond to
distributions with a median of 1.0.

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

Figure 4.2

Typical Hazard Functions for the Log-Logistic Model

When σ < 1, the log-logistic hazard is similar to the log-normal hazard:
it starts at 0, rises to a peak, and then declines toward 0. When σ > 1, the
hazard behaves like the decreasing Weibull hazard: it starts at infinity and
declines toward 0. When σ =1, the hazard has a value of λ at t=0 and then
declines toward 0 as t goes to infinity.
Despite the complexity of its hazard function, the log-logistic model
has a rather simple survivor function:

S (t ) =

1
1 + (λ t )γ

As before, γ = 1/σ and λ = exp{–[β0 + β1x1 + … + βkxk]}. A little algebra
shows that this can be written as

 S (t ) 
•
•
•
log 
 = β 0 + β1 x1 +  + β k xk − γ log t
−
S
t
1
(
)


•

where βi = βi/σ for i = 1, ..., k. This is just a logistic regression model for
the probability that an event occurs prior to t. Thus, for the recidivism
data, the log-logistic model can be estimated by fitting a logistic regression
model to the dichotomy arrested versus not arrested in the first year after
release. (Because t is a constant 52 weeks, the term γ log t gets absorbed
into the intercept.) Of course, this estimation method is not fully efficient
because we are not using the information on the exact timing of the arrests,
and we certainly will not get the same estimates. The point is that the two
apparently different methods are actually estimating the same underlying
model.

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

To fit the log-logistic model with PROC LIFEREG, you specify
DISTRIBUTION=LLOGISTIC as an option in the MODEL statement.
Output 4.4 shows the results for the recidivism data. The first thing to
notice is that the estimate of σ (labeled “Scale”) is less than 1.0, implying
that the estimated hazard function follows the inverted U-shaped form
shown in Figure 4.2. Given what I just said about the similarity of the lognormal and log-logistic hazards, you might expect the other results to be
similar to the log-normal output in Output 4.1. But the coefficients and test
statistics actually appear to be closer to those for the Weibull model in
Output 4.3.

Output 4.4 Log-Logistic Model Applied to Recidivism Data

Analysis of Maximum Likelihood Parameter Estimates

Parameter
Intercept
fin
age
race
wexp
mar
paro
prio
Scale

DF Estimate
1
1
1
1
1
1
1
1
1

3.9183
0.2889
0.0364
-0.2791
0.1784
0.3473
0.0508
-0.0692
0.6471

Standard
Error
0.4274
0.1456
0.0156
0.2297
0.1572
0.2697
0.1496
0.0227
0.0559

95% Confidence
Limits
3.0805
0.0035
0.0058
-0.7293
-0.1297
-0.1812
-0.2424
-0.1138
0.5463

4.7561
0.5742
0.0669
0.1710
0.4865
0.8758
0.3440
-0.0246
0.7666

ChiSquare Pr > ChiSq
84.03
3.94
5.45
1.48
1.29
1.66
0.12
9.25

<.0001
0.0472
0.0195
0.2242
0.2563
0.1978
0.7341
0.0023

The Gamma Model

The generalized gamma model has one more parameter than any of
the other models we have considered, and that fact implies that its hazard
function can take on a wide variety of shapes. In particular, the
exponential, Weibull, and log-normal models (but not the log-logistic) are
all special cases of the generalized gamma model. This fact is exploited
later in this chapter when we consider likelihood ratio tests for comparing
different models. But the generalized gamma model can also take on
shapes that are unlike any of these special cases. Most important, it can
have hazard functions with U or bathtub shapes in which the hazard
declines, reaches a minimum, and then increases. It is well known that the
hazard for human mortality, considered over the whole life span, has such
a shape. On the other hand, the generalized gamma model cannot
represent hazard functions that have more than one reversal of direction.

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

Given the richness of the generalized gamma model, why not always
use it instead of the other models? There are two reasons. First, the formula
for the hazard function for the generalized gamma model is rather
complicated, involving the gamma function and the incomplete gamma
function. Consequently, you may often find it difficult to judge the shape of
the hazard function from the estimated parameters. By contrast, hazard
functions for the specific submodels can be rather simply described, as we
have already seen. Second, computation for the generalized gamma model is
more difficult. For example, it took more than five times as much computer
time to estimate the generalized gamma model for the recidivism data as
compared with the exponential model. This fact can be an important
consideration when you are working with very large data sets. The
generalized gamma model also has a reputation for convergence problems,
although the parameterization and numerical algorithms used by PROC
LIFEREG seem to have reduced these to a minimum.
To fit the generalized gamma model with PROC LIFEREG, you
specify D=GAMMA as an option in the MODEL statement. Output 4.5
shows the results from fitting this model to the recidivism data. As usual,
the Scale parameter is the estimate of σ in equation (4.2). The estimate
labeled “Shape” is the additional shape parameter that is denoted by δ in
the PROC LIFEREG documentation. (In the output for earlier releases of
PROC LIFEREG, this parameter is labeled “Gamma.”) When the shape
parameter is 0, we get the log-normal distribution. When it is 1.0, we have
the Weibull distribution. In Output 4.5, the shape estimate is almost
exactly 1.0, so we are very close to the Weibull distribution. Later, we’ll
make this comparison more rigorous.

Output 4.5 Generalized Gamma Model Applied to the Recidivism Data

Parameter
Intercept
fin
age
race
wexp
mar
paro
prio
Scale
Shape

DF Estimate
1
1
1
1
1
1
1
1
1
1

3.9915
0.2724
0.0407
-0.2255
0.1073
0.3118
0.0588
-0.0659
0.7151
0.9943

Standard
Error
0.4349
0.1401
0.0165
0.2280
0.1659
0.2769
0.1398
0.0213
0.2396
0.4849

95% Confidence
Limits
3.1391
-0.0022
0.0082
-0.6723
-0.2179
-0.2309
-0.2152
-0.1076
0.3708
0.0439

4.8439
0.5471
0.0731
0.2213
0.4326
0.8545
0.3328
-0.0241
1.3790
1.9446

ChiSquare Pr > ChiSq
84.23
3.78
6.04
0.98
0.42
1.27
0.18
9.56

<.0001
0.0518
0.0140
0.3226
0.5177
0.2602
0.6741
0.0020

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

CATEGORICAL VARIABLES AND THE CLASS STATEMENT
In the recidivism example, several of the covariates—race, marital
status, work experience, and parole status—are dichotomous variables that
are coded as indicator (dummy) variables. For categorical covariates with
more than two categories, the standard approach is to create a set of
indicator variables, one for each category (except for one). You can do this
in the DATA step, but PROC LIFEREG does it automatically if the variable
(or variables) is listed in a CLASS statement. Here’s an example. Another
covariate in the recidivism data set is education, which was originally
coded like this:
2 = 6th grade or less
3 = 7th to 9th grade
4 = 10th to 11th grade
5 = 12th grade
6 = some college

24 cases
239 cases
119 cases
39 cases
11 cases

Due to the small numbers of cases in the two extreme categories, I
combined them (in a DATA step) with the adjacent categories to produce a
variable EDUC with values of 3 (9th or less), 4 (10th to 11th), and 5 (12th
or more). I then specified a Weibull model in PROC LIFEREG with the
following statements:
PROC LIFEREG DATA=recid;
CLASS educ;
MODEL week*arrest(0)=fin age race wexp mar paro
prio educ / D=WEIBULL;
RUN;

Output 4.6 shows the results (some output lines have been omitted).
For any variables listed in the CLASS statement, PROC LIFEREG first
reports the number of levels found and the values for those levels. The
Type III Analysis of Effects table, which really served no purpose in our
previous models, now gives us a very useful overall test for EDUC, which
(with a p-value of .20) is not statistically significant. This test has two
degrees of freedom (df), corresponding to the two coefficients that are
estimated for EDUC. What is particularly attractive about this test is that it
does not depend on which category of EDUC is the omitted category.
In the table of estimates, there are three lines for the EDUC variable.
The first two lines contain coefficients, standard errors, and hypothesis
tests for levels 3 and 4 of EDUC, while the last line merely informs us that
level 5 is the omitted category. Hence, each of the estimated coefficients is
a contrast with level 5. (The default in PROC LIFEREG is to take the

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

highest formatted value as the omitted category, but you can get some
control over this with the ORDER option in the PROC LIFEREG statement.)

Output 4.6 Recidivism Model with Education as a CLASS Variable

Class Level Information
Name

Levels

educ

Values
3 4 5

Type III Analysis of Effects

Effect
fin
age
race
wexp
mar
paro
prio
educ

Wald
Chi-Square

Pr > ChiSq

1
1
1
1
1
1
1
2

3.8069
6.0426
1.2819
0.2581
1.2160
0.2221
7.5308
3.2023

0.0510
0.0140
0.2575
0.6114
0.2701
0.6374
0.0061
0.2017

Analysis of Maximum Likelihood Parameter Estimates

Parameter
Intercept
fin
age
race
wexp
mar
paro
prio
educ
3
educ
4
educ
5
Scale
Weibull Shape

DF Estimate
1
1
1
1
1
1
1
1
1
1
0
1
1

4.4680
0.2690
0.0392
-0.2524
0.0773
0.3013
0.0658
-0.0585
-0.5116
-0.3536
0.0000
0.7119
1.4047

Standard
Error
0.5171
0.1379
0.0159
0.2229
0.1522
0.2732
0.1396
0.0213
0.3090
0.3243
.
0.0634
0.1251

95% Confidence
Limits
3.4544
-0.0012
0.0079
-0.6893
-0.2209
-0.2342
-0.2078
-0.1004
-1.1172
-0.9892
.
0.5979
1.1798

5.4816
0.5392
0.0705
0.1845
0.3755
0.8368
0.3394
-0.0167
0.0941
0.2819
.
0.8476
1.6726

ChiSquare Pr > ChiSq
74.65
3.81
6.04
1.28
0.26
1.22
0.22
7.53
2.74
1.19
.

Unlike most other SAS procedures that have a CLASS statement,
PROC LIFEREG does not have a TEST or CONTRAST statement that
would enable you to test whether EDUC 3 has the same coefficient as

<.0001
0.0510
0.0140
0.2575
0.6114
0.2701
0.6374
0.0061
0.0978
0.2755
.

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

EDUC 4. Methods for constructing such a test by hand calculation are
described in the Hypothesis Tests section below.
In earlier releases of PROC LIFEREG, it was not possible to include
interaction terms directly in the MODEL statement. SAS 9.2 allows
interactions for any set of variables using the familiar notation X1*X2.

MAXIMUM LIKELIHOOD ESTIMATION
All models in PROC LIFEREG are estimated by the method of
maximum likelihood. This section explores some of the basics of ML
estimation, with an emphasis on how it handles censored observations.
The discussion is not intended to be rigorous. If you want a more complete
and careful treatment of ML, you should consult one of the many texts
available on the subject. For example, Kalbfleisch and Prentice (2002) give
a more detailed introduction in the context of survival analysis.
ML is a quite general approach to estimation that has become
popular in many different areas of application. There are two reasons for
this popularity. First, ML produces estimators that have good large-sample
properties. Provided that certain regularity conditions are met, ML
estimators are consistent, asymptotically efficient, and asymptotically
normal. Consistency means that the estimates converge in probability to
the true values as the sample gets larger, implying that the estimates will
be approximately unbiased in large samples. Asymptotically efficient
means that, in large samples, the estimates will have standard errors that
are (approximately) at least as small as those for any other estimation
method. And, finally, asymptotically normal means that the sampling
distribution of the estimates will be approximately normal in large
samples, which implies that you can use the normal and chi-square
distributions to compute confidence intervals and p-values.
All these approximations get better as the sample size gets larger.
The fact that these desirable properties have been proven only for large
samples does not mean that ML has bad properties for small samples. It
simply means that we often don’t know what the small-sample properties
are. And in the absence of attractive alternatives, researchers routinely use
ML estimation for both large and small samples. Although I won’t argue
against that practice, I do urge caution in interpreting p-values and
confidence intervals when samples are small. Despite the temptation to
accept larger p-values as evidence against the null hypothesis in small
samples, it is actually more reasonable to demand smaller values to
compensate for the fact that the approximation to the normal or chi-square
distributions may be poor.

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

The other reason for ML’s popularity is that it is often
straightforward to derive ML estimators when there are no other obvious
possibilities. As we will see, one case that ML handles nicely is data with
censored observations. Although you can use least squares with certain
adjustments for censoring (Lawless, 2002), such estimates often have much
larger standard errors, and there is little available theory to justify the
construction of hypothesis tests or confidence intervals.
The basic principle of ML is to choose as estimates those values that
will maximize the probability of observing what we have, in fact, observed.
There are two steps to this: (1) write down a formula for the probability of
the data as a function of the unknown parameters, and (2) find the values
of the unknown parameters that make the value of this formula as large as
possible.
The first step is known as constructing the likelihood function. To
accomplish this, you must specify a model, which amounts to choosing a
probability distribution for the dependent variable and choosing a
functional form that relates the parameters of this distribution to the values
of the covariates. We have already considered those two choices. The
second step—maximization—typically requires an iterative numerical
method that involves successive approximations.
In the next section, I work through the basic mathematics of
constructing and maximizing the likelihood function. You can skip this
part without loss of continuity if you’re not interested in the details or if
you simply want to postpone the effort. Immediately after this section, I
discuss some of the practical details of ML estimation with PROC
LIFEREG.
Maximum Likelihood Estimation: Mathematics

Assume that we have n independent individuals (i = 1, …, n). For
each individual i, the data consist of three parts: ti, δi, and xi, where ti is
the time of the event or the time of censoring, δi, is an indicator variable
with a value of 1 if ti is uncensored or 0 if right censored, and xi = [1 xi1 …
xik]′ is a vector of covariate values (the 1 is for the intercept). For
simplicity, we treat xi as fixed rather than random. We could get equivalent
results if xi were random and the distributions of δi and ti were expressed
conditional on the values of xi. But that would just complicate the
notation. We also assume that censoring is non-informative.
For the moment, suppose that all the observations are uncensored.
Because we are assuming independence, it follows that the probability of
the entire data is found by taking the product of the probabilities of the
data for every individual. Because ti is assumed to be measured on a
continuum, the probability that it will take on any specific value is 0.

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

Instead, we represent the probability of each observation by the probability
density function (p.d.f.), f (ti). Thus, the probability (or likelihood) of the
data is given by the following expression, where Π indicates repeated
multiplication:
n

L = ∏ f i (ti ) .
i =1

Notice that fi is subscripted to indicate that each individual has a different
p.d.f. that depends on the covariates.
To proceed further, we need to substitute an expression for fi(ti) that
involves the covariates and the unknown parameters. Before we do that,
however, let’s see how this likelihood is altered if we have censored cases.
If an individual is censored at time ti, all we know is that the individual’s
event time is greater than ti. But the probability of an event time greater
than ti is given by the survivor function S(t) evaluated at time ti. Now
suppose that we have r uncensored observations and n – r censored
observations. If we arrange the data so that all the uncensored cases come
first, we can write the likelihood as
r

i =1

i = r +1

L = ∏ f i (ti ) ∏ Si (ti )
where, again, we subscript the survivor function to indicate that it
depends on the covariates. Using the censoring indicator δ, we can
equivalently write this as
n

L = ∏ [ f i (ti )] i [Si (ti )]
δ

1−δ i

i =1

Here δi acts as a switch, turning the appropriate function on or off,
depending on whether the observation is censored. As a result, we do not
need to order the observations by censoring status. This last expression,
which applies to all the models that PROC LIFEREG estimates with rightcensored data, shows how censored and uncensored cases are combined in
ML estimation.
Once we choose a particular model, we can substitute appropriate
expressions for the p.d.f. and the survivor function. Let’s take the simplest
case—the exponential model. We have

f i (ti ) = λi e − λi t i and Si (ti ) = e − λi t i
where λi = exp(−βx i ) and β is a vector of coefficients. Substituting, we get
n

[

L = ∏ λi e −λiti
i =1

] [e ]
δi

−λiti 1−δ i

= ∏ λδi i e −λiti .
i =1

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

Although this expression can be maximized directly, it is generally
easier to work with the natural logarithm of the likelihood function
because products get converted into sums and exponents become
coefficients. Because the logarithm is an increasing function, whatever
maximizes the logarithm also maximizes the original function.
Taking the logarithm of the likelihood, we get
n

log L = ∑ δ i log λi − ∑ λi ti
i =1

i =1

= −β∑ δ i x i − ∑ ti e −βxi
Now we are ready for step 2, finding values of β that make this
expression as large as possible. There are many different methods for
maximizing functions like this. One well-known approach is to find the
derivative of the function with respect to β , set the derivative equal to 0,
and then solve for β . Taking the derivative and setting it equal to 0 gives
us
n

∑ δ i xi = ∑ xiti e −βx .
i

i =1

Because xi is a vector, this is actually a system of k + 1 equations,
one for each element of β . While these equations are not terribly
complicated, the problem is that they involve nonlinear functions of β .
Consequently, except in special cases (like a single dichotomous x
variable), there is no explicit solution. Instead, we have to rely on iterative
methods, which amount to successive approximations to the solution until
the approximations converge to the correct value. Again, there are many
different methods for doing this. All give the same solution, but they differ
in such factors as speed of convergence, sensitivity to starting values, and
computational difficulty at each iteration.
PROC LIFEREG uses the Newton-Raphson algorithm (actually a
ridge stabilized version of the algorithm), which is by far the most popular
numerical method for solving for β . The method is named after Sir Isaac
Newton, who devised it for a single equation and a single unknown. But
who was Raphson? Joseph Raphson was a younger contemporary of
Newton who generalized the algorithm to multiple equations with
multiple unknowns.
The Newton-Raphson algorithm can be described as follows. Let
U(β) be the vector of first derivatives of log L with respect to β and let
I (β) be the matrix of second derivatives of log L with respect to β . That is,

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

U(β) =
I (β) =

∂ log L
= −∑ δ i x i + ∑ ti exp(−βx i )x i
∂β
i =1
i =1
n

n
∂ 2 log L
= −∑ ti exp(−βx i )x i x′i
∂β∂β′
i =1

The vector of first derivatives U(β) is sometimes called the gradient
or score, while the matrix of second derivatives I (β) is called the Hessian.
The Newton-Raphson algorithm is then

β j +1 = β j − I −1 (β j )U(β j )

(4.4)

-1

where I is the inverse of I. In practice, we need a set of starting values
β 0, which PROC LIFEREG calculates by using ordinary least squares,
treating the censored observations as though they were uncensored. These
starting values are substituted into the right side of equation (4.4), which
yields the result for the first iteration, β 1. These values are then substituted
back into the right side, the first and second derivatives are recomputed,
and the result is β 2. This process is repeated until the maximum change in
the parameter estimates from one step to the next is less than .00000001.
(This is an absolute change if the current parameter value is less than .01;
otherwise, it is a relative change.)
Once the solution is found, a convenient by-product of the NewtonRaphson algorithm is an estimate of the covariance matrix of the
coefficients, which is just − I −1 (βˆ j ) . This matrix, which can be printed by
listing COVB as an option in the MODEL statement, is often useful for
constructing hypothesis tests about linear combinations of coefficients.
PROC LIFEREG computes standard errors of the parameters by taking the
square roots of the main diagonal elements of this matrix.
Maximum Likelihood Estimation: Practical Details

PROC LIFEREG chooses parameter estimates that maximize the
logarithm of the likelihood of the data. For the most part, the iterative
methods used to accomplish this task work quite well with no attention
from the data analyst. If you’re curious to see how the iterative process
works, you can request ITPRINT as an option in the MODEL statement.
Then, for each iteration, PROC LIFEREG will print out the log-likelihood
and the parameter estimates. When the iterations are complete, the final
gradient vector and the negative of the Hessian matrix will also be printed
(see the preceding section for definitions of these quantities).
When the exponential model was fitted to the recidivism data, the
ITPRINT output revealed that it took six iterations to reach a solution. The
log-likelihood for the starting values was –531.1, which increased to

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

–327.5 at convergence. Examination of the coefficient estimates showed
only slight changes after the fourth iteration. By comparison, the
generalized gamma model took 13 iterations to converge.
Occasionally the algorithm fails to converge, although this seems to
occur much less frequently than it does with logistic regression. In general,
nonconvergence is more likely to occur when samples are small, when
censoring is heavy, or when many parameters are being estimated. There is
one situation, in particular, that guarantees nonconvergence (at least in
principle). If all the cases at one value of a dichotomous covariate are
censored, the coefficient for that variable becomes larger in magnitude at
each iteration. Here’s why: the coefficient of a dichotomous covariate is a
function of the logarithm of the ratio of the hazards for the two groups. But
if all the cases in a group are censored, the ML estimate for the hazard in
that group is 0. If the 0 is in the denominator of the ratio, then the
coefficient tends toward plus infinity. If it’s in the numerator, taking the
logarithm yields a result that tends toward minus infinity. By extension, if
a covariate has multiple values that are treated as a set of dichotomous
variables (for example, with a CLASS statement) and all cases are censored
for one or more of the values, nonconvergence should result. When this
happens, there is no ideal solution. You can remove the offending variable
from the model, but that variable may actually be one of the strongest
predictors. When the variable has more than two values, you can combine
adjacent values or treat the variable as quantitative.
PROC LIFEREG has two ways of alerting you to convergence
problems. If the number of iterations exceeds the maximum allowed (the
default is 50), SAS issues the message: WARNING: Convergence not
attained in 50 iterations.

WARNING: The procedure is

continuing but the validity of the model fit is
questionable. If it detects a problem before the iteration limit is reached,

the software says WARNING: The negative of the Hessian is not
positive definite. The convergence is questionable.

Unfortunately, PROC LIFEREG sometimes reports estimates and gives no
warning message in situations that are fundamentally nonconvergent. The
only indication of a problem is a coefficient that is large in magnitude
together with a huge standard error.
It’s tempting to try to get convergence by raising the default
maximum number of iterations or by relaxing the convergence criterion.
This rarely works, however, so don’t get your hopes up. You can raise the
maximum with the MAXITER= option in the MODEL statement. You can
alter the convergence criterion with the CONVERGE= option, but I don’t
recommend this unless you know what you’re doing. Too large a value

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

could make it seem that convergence had occurred when there is actually
no ML solution.

HYPOTHESIS TESTS
PROC LIFEREG is somewhat skimpy in its facilities for hypothesis
tests. It automatically reports a chi-square test for the hypothesis that each
coefficient is 0. These are Wald tests that are calculated simply by dividing
each coefficient by its estimated standard error and squaring the result. For
models like the exponential that restrict the scale parameter to 1.0, PROC
LIFEREG reports a Lagrange multiplier chi-square statistic (also known as
a score statistic) for the hypothesis that the parameter is, indeed, equal to
1.0. Finally, as we’ve seen, for categorical variables named in a CLASS
statement, PROC LIFEREG gives a Wald chi-square statistic for the null
hypothesis that all the coefficients associated with the variable are 0.
To test other hypotheses, you have to construct the appropriate
statistic yourself. Before describing how to do this, I’ll first present some
background. For all the regression models considered in this book, there
are three general methods for constructing test statistics: Wald statistics,
score statistics, and likelihood-ratio statistics. Wald statistics are calculated
using certain functions (quadratic forms) of parameter estimates and their
estimated variances and covariances. Score statistics are based on similar
functions of the first and second derivatives of the log-likelihood function.
Finally, likelihood-ratio statistics are calculated by maximizing the
likelihood twice: under the null hypothesis and with the null hypothesis
relaxed. The statistic is then twice the positive difference in the two loglikelihoods.
You can use all three methods to test the same hypotheses, and all
three produce chi-square statistics with the same number of degrees of
freedom. Furthermore, they are asymptotically equivalent, meaning that
their approximate large-sample distributions are identical. Hence,
asymptotic theory gives no basis for preferring one method over another.
There is some evidence that likelihood-ratio statistics may more closely
approximate a chi-square distribution in small- to moderate-sized samples,
however, and some authors (for example, Collett, 2002) express a strong
preference for these statistics. On the other hand, Wald tests and score tests
are often more convenient to calculate because they don’t require reestimation of the model for each hypothesis tested.
Let’s first consider a likelihood-ratio test of the null hypothesis that
all the covariates have coefficients of 0. This is analogous to the usual Ftest that is routinely reported for linear regression models. (Many

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

authorities hold that if this hypothesis is not rejected, then there is no
point in examining individual coefficients for statistical significance.) To
calculate this statistic, we need only to fit a null model that includes no
covariates. For a Weibull model, we can accomplish that with the
following statement:
MODEL week*arrest(0)= / D=WEIBULL;

For the recidivism data, this produces a log-likelihood of
–338.59. By contrast, the Weibull model with seven covariates displayed
in Output 4.3 has a log-likelihood of –321.85. Taking twice the positive
difference between these two values yields a chi-square value of 33.48.
With seven degrees of freedom (the number of covariates excluded from
the null model), the p-value is less than .001. So we reject the null
hypothesis and conclude that at least one of the coefficients is nonzero.
You can also test the same hypothesis with a Wald statistic, but that
involves the following steps:
1. request that the parameter estimates and their covariance matrix
be written to a SAS data set
2. read that data set into PROC IML, the SAS matrix algebra
procedure
3. use PROC IML to perform the necessary matrix calculations.
(These calculations include inverting the appropriate submatrix of the
covariance matrix and premultiplying and postmultiplying that matrix by
a vector containing appropriate linear combinations of the coefficients.)
That’s clearly a much more involved procedure.
Wald statistics for testing the equality of any two coefficients are
simple to calculate. The method is particularly useful for doing post-hoc
comparisons of the coefficients of CLASS variables. Earlier we used a
CLASS statement to include a three-category education variable in the
model. As shown in Output 4.6, there is one chi-square test comparing
category 3 with category 5 and another chi-square test comparing category
4 with category 5. But there is no test reported for comparing category 3
with category 4. The appropriate null hypothesis is that β3 = β4, where the
subscripts refer to the values of categories. A Wald chi-square for testing
this hypothesis can be computed by
( βˆ3 − βˆ4 ) 2
.
Var ( βˆ3 ) + Var ( βˆ4 ) − 2Cov( βˆ3, βˆ4 )

(4.5)

Estimates of the variances and covariances in the denominator are
easily obtained from the covariance matrix that was requested in the
MODEL statement. Output 4.7 shows a portion of the printed matrix.

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

Output 4.7

A Portion of the Covariance Matrix for a Model with a CLASS Variable
Estimated Covariance Matrix

Intercept
fin
age
race
wexp
mar
paro
prio
educ3
educ4

 

educ3

educ4

Scale

-0.094637
-0.000920
0.000137
0.002250
0.002200
0.000551
-0.000617
-0.000593
0.095495
0.086663

-0.090901
-0.001186
0.000285
-0.003371
-0.001100
0.000826
0.000704
-0.000131
0.086663
0.105149

0.003377
0.001338
0.000202
-0.001341
0.000398
0.001593
0.000383
-0.000260
-0.002738
-0.001756

Var bˆ3 is found to be .095495 at the intersection of educ3 with
itself, and similarly Var bˆ4 is .105149 at the intersection of educ4 with
itself. The covariance is .086663 at the intersection of educ3 and educ4.
Combining these numbers with the coefficient estimates in Output 4.7, we
get

 

[.5116 − (−.3536)]2
.09549 + .1051 − 2(.08666)

= .9154 .

With 1 degree of freedom, the chi-square value is far from the .05 critical
value of 3.84. We conclude that there is no difference in the time to arrest
between those with 9th grade or less and those with 10th or 11th grade
education. This should not be surprising because the overall chi-square
test is not significant, nor is the more extreme comparison of category 3
with category 5. Of course, another way to get this same test statistic is
simply to re-run the model with category 4 as the omitted category rather
than category 5 (which can be accomplished by recoding the variable so
that category 4 has the highest value). The chi-square statistic for category
3 will then be equal to .9154.
When performing post-hoc comparisons like this, it is generally
advisable to adjust the alpha level for multiple comparisons. The simplest
approach is the well-known Bonferroni method: For k tests and an overall
Type I error rate of α, each test uses α/k as the criterion value.
We can also test the hypothesis that β3=β4 with the likelihood ratio
statistic. To do that, we must re-estimate the model while imposing the
constraint that β3=β4. We can do this by recoding the education variable so

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

that levels 3 and 4 have the same value. For example, the DATA step can
contain a statement like
IF educ = 3 THEN educ = 4;

When I estimated the model with this recoding, the log-likelihood
was –317.97, compared with –317.50 with the original coding. Twice the
positive difference is .94, which, again, is far from statistically significant.

GOODNESS-OF-FIT TESTS WITH THE LIKELIHOOD-RATIO STATISTIC
As we have seen, the AFT model encompasses a number of
submodels that differ in the assumed distribution for T, the time of the
event. When we tried out those models on the recidivism data, we found
that they produced generally similar coefficient estimates and p-values. A
glaring exception is the log-normal model, which yields qualitatively
different conclusions for some of the covariates. Clearly, we need some
way of deciding between the log-normal and the other models. Even if all
the models agree on the coefficient estimates, they still have markedly
different implications for the shape of the hazard function. Again we may
need methods for deciding which of these shapes is the best description of
the true hazard function.
In the next section, we’ll consider some graphical methods for
comparing models. Here, we examine a simple and often decisive method
based on the likelihood-ratio statistic. In general, likelihood-ratio statistics
can be used to compare nested models. A model is said to be nested within
another model if the first model is a special case of the second. More
precisely, model A is nested within model B if A can be obtained by
imposing restrictions on the parameters in B. For example, the exponential
model is nested within the Weibull model. You get the exponential from
the Weibull by forcing the scale parameter σ equal to 1, and you get the
exponential from the gamma by forcing both the shape and scale
parameters equal to 1.
If model A is nested within model B, we can evaluate the fit of A by
taking twice the positive difference in the log-likelihoods for the two
models. Of course, to evaluate a model in this way, you need to find
another model within which it is nested. As previously noted, the Weibull
and log-normal models (but not the log-logistic) are both nested within the
generalized gamma model, making it a simple matter to evaluate them with
the likelihood-ratio test.

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

Here are the restrictions on the generalized gamma that are implied
by its submodels:
δ=1
σ = 1, δ = 1
δ=0

Weibull
exponential
log-normal

Remember that σ is the scale parameter and δ is the shape parameter.
The likelihood-ratio test for each of these models is, in essence, a test for
the null hypothesis that the particular restriction is true. Hence, these tests
should be viewed not as omnibus tests of the fit of a model but rather as
tests of particular features of a model and its fit to the data.
Let’s calculate the likelihood-ratio tests for the recidivism data. The
log-likelihoods for the models fitted earlier in this chapter are
–325.83
–319.38
–322.69
–319.40
–319.38

exponential
Weibull
log-normal
log-logistic
generalized gamma

Because these log-likelihoods are all negative (which will virtually
always be the case), lower magnitudes correspond to better fits. Taking the
differences between nested models and multiplying by 2 yields the
following likelihood-ratio chi-square statistics:
12.90
12.90
.00
6.62

exponential vs. Weibull
exponential vs. g. gamma
Weibull vs. g. gamma
log-normal vs. g. gamma

With the exception of the exponential model versus the generalized
gamma model (which has 2 d.f.), all these tests have a single degree of
freedom corresponding to the single restriction being tested.
The conclusions are clear. The exponential model must be rejected
(p=.002), implying that the hazard of arrest is not constant over the 1-year
interval. This is consistent with the results we saw earlier for the Lagrange
multiplier test, although that test produced a chi-square value of 24.93,
more than twice as large as the likelihood-ratio statistic. The log-normal
model must also be rejected, although somewhat less decisively (p=.01).
On the other hand, the Weibull model fits the data very well. Apparently,
we can safely disregard the discrepant coefficient estimates for the lognormal model because the model is not consistent with the data.

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

100

Before embracing the Weibull model, however, remember that the
log-logistic model, which has a nonmonotonic hazard function, does not
fit into our nesting scheme. For the recidivism data, its log-likelihood is
only trivially lower than that for the gamma model, suggesting a very good
fit to the data. While this fact should lead us to retain the log-logistic
model as one of our possible candidates, we cannot use it in a formal test
of significance.
In interpreting these likelihood-ratio statistics, you should keep in
mind that the validity of each test rests on the (at least approximate) truth
of the more general model. If that model does not fit the data well, then the
test can be quite misleading. I have seen several examples in which the
test for the exponential model versus the Weibull model is not significant,
but the test for the Weibull model versus the generalized gamma model is
highly significant. Without seeing the second test, you might conclude that
the hazard is constant when, in fact, it is not. But what about the
generalized gamma model itself? How do we know that it provides a
decent fit to the data? Unfortunately, we can’t get a likelihood-ratio test of
the generalized gamma model unless we can fit an even more general
model. And even if we could fit a more general model, how would we
know that model was satisfactory? Obviously you have to stop somewhere.
As noted earlier, the generalized gamma model is a rich family of
distributions, so we expect it to provide a reasonably good fit in the
majority of cases.

GRAPHICAL METHODS

FOR

EVALUATING MODEL FIT

Another way to discriminate between different probability
distributions is to use graphical diagnostics. In Chapter 3, we saw how to
use PROC LIFETEST to get plots of the estimated survivor function, which
can be used to evaluate two of the distributional models considered in this
chapter. Specifically, the PLOTS=LS option produces a plot of
− log Sˆ (t ) versus t. If the true distribution is exponential, this plot should
yield a straight line with an origin at 0. The LLS option produces a plot of
log[− log Sˆ (t )] versus log t, which should be a straight line if the true
distribution is Weibull.
There are two limitations to these methods. First, PROC LIFETEST
cannot produce any graphs suitable for evaluation of the gamma, lognormal or log-logistic. Second, these graphs do not adjust for the effects of
covariates. Both limitations are removed by the PROBPLOT statement in
PROC LIFEREG.

CHAPTER 4

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101

The PROBPLOT statement could hardly be easier to use. After the
MODEL statement, simply write
PROBPLOT;

Although several options are available, they are usually not necessary.
The PROBPLOT statement produces non-parametric estimates of the
survivor function using a modified Kaplan-Meier method that adjusts for
covariates. It then applies a transformation to the survivor estimates that,
when graphed against the log of time, should appear as a straight line if the
specified model is correct.
Output 4.8 displays a graph produced by the PROBPLOT statement
for the exponential model applied to the recidivism data. The upward
sloping straight line represents the survival function predicted by the
model. The shaded bands around that line are the 95% confidence bands.
The circles are the non-parametric survival function estimates. Ideally, all
the non-parametric estimates should lie within the confidence bands.

Output 4.8 Probability Plot for Exponential Model Applied to Recidivism Data

CHAPTER 4

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102

Output 4.9 displays the probability plot for the log-normal model.
Again, we see evidence that the model doesn’t fit the data well, although
it’s not nearly as bad as the exponential model. Finally, Output 4.10 shows
the probability plot for the Weibull model. Here all the non-parametric
estimates fall within the 95% confidence bands.

Output 4.9 Probability Plot for Log-Normal Model Applied to Recidivism Data

The probability plots for these three models are consistent with the
likelihood-ratio tests reported earlier. The Weibull model fits well but the
exponential and log-normal models do not. Such consistency will not
always occur, however. Unlike the likelihood-ratio tests, which compare
one model with another, the probability plots compare the model against
the data and thus allow us to evaluate the overall goodness of fit of the
model.

CHAPTER 4

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103
Output 4.10 Probability Plot for Weibull Model Applied to Recidivism Data

LEFT CENSORING AND INTERVAL CENSORING
One of PROC LIFEREG’s more useful features is its ability to handle
left censoring and interval censoring. Recall that left censoring occurs
when we know that an event occurred earlier than some time t, but we
don’t know exactly when. Interval censoring occurs when the time of
event occurrence is known to be somewhere between times a and b, but
we don’t know exactly when. Left censoring can be seen as a special case
of interval censoring in which a = 0; right censoring is a special case in
which b=∞ .
Interval-censored data are readily incorporated into the likelihood
function. The contribution to the likelihood for an observation censored
between a and b is just Si(a) - Si(b), where Si(.) is the survivor function for
observation i. (This difference is always greater than or equal to 0 because
Si(t) is a nonincreasing function of t.) In other words, the probability of an
event occurring in the interval (a, b) is the probability of an event
occurring after a minus the probability of it occurring after b. For leftcensored data, Si(a) = 1; for right-censored data, Si(b)=0.

CHAPTER 4

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104

PROC LIFEREG can handle any combination of left-censored, rightcensored, and interval-censored data, but a different MODEL statement
syntax is required if there are any left-censored or interval-censored
observations. Instead of a time variable and a censoring variable, PROC
LIFEREG needs two time variables, an upper time and a lower time. Let’s
call them UPPER and LOWER. The MODEL statement then reads as
follows:
MODEL (lower,upper)=list of covariates;

The censoring status is determined by whether the two values are
equal and whether either is coded as missing data:
Uncensored:
Interval Censored:
Right Censored:
Left Censored:

LOWER and UPPER are both present and equal.
LOWER and UPPER are present and different.
LOWER is present, but UPPER is missing.
LOWER is missing, but UPPER is present.

You might think that left censoring could also be indicated by coding
LOWER as 0, but PROC LIFEREG excludes any observations with times
that are 0 or negative. Observations are also excluded if both UPPER and
LOWER are missing or if LOWER > UPPER. Here are some examples:
Observation
1
2
3
4
5
6

Lower
3.9
7.2
4.1
.
0
3.2

Upper
3.9
.
5.6
2.0
5.8
1.9

Status
Uncensored
Right Censored
Interval Censored
Left Censored
Excluded
Excluded

Let’s look at an example of left censoring for the recidivism data.
Suppose that of the 114 arrests, the week of arrest was unknown for 30
cases. In other words, we know that an arrest occurred between 0 and 52
weeks, but we don’t know when. To illustrate this, I modified the
recidivism data by recoding the WEEK variable as missing for the first 30
arrests in the data set. This was accomplished with the following program:
PROC SORT DATA=recid OUT=recid2;
BY DESCENDING arrest;
DATA recidlft;
SET recid2;
IF _N_ LE 30 THEN week = .;
RUN;

CHAPTER 4

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105

The following program further modifies the data set to create the
UPPER and LOWER variables needed for PROC LIFEREG:
DATA recid3;
SET recidlft;
/* uncensored cases: */
IF arrest=1 AND week ne . THEN DO;
upper=week;
lower=week;
END;
/* left-censored cases: */
IF arrest=1 AND week = . THEN DO;
upper=52;
lower=.;
END;
/* right-censored cases: */
IF arrest=0 THEN DO;
upper=.;
lower=52;
END;
RUN;

The code for estimating a Weibull model is then
PROC LIFEREG DATA=recid3;
MODEL (lower,upper)=fin age race wexp mar paro prio
/ D=WEIBULL;
RUN;

Results in Output 4.11 should be compared with those in Output 4.3,
for which there were no left-censored cases. Although the results are quite
similar, the chi-square statistics are nearly all smaller when some of the
data are left censored. This is to be expected because left censoring entails
some loss of information. Note that you cannot compare the log-likelihood
for this model with the log-likelihood for the model with no left censoring.
Whenever you alter the data, the log-likelihoods are no longer comparable.

CHAPTER 4

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106

Output 4.11 Results for the Weibull Model with Left-Censored Data

Model Information
Data Set
Dependent Variable
Dependent Variable
Number of Observations
Noncensored Values
Right Censored Values
Left Censored Values
Interval Censored Values
Name of Distribution
Log Likelihood

WORK.DATA1
Log(lower)
Log(upper)
432
84
318
30
0
Weibull
-294.0611169

Analysis of Maximum Likelihood Parameter Estimates

Parameter
Intercept
fin
age
race
wexp
mar
paro
prio
Scale
Weibull Shape

DF Estimate
1
1
1
1
1
1
1
1
1
1

3.9567
0.2490
0.0413
-0.2191
0.0777
0.3001
0.0682
-0.0623
0.6911
1.4469

Standard
Error
0.4078
0.1348
0.0158
0.2145
0.1484
0.2658
0.1356
0.0207
0.0717
0.1501

95% Confidence
Limits
3.1574
-0.0152
0.0102
-0.6395
-0.2132
-0.2208
-0.1975
-0.1030
0.5639
1.1806

4.7560
0.5133
0.0723
0.2013
0.3686
0.8211
0.3339
-0.0217
0.8470
1.7732

ChiSquare Pr > ChiSq
94.14
3.41
6.79
1.04
0.27
1.27
0.25
9.03

<.0001
0.0647
0.0092
0.3070
0.6006
0.2588
0.6151
0.0027

The recidivism data can also be used to illustrate interval censoring.
Because we know only the week of the arrest and not the exact day, we can
actually view each arrest time as an interval-censored observation.
Although Petersen (1991) has shown that some bias can result from
treating discrete data as continuous, there’s probably little danger of bias
with 52 different values for the measurement of arrest time. Nonetheless,
we can use the interval-censoring option to get a slightly improved
estimate. For an arrest that occurs in week 2, the actual interval in which
the arrest occurred is (1, 2). Similarly, the interval is (2, 3) for an arrest
occurring in week 3. This suggests the following recoding of the data:

CHAPTER 4

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107

DATA recidint;
SET recid;
/* interval-censored cases: */
IF arrest=1 THEN DO;
upper=week;
lower=week-.9999;
END;
/* right-censored cases: */
IF arrest=0 THEN DO;
upper=.;
lower=52;
END;
RUN;
PROC LIFEREG DATA=recidint;
MODEL (lower, upper) = fin age race wexp mar paro prio
/ D=WEIBULL;
RUN;

To get the lower value for the interval-censored cases, I subtracted
.9999 instead of 1 so that the result is not 0 for those persons with
WEEK=1 (which would cause PROC LIFEREG to exclude the observation).
The results in Output 4.12 are very close to those in Output 4.3,
which assumed that time was measured exactly. (Some output lines are
deleted.) If the intervals had been larger, we might have found more
substantial differences. The magnitude of the log-likelihood is nearly
doubled for the interval-censored version but, again, log-likelihoods are
not comparable when the data are altered.

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

108
Output 4.12 Results Treating Recidivism Data as Interval Censored

WORK.RECIDINT
Log(lower)
Log(upper)
432
0
318
0
114
Weibull
-680.995873

Analysis of Maximum Likelihood Parameter Estimates

Parameter
Intercept
fin
age
race
wexp
mar
paro
prio
Scale
Weibull Shape

DF Estimate
1
1
1
1
1
1
1
1
1
1

3.9906
0.2837
0.0425
-0.2343
0.1106
0.3246
0.0618
-0.0685
0.7435
1.3449

Standard
Error
0.4374
0.1440
0.0167
0.2298
0.1582
0.2853
0.1457
0.0218
0.0665
0.1204

95% Confidence
Limits
3.1333
0.0015
0.0098
-0.6846
-0.1994
-0.2345
-0.2238
-0.1113
0.6239
1.1285

4.8479
0.5659
0.0753
0.2161
0.4206
0.8837
0.3474
-0.0257
0.8861
1.6028

ChiSquare Pr > ChiSq
83.23
3.88
6.48
1.04
0.49
1.29
0.18
9.83

<.0001
0.0488
0.0109
0.3079
0.4845
0.2552
0.6714
0.0017

GENERATING PREDICTIONS AND HAZARD FUNCTIONS
After fitting a model with PROC LIFEREG, it’s sometimes desirable
to generate predicted survival times for the observations in the data set. If
you want a single point estimate for each individual, the predicted median
survival time is probably the best. You can get this easily with the
OUTPUT statement, as shown in the following example:
PROC LIFEREG DATA=recid;
MODEL week*arrest(0)=fin age race wexp mar paro prio
/ D=WEIBULL;
OUTPUT OUT=a P=median STD=s;
RUN;

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

109

PROC PRINT DATA=a;
VAR week arrest _prob_ median s;
RUN;

The P= option in the OUTPUT statement requests percentiles. By
default, PROC LIFEREG calculates the 50th percentile (that is, the median).
(The word median in the OUTPUT statement is just the variable name I
chose to hold the quantiles.) You can request other percentiles with the
QUANTILE keyword, as described in the PROC LIFEREG documentation.
The STD keyword requests the standard errors of the medians.
Output 4.13 shows the first 20 cases in the new data set. In the output,
_PROB_ is the quantile (that is, the percentile divided by 100) requested, and S
is the standard error of the median. Note that many of these predicted medians
are much greater than the observed event (or censoring times) and often well
beyond the observation limit of 52 weeks. This is not surprising given that
nearly 75 percent of the observations are censored. Although we observed
individuals for only 52 weeks, the fitted distribution is not at all limited by the
censoring times. Therefore, you should be very cautious in interpreting these
predicted medians because the model is being extrapolated to times that are far
beyond those that are actually observed.

Output 4.13

Predicted Median Survival Times for Recidivism Data (First 20 Cases)
Obs

week

arrest

_PROB_

median

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

20
17
25
52
52
52
23
52
52
52
52
52
37
52
25
46
28
52
52
52

1
1
1
0
0
0
1
0
0
0
0
0
1
0
1
1
1
0
0
0

0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5

86.910
43.353
45.257
167.947
87.403
86.166
148.225
93.083
54.873
83.518
109.544
165.290
56.411
146.285
65.358
110.592
45.471
65.816
44.305
122.208

14.532
6.646
14.654
50.293
21.731
13.989
42.416
15.532
7.694
15.483
18.986
50.312
10.132
38.768
8.360
18.115
6.814
10.306
9.429
30.552

CHAPTER 4

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110

You can also get predicted values for sets of covariate values that are
not in the original data set. Before estimating the model, simply append to
the data set additional observations with the desired covariate values and
with the event time set to missing. These added observations are not used
in estimating the model, but predicted values will be generated for them.
Instead of predicted survival times, researchers often want to predict
the probability of surviving to some specified time (for example, 5-year
survival probabilities). While these are not directly computed by PROC
LIFEREG, it’s fairly straightforward to calculate them. This is
accomplished by substituting linear predictor values that can be produced
by the OUTPUT statement into formulas for the survivor function (given in
the PROC LIFEREG documentation). To make it easy, I’ve written a macro
called PREDICT, which is described in detail in Appendix 1, “Macro
Programs.” This macro is used in the following way. When specifying the
model in PROC LIFEREG, you must request that the parameter estimates
be written to a data set using the OUTEST= option. Next, use the XBETA=
option in the OUTPUT statement to request that the linear predictor be
included in a second data set. Finally, call the macro, indicating the names
of the two data sets, the name assigned to the linear predictor, and the time
for calculating the survival probabilities. For example, to produce 30-week
survival probabilities for the recidivism data, submit these statements:
PROC LIFEREG DATA=recid OUTEST=a;
MODEL week*arrest(0) = fin age race wexp mar paro prio
/ D=WEIBULL;
OUTPUT OUT=b XBETA=lp;
RUN;
%PREDICT(OUTEST=a,OUT=b,XBETA=lp,TIME=30)

Output 4.14 shows the first 20 cases of the new data set (_PRED_).
The last column (prob) contains the 30-week survival probabilities based
on the fitted model.

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111

Output 4.14

Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

Predicted 30-Week Survival Probabilities for Recidivism Data

week

fin

20
17
25
52
52
52
23
52
52
52
52
52
37
52
25
46
28
52
52
52

0
0
0
1
0
0
0
1
0
0
1
0
0
0
0
1
0
0
0
0

age
27
18
19
23
19
24
25
21
22
20
26
40
17
37
20
22
19
20
25
24

race
1
1
0
1
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
0

wexp
0
0
1
1
1
1
1
1
0
1
0
1
1
1
0
1
0
0
0
1

mar

paro

prio

educ

arrest

prob

0
0
0
1
0
0
1
0
0
0
0
0
0
0
0
0
0
0
0
0

1
1
1
1
1
0
1
1
0
0
1
0
1
0
1
1
0
0
1
1

3
8
13
1
3
2
0
4
6
0
3
2
5
2
3
2
7
2
12
1

3
4
3
5
3
4
4
3
3
5
3
5
3
3
4
3
3
3
3
3

1
1
1
0
0
0
1
0
0
0
0
0
1
0
1
1
1
0
0
0

30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30
30

0.85579
0.66139
0.67760
0.94010
0.85684
0.85417
0.92903
0.86811
0.74306
0.84816
0.89356
0.93878
0.75151
0.92776
0.79267
0.89490
0.67934
0.79448
0.66965
0.90800

Because every model estimated in PROC LIFEREG has an implicit
hazard function, it would be nice to see what that hazard function looks
like. I’ve written another macro called LIFEHAZ (also described in
Appendix 1) that produces a graph of the hazard as a function of time. As
with the PREDICT macro, you first need to fit a PROC LIFEREG model that
includes the OUTPUT statement and the OUTEST= option in the PROC
statement. Using the same PROC LIFEREG specification that was used
before with the PREDICT macro, you then submit the following:
%LIFEHAZ(OUTEST=a,OUT=b,XBETA=lp)

For the recidivism data using the Weibull model, this macro
produces the graph shown in Output 4.15. Keep in mind that this graph
depends heavily on the specified model. If we specify a log-normal model
before using the LIFEHAZ macro, the graph will look quite different.

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

112
Output 4.15 Graph of Hazard Function for Recidivism Data

0.008

0.007

0.006

0.005

0.004

0.003

0.002

0.001

0.000
0

THE PIECEWISE EXPONENTIAL MODEL
All the AFT models we have considered so far assume that the
hazard is a smooth, relatively simple function of time. The Cox model
(estimated with the PHREG procedure) is much less restrictive in this
regard, but it lacks the facility to test hypotheses about the shape of the
hazard function. One way to get some of the flexibility of the Cox model
without losing the hypothesis testing capability is to employ the piecewise
exponential model, a method that is widely used in several fields. We can
easily estimate it with PROC LIFEREG, although it requires some
preliminary restructuring of the data. A bonus of this method is the ability
to incorporate time-dependent covariates.
The basic idea is simple. Divide the time scale into intervals.
Assume that the hazard is constant within each interval but can vary
across intervals. In symbols, we define a set of J intervals, with cut points
a0, a1, … , aJ, where a0 = 0, and aJ = ∞. Thus, interval j is given by [aj-1, aj).
The hazard for individual i is assumed to have the form
hi (t ) = λ j eβxi

for aj-1 ≤ t < aj

or equivalently
log hi (t ) = α j + βx i

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

where αj = log λj. Thus, the intercept in the log-hazard equation is allowed
to vary in an unrestricted fashion from one interval to another. The choice
of intervals is arbitrary, however, leading to some arbitrariness in the
estimates.
The procedure for estimating this model is best explained by way of
example. For the recidivism data, let’s break up the 52-week observation
period into four quarters of 13 weeks each and assume that the hazard is
constant within each quarter. We then create a new data set with possibly
multiple records for each person. One record is created for each quarter
during which an individual was at risk of the first arrest. Four records are
created for persons who were arrested in the fourth quarter or who were
not arrested at all. Three records are created for those arrested in the third
quarter, two records are created for those arrested in the second quarter,
and one record is created for those arrested in the first quarter. This yields
a total of 1,573 person-quarters for the 432 ex-convicts.
Each record is treated as a distinct observation, with the time reset to
0 at the beginning of the quarter. If an arrest occurred in the quarter, a
censoring variable for that person-quarter is set to 1; otherwise, it is set to
0. If no arrest occurred in the quarter, a time variable is assigned the full 13
weeks. If an arrest occurred, the time variable is coded as the length of
time from the start of the quarter until the arrest.
For the recidivism data, all censoring is at the end of the fourth
quarter. Had there been any censoring within quarters, the time variable
would be coded as the length of time from the beginning of the quarter
until censoring occurred. The fixed covariates are simply replicated for
each quarter. If there were any time-dependent covariates, their values at
the beginning of each quarter could be assigned to the records for that
quarter.
Here’s a DATA step for creating such a data set:
DATA quarter;
SET recid;
quarter=CEIL(week/13);
DO j=1 TO quarter;
time=13;
event=0;
IF j=quarter AND arrest=1 THEN DO;
event=1;
time=week-13*(quarter-1);
END;
OUTPUT;
END;
RUN;

113

CHAPTER 4

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114

The CEIL function, which produces the smallest integer greater than its
argument, yields values of 1, 2, 3, or 4, corresponding to the quarter. The
DO loop produces a record for each quarter at risk. The TIME and EVENT
variables are initialized at the values appropriate for quarters in which
arrests did not occur. The IF statement checks to see if an arrest occurred
in the quarter. If an arrest did occur, then EVENT and TIME are
appropriately recoded. Finally, a new record is output containing all the
original variables plus the newly created ones.
We then call the LIFEREG procedure and specify an exponential
model:
PROC LIFEREG DATA=quarter;
CLASS j;
MODEL time*event(0)=fin age race wexp mar paro prio j
/ D=EXPONENTIAL COVB;
RUN;

The variable J—the index variable in the DO loop of the DATA step—has
values of 1, 2, 3, or 4, corresponding to the quarter covered by each record.
It is specified as a CLASS variable so that PROC LIFEREG will set up an
appropriate set of indicator variables to estimate the αj’s in the piecewise
exponential model (actually we estimate contrasts between the αj’s).
Results in Output 4.16 show a significant effect of J (quarter),
implying that the hazard is not constant over time. The Wald chi-square
value is 8.70 on 3 d.f., which is corroborated by a likelihood-ratio test with
a chi-square value of 9.52. (The likelihood-ratio test is calculated by
rerunning the model without J and taking twice the positive difference in
the log-likelihoods.) The coefficients for the three indicator variables are
all contrasts with the fourth quarter. To interpret these coefficients, it’s
probably best to change their signs so that they reflect hazards rather than
survival times. In contrast to the Weibull model, which imposed a
monotonically increasing hazard, the pattern displayed here is not
monotonic. The estimated hazard increases from first to second quarter,
then decreases, and then increases again.

CHAPTER 4

Estimating Parametric Regression Models with PROC LIFEREG

115

Output 4.16

Results for Piecewise Exponential Model Applied to Recidivism Data

Type III Analysis of Effects

Effect
j

Wald
Chi-Square

Pr > ChiSq

8.6954

0.0336

Analysis of Maximum Likelihood Parameter Estimates

Parameter
Intercept
fin
age
race
wexp
mar
paro
prio
j
j
j
j
Scale

DF Estimate

1
2
3
4

1
1
1
1
1
1
1
1
1
1
1
0
0

3.7226
0.3774
0.0570
-0.3126
0.1489
0.4331
0.0836
-0.0909
0.8202
0.1883
0.3134
0.0000
1.0000

Standard
Error
0.6082
0.1913
0.0220
0.3080
0.2122
0.3818
0.1957
0.0286
0.2841
0.2446
0.2596
.
0.0000

95% Confidence
Limits
2.5307
0.0023
0.0140
-0.9162
-0.2670
-0.3152
-0.3000
-0.1470
0.2633
-0.2911
-0.1953
.
1.0000

4.9146
0.7524
0.1000
0.2910
0.5648
1.1814
0.4672
-0.0347
1.3771
0.6677
0.8221
.
1.0000

ChiSquare Pr > ChiSq
37.47
3.89
6.73
1.03
0.49
1.29
0.18
10.07
8.33
0.59
1.46
.

<.0001
0.0486
0.0095
0.3101
0.4828
0.2566
0.6692
0.0015
0.0039
0.4414
0.2273
.

The chi-square tests for the individual indicator variables show that
the hazard of arrest in the first quarter is significantly lower than the
hazard in the last quarter. Although the two middle quarters have lower
estimated hazards than the last, the differences are not significant. A Wald
test (constructed from the covariance matrix produced by the COVB
option) comparing the first and second quarters is also significant at about
the .03 level. The coefficients and p-values for the remaining variables are
consistent with those found with the conventional exponential and
Weibull models.
Of course, there is a certain arbitrariness that arises from the division
of the observation period into quarters. To increase confidence in the
results, you may want to try different divisions and see if the results are
stable. I re-estimated the model for the recidivism data with a division into
13 “months” of four weeks each, simply by changing all the 13’s to 4’s in

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the DATA step. This produces a data set with 4,991 records. Results for the
fixed covariates are virtually identical. The Wald chi-square test for the
overall effect of J (month) is also about the same, but, with 12 degrees of
freedom, the p-value is well above conventional levels for statistical
significance.
Here are some final observations about the piecewise exponential
model:
 You do not need to be concerned about the fact that the working
data set has multiple records for each individual. In particular,
there is no inflation of test statistics resulting from lack of
independence. The fact that the results are so similar regardless
of how many observations are created should reassure you on
this issue. The reason it’s not a problem is that the likelihood
function actually factors into a distinct term for each individual
interval. This conclusion does not apply, however, when the
data set includes multiple events for each individual.
 The piecewise model can be easily fit in a Bayesian framework
using PROC PHREG. This option will be discussed in the next
chapter.
 The piecewise exponential model is very similar to the discretetime methods described in Chapter 7, “Analysis of Tied or
Discrete Data with PROC LOGISTIC,” The principal difference is
that estimation of the piecewise exponential model uses
information on the exact timing of events, while the discrete-time
methods are based on interval-censored data.
 There is no requirement that the intervals have equal length,
although that simplifies the DATA step somewhat. Because
there’s some benefit in having roughly equal numbers of events
occurring in each interval, this sometimes requires unequal
interval lengths.
 The use of time-dependent covariates in the piecewise
exponential model can substantially complicate the DATA step
that creates the multiple records. For examples of how to do this,
see Chapter 7 on discrete-time methods. There are also a number
of issues about design and interpretation of studies with timedependent covariates that are discussed in detail in Chapters 5
and 7.

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BAYESIAN ESTIMATION AND TESTING
Beginning with SAS 9.2, PROC LIFEREG can do a Bayesian analysis
for any of the models discussed in this chapter. In a Bayesian analysis, the
parameters of the model are treated as random variables rather than as
fixed characteristics of the population. A prior distribution must be
specified for these parameters. This is a probability distribution that
incorporates any previous knowledge or beliefs about the parameters. The
goal of the analysis is to produce the posterior distribution of the
parameters. The posterior distribution is obtained by using Bayes’ theorem
to combine the likelihood function of the data with the prior distribution.
Once you have the posterior distribution for a particular parameter, the
mean or median of that distribution can be used as a point estimate of the
parameter, and appropriate percentiles of the distribution can be used as
interval estimates. For a useful primer on Bayesian analysis in SAS, see
the SAS/STAT 9.2 User’s Guide, “Introduction to Bayesian Analysis
Procedures.”
A Bayesian approach has several attractions that are relevant here:
 Inferences are exact, in the sense that they do not rely on largesample approximations, and thus may be more accurate for small
samples.
 Information from prior research studies can be readily
incorporated into the analysis.
 Comparison of non-nested models can be accomplished in a
systematic framework.
These advantages come with a major computational cost. The Markov
Chain Monte Carlo (MCMC) methods used by PROC LIFEREG to simulate
the posterior distribution are very intensive. For example, it took only
about a tenth of a second on my laptop to estimate the log-normal model
reported in Output 4.1. By contrast, a Bayesian analysis (with the default
non-informative prior distributions) took more than 5 minutes.
The other potential disadvantage of a Bayesian approach is the
difficulty in coming up with suitable prior distributions for the parameters
of the model. The default is to use non-informative priors that embody
little or no information about the parameters. PROC LIFEREG uses uniform
(constant) priors for the coefficients and a “just informative” gamma prior
for the scale parameter. But if you use these defaults, results typically don’t
differ much from conventional maximum likelihood estimation. In fact, for
the exponential model (which has the scale parameter fixed at 1), the
maximum likelihood estimator for the coefficients is simply the mode of

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the posterior distribution. Although you have the option of choosing
informative priors, this is not such an easy or straightforward task.
As with most software for Bayesian analysis, PROC LIFEREG uses
the MCMC algorithm (more specifically, the Gibbs sampler) to simulate a
large number of sequential random draws from the posterior distribution—
by default, 12,000 draws. The first 2,000 (the “burn-in” iterations) are
discarded in order to ensure that that the algorithm has converged to the
correct posterior distribution. Even then, however, there is no guarantee
that convergence has been attained. For that reason, PROC LIFEREG
provides diagnostic plots and several test statistics to evaluate the
convergence of the distribution.
If you are OK with the defaults, requesting a Bayesian analysis is
very simple. Just include the BAYES statement. Here is how to do it for the
Weibull model that we estimated earlier (Output 4.3), using the recidivism
data:
ODS HTML;
ODS GRAPHICS ON;
PROC LIFEREG DATA=recid;
MODEL week*arrest(0)=fin age race wexp mar paro prio /D=WEIBULL;
BAYES;
RUN;
ODS GRAPHICS OFF;
ODS HTML CLOSE;

ODS GRAPHICS is used here so that PROC LIFEREG will produce a
plot of the posterior distribution for each parameter, along with two
diagnostic plots, a trace plot and an autocorrelation plot. Selected portions
of the tabular results are shown in Output 4.17. Plots for one of the
parameters (the coefficient of AGE) are shown in Output 4.18.
The first two tables in Output 4.17 describe the prior distributions
for all the parameters. By default, the regression coefficients all have the
“Constant” or uniform prior, and they are assumed to be independent of
each other and the scale parameter. The scale parameter is assigned a
gamma distribution with “hyperparameters” of .001 for both the scale and
inverse-shape parameters. This is equivalent to using a gamma distribution
with a mean of 1 and a very large standard deviation (approximately 32).
The “Initial Values of the Chain” table displays the starting values
for the parameters. These are “posterior mode” estimates, which means
that they are obtained by maximizing the likelihood function weighted by
the prior distribution. The “Fit Statistics” table shows three fit measures
that were described earlier in this chapter: AIC, AICC, and BICC. In fact,

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the values shown here are exactly the same as those reported for
conventional maximum likelihood estimation.
What’s new are the DIC and pD statistics. DIC stands for deviance
information criterion, a widely used measure of fit for Bayesian analysis.
It’s essentially a Bayesian analog to the AIC statistic, calculated as

DIC = −2 log L(θ ) + 2 pD
where L is the likelihood function, θ is the mean of the posterior
distribution of the parameter vector θ, and pD is the “effective number of
parameters.” It is calculated as

pD = Mean[−2 log L(θ )] + 2 log L(θ )
where the mean is taken over the posterior distribution of θ. As with the
AIC, the DIC is used primarily to compare models with different numbers
of parameters.
The next table, “Posterior Summaries,” is the heart of the analysis.
For each parameter, we get the mean of the posterior distribution, based on
the sample of 10,000 random draws from that distribution. Keep in mind
that all the numbers in this table are simulation-based estimates, and
therefore will be slightly different if we run the model again. But with
10,000 cases, the sampling error should be small. The next column is the
standard deviation, which plays the same role as the standard error in
conventional frequentist methodology. Finally, we get the 25th, 50th, and
75th percentiles of the posterior distribution. The 50th percentile is, of
course, the median, and some may prefer this to the mean as a point
estimate for the parameter of interest.
The “Posterior Intervals” table displays what are usually called
credible intervals in Bayesian analysis, and by default, they are 95%
intervals. Credible intervals play the same role as confidence intervals in
conventional statistics, although the interpretation is somewhat different.
A 95% confidence interval is interpreted by saying that if we repeated the
sampling and estimation process many times, 95% of the constructed
confidence intervals would include the true parameter. A 95% credible
interval says simply that there is a .95 probability that the true parameter is
included in the reported interval.
The “Equal Tail Intervals” are merely the 2.5th percentiles and the
97.5th percentiles of the posterior distribution. Alternatively, one can use
the “highest posterior density” or HPD intervals. These are the narrowest
possible intervals that contain, by default, 95% of the posterior
distribution. Although p-values are not reported for the estimates, a 95%
credible interval that does not include 0 is the Bayesian equivalent of an

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estimate that is significantly different from 0 at the .05 level (by a twosided test).
The two remaining tables are used as diagnostics for the performance
of the MCMC algorithm. In the sequence of random draws from the
posterior distribution, each new draw depends, in part, on the value of the
preceding draw. That induces an autocorrelation in the drawn values that,
ideally, should decline to 0 as the distance between draws gets larger.
Persistently high values of the autocorrelation suggest that the algorithm is
not mixing well. These autocorrelations can vary substantially from one
parameter to another. For example, the autocorrelation for adjacent values
(lag 1) of the “fin” coefficient is only .04, while for “age” it is .79. The
“age” autocorrelation declines to .29 for values that are five draws apart
(lag 5) and then to .08 for lag 10. A graph of the autocorrelations for the
“age” coefficient is shown in Output 4.18.
Although autocorrelation should not cause any problem for
estimates of the means, medians, and credible intervals, it may induce
some bias in estimates of the standard deviations of the posterior
distribution (Daniels and Hogan, 2008). This can be fixed by thinning the
sample (for example, retaining only every 10th sample instead of every
sample). To do this in PROC LIFEREG, you simply use THIN=10 as an
option in the BAYES statement. Of course, the disadvantage of this is that
you would need to generate 10 times as many samples to get the same
degree of accuracy. And in my (limited) experience, using this option
typically has very little impact on the standard deviations. If you do
choose to thin, the autocorrelation graph should provide useful guidance
on what level of thinning is needed.
The last table displays the “Geweke Diagnostics” for convergence of
MCMC. The idea behind these statistics is very simple. After discarding
the 2,000 burn-in iterations, we divide the sample into equal subsamples
of early and late iterations. For the two subsamples, we calculate the
means for each parameter. If the distribution has converged, these means
should be equal, apart from sampling error. The Geweke statistic is just a
standard z-statistic for differences between the two means. For the
recidivism data, none of the differences approaches statistical significance.
PROC LIFEREG also offers several other optional tests of convergence.

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Output 4.17

Results for Bayesian Analysis of the Weibull Model for Recidivism
Bayesian Analysis
Uniform Prior for Regression Coefficients
Parameter

Prior

Intercept
fin
age
race
wexp
mar
paro
prio

Constant
Constant
Constant
Constant
Constant
Constant
Constant
Constant

Independent Prior Distributions for Model Parameters

Parameter

Prior
Distribution

Scale

Gamma

Hyperparameters
Shape

0.001

Inverse Scale

0.001

Initial Values of the Chain
Chain

Seed

Intercept

fin

age

race

wexp

681742001

3.989018

0.27029

0.040422

-0.22319

0.105745

Initial Values of the Chain
mar

paro

prio

Scale

0.308981

0.058355

-0.06541

0.706806

Fit Statistics
AIC (smaller is better)
AICC (smaller is better)
BIC (smaller is better)
DIC (smaller is better)
pD (effective number of parameters)

656.753
657.180
693.369
657.110
8.753

(continued)

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Output 4.17 (continued)

Posterior Summaries

Parameter

Mean

Standard
Deviation

25%

Intercept
fin
age
race
wexp
mar
paro
prio
Scale

10000
10000
10000
10000
10000
10000
10000
10000
10000

4.0183
0.2927
0.0444
-0.2661
0.1171
0.3767
0.0628
-0.0675
0.7663

0.4505
0.1502
0.0171
0.2408
0.1648
0.3060
0.1523
0.0228
0.0708

3.7214
0.1905
0.0325
-0.4222
0.00599
0.1647
-0.0384
-0.0823
0.7166

Percentiles
50%
4.0233
0.2909
0.0438
-0.2565
0.1161
0.3613
0.0624
-0.0672
0.7619

75%
4.3152
0.3902
0.0554
-0.1008
0.2246
0.5749
0.1625
-0.0527
0.8109

Posterior Intervals
Parameter

Alpha

Equal-Tail Interval

Intercept
fin
age
race
wexp
mar
paro
prio
Scale

0.050
0.050
0.050
0.050
0.050
0.050
0.050
0.050
0.050

3.1265
0.00557
0.0130
-0.7700
-0.2005
-0.1900
-0.2351
-0.1137
0.6383

4.8962
0.5971
0.0802
0.1813
0.4436
0.9996
0.3628
-0.0225
0.9201

HPD Interval
3.1210
0.000220
0.0126
-0.7401
-0.1940
-0.1935
-0.2255
-0.1139
0.6295

4.8881
0.5888
0.0796
0.2044
0.4487
0.9931
0.3686
-0.0229
0.9067

Posterior Autocorrelations
Parameter

Lag 1

Lag 5

Lag 10

Lag 50

Intercept
fin
age
race
wexp
mar
paro
prio
Scale

0.5590
0.0437
0.7862
0.6416
0.2374
0.7769
0.0808
0.7405
0.6061

0.0933
0.0064
0.2927
0.1024
0.0563
0.2421
0.0139
0.1581
0.1295

-0.0033
0.0134
0.0821
0.0067
0.0047
0.0207
0.0183
0.0114
0.0677

-0.0137
-0.0067
-0.0099
0.0076
-0.0096
-0.0073
-0.0040
-0.0093
-0.0145

(continued)

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Output 4.17 (continued)

Geweke Diagnostics
Parameter

Pr > |z|

Intercept
fin
age
race
wexp
mar
paro
prio
Scale

0.8086
1.4737
-0.0764
-0.4365
-0.6223
0.7653
-0.0576
-1.0156
1.3244

0.4188
0.1406
0.9391
0.6625
0.5337
0.4441
0.9541
0.3098
0.1854

Output 4.18 displays the plots that are produced by default when the
BAYES statement is used. Only the plots for the AGE coefficient are
shown. We have already considered the autocorrelation plot. The top plot
is a trace plot of the sampled values of the AGE coefficient across the
sequence of iterations, from 2,000 to 12,000. The first 2,000 values are not
shown because they are considered burn-in iterations and are not used in
calculating any of the estimates. Ideally, the trace plot should look like
white noise (that is, it should have no pattern whatsoever). Certainly, we
don’t want to see any upward or downward trends, nor any evidence that
the variability is increasing or decreasing. The plot shown here is
consistent with the relatively high levels of autocorrelation that persist up
to about lag 10.
The last plot is a graph of the empirical density function for the
posterior distribution. These graphs typically look similar to normal
distributions, especially if the sample size is large. In this example, the
bulk of the distribution is clearly well above 0, consistent with the 95%
credible intervals in Output 4.17 that do not include zero.

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Output 4.18

Graphs for Bayesian Analysis of Recidivism Data

CONCLUSION
PROC LIFEREG provides effective methods for regression analysis of
censored survival data, especially data with left censoring or interval
censoring. These methods are somewhat less robust than the more widely
used Cox regression analysis performed by PROC PHREG but, in most
cases, the results produced by the two approaches are very similar.
Moreover, unlike PROC PHREG, PROC LIFEREG makes it possible to test
certain hypotheses about the shape of the hazard function.
The biggest limitation of the PROC LIFEREG models is the inability
to incorporate time-dependent covariates, although you can accomplish
this to some degree with the piecewise exponential model. We now turn to
PROC PHREG, which excels at this particular task.

CHAPTER 5
Estimating Cox Regression Models with PROC PHREG
Introduction 125
The Proportional Hazards Model 126
Partial Likelihood 128
Tied Data 142
Time-Dependent Covariates 153
Cox Models with Nonproportional Hazards 172
Interactions with Time as Time-Dependent Covariates 177
Nonproportionality via Stratification 179
Left Truncation and Late Entry into the Risk Set 183
Estimating Survivor Functions 186
Testing Linear Hypotheses with CONTRAST or TEST Statements 192
Customized Hazard Ratios 195
Bayesian Estimation and Testing 197
Conclusion 200

INTRODUCTION
PROC PHREG (pronounced P-H-REG, not FREG) implements the
regression method first proposed in 1972 by the British statistician Sir
David Cox in his famous paper “Regression Models and Life Tables”
(Journal of the Royal Statistical Society, Series B). It’s difficult to
exaggerate the impact of this paper. In the 2009 ISI Web of Science, it was
cited over 1,000 times, making it the most highly cited journal article in
the entire literature of statistics. In fact, Garfield (1990) reported that its
cumulative citation count placed it among the top 100 papers in all of
science. These citation counts undoubtedly underestimate the actual use
of the method because many authors don’t bother citing the original paper.
What explains this enormous popularity? Perhaps the most
important reason is that, unlike the parametric methods discussed in
Chapter 4, “Estimating Parametric Regression Models with PROC
LIFEREG,” Cox’s method does not require that you choose some particular
probability distribution to represent survival times. That’s why it’s called
semiparametric. As a consequence, Cox’s method (often referred to as Cox

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regression) is considerably more robust. A second reason for the paper’s
popularity is that Cox regression makes it relatively easy to incorporate
time-dependent covariates (that is, covariates that may change in value
over the course of the observation period).
There are other attractive features of Cox regression that are less
widely known or appreciated. Cox regression permits a kind of stratified
analysis that is very effective in controlling for nuisance variables. And
Cox regression makes it easy to adjust for periods of time in which an
individual is not at risk of an event. Finally, Cox regression can readily
accommodate both discrete and continuous measurement of event times.
Despite all these desirable qualities, Cox regression should not be
viewed as the universal method for regression analysis of survival data. As
I indicated in Chapter 4, there are times when a parametric method is
preferable. And for most applications, you can do a reasonably good job of
survival analysis using only PROC LIFEREG and PROC LIFETEST. Still, if
I could have only one SAS procedure for doing survival analysis, it would
be PROC PHREG.
All implementations of Cox regression are not created equal. Among
the many available commercial programs, PROC PHREG stands out for its
toolkit of powerful features. While some programs don’t allow
stratification—a fatal deficit in my view—PROC PHREG has a very flexible
stratification option. Many programs don’t handle time-dependent
covariates at all; those that do often have severe restrictions on the number
or kinds of such covariates. In contrast, PROC PHREG has by far the most
extensive and powerful capabilities for incorporating time-dependent
covariates. And while Cox regression can theoretically deal with discrete
(tied) data, most programs use approximations that are inadequate in many
cases. In contrast, PROC PHREG offers two exact algorithms for tied data.

THE PROPORTIONAL HAZARDS MODEL
In his 1972 paper, Cox made two significant innovations. First, he
proposed a model that is usually referred to as the proportional hazards
model. That name is somewhat misleading, however, because the model
can readily be generalized to allow for nonproportional hazards. Second,
he proposed a new estimation method that was later named partial
likelihood or, more accurately, maximum partial likelihood. The term Cox
regression refers to the combination of the model and the estimation

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method. It didn’t take any great leap of imagination to formulate the
proportional hazards model—it’s a relatively straightforward
generalization of the Weibull and Gompertz models that we considered in
Chapter 2, “Basic Concepts of Survival Analysis.” But the partial
likelihood method is something completely different. It took years for
statisticians to fully understand and appreciate this novel approach to
estimation.
Before discussing partial likelihood, let’s first examine the model
that it was designed to estimate. We’ll start with the basic model that does
not include time-dependent covariates or nonproportional hazards. The
model is usually written as

hi (t ) = λ0 (t ) exp( β1 xi1 +  + β k xik )

(5.1)

This equation says that the hazard for individual i at time t is the product
of two factors:
 a function λ0(t) that is left unspecified, except that it can’t be
negative
 a linear function of a set of k fixed covariates, which is then
exponentiated.
The function λ0(t) can be regarded as the hazard function for an individual
whose covariates all have values of 0. It is often called the baseline hazard
function.
Taking the logarithm of both sides, we can rewrite the model as

log hi (t ) = α (t ) + β1 xi1 +  + β k xik

(5.2)

where α(t) = log λ0(t). If we further specify α(t) = α, we get the exponential
model. If we specify α(t) = αt, we get the Gompertz model. Finally, if we
specify α(t) = α log t, we have the Weibull model. As we will see, however,
the great attraction of Cox regression is that such choices are unnecessary.
The function α(t) can take any form whatever, even that of a step function.
Why is equation (5.1) called the proportional hazards model?
Because the hazard for any individual is a fixed proportion of the hazard
for any other individual. To see this, take the ratio of the hazards for two
individuals i and j, and apply equation (5.1):

hi (t )
= exp{β1 ( xi1 − x j ) +  + β k ( xik − x jk )}
h j (t )

(5.3)

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128

Estimating Cox Regression Models with PROC PHREG

What’s important about this equation is that λ0(t) cancels out of the
numerator and denominator. As a result, the ratio of the hazards is
constant over time. If we graph the log hazards for any two individuals, the
proportional hazards property implies that the hazard functions should be
strictly parallel, as in Figure 5.1.

Figure 5.1

Parallel Log-Hazard Functions from Proportional Hazards Model

log h(t)
Case 1
Case 2

PARTIAL LIKELIHOOD
What’s remarkable about partial likelihood is that you can estimate
the β coefficients of the proportional hazards model without having to
specify the baseline hazard function λ0(t). In this section, we will
 consider some general properties of partial likelihood
 look at two simple examples using PROC PHREG
 examine the mathematics of the method in some detail.
The likelihood function for the proportional hazards model of
equation (5.1) can be factored into two parts:
 one part depends on both λ0(t) and β = [β1 β2 … βk]', the vector of
coefficients
 the other part depends on β alone.
What partial likelihood does, in effect, is discard the first part and
treat the second part—the partial likelihood function—as though it were
an ordinary likelihood function. You get estimates by finding values of
β that maximize the partial likelihood. Because there is some information
about β in the discarded portion of the likelihood function, the resulting

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estimates are not fully efficient. Their standard errors are larger than they
would be if you used the entire likelihood function to obtain the estimates.
In most cases, however, the loss of efficiency is quite small (Efron, 1977).
What you gain in return is robustness because the estimates have good
properties regardless of the actual shape of the baseline hazard function.
To be specific, partial likelihood estimates still have two of the three
standard properties of ML estimates: they are consistent and
asymptotically normal. In other words, in large samples they are
approximately unbiased and their sampling distribution is approximately
normal.
Another interesting property of partial likelihood estimates is that
they depend only on the ranks of the event times, not their numerical
values. This implies that any monotonic transformation of the event times
will leave the coefficient estimates unchanged. For example, we could add
a constant to everyone’s event time, multiply the result by a constant, take
the logarithm, and then take the square root—all without producing the
slightest change in the coefficients or their standard errors.
Partial Likelihood: Examples

Let’s first apply the partial likelihood method to the recidivism data
that we introduced in The Life-Table Method in Chapter 3, “Estimating
and Comparing Survival Curves with PROC LIFETEST,” and that was
repeatedly analyzed in Chapter 4. The syntax for PROC PHREG is almost
identical to that for PROC LIFEREG, except that you do not need to specify
a distribution:
PROC PHREG DATA=recid;
MODEL week*arrest(0)=fin age race wexp mar paro prio;
RUN;

Output 5.1 shows the results.

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Output 5.1 Results from Partial Likelihood Estimation with Recidivism Data

The PHREG Procedure
Model Information
Data Set
Dependent Variable
Censoring Variable
Censoring Value(s)
Ties Handling

MY.RECID
week
arrest
0
BRESLOW

Number of Observations Read
Number of Observations Used

432
432

Summary of the Number of Event and Censored Values

Total

Event

Censored

Percent
Censored

432

114

318

73.61

Convergence Status
Convergence criterion (GCONV=1E-8) satisfied.

Model Fit Statistics

Criterion
-2 LOG L
AIC
SBC

Without
Covariates

With
Covariates

1351.367
1351.367
1351.367

1318.241
1332.241
1351.395

(continued)

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Output 5.1 (continued)

Testing Global Null Hypothesis: BETA=0
Test
Likelihood Ratio
Score
Wald

Chi-Square

Pr > ChiSq

33.1256
33.3828
31.9875

7
7
7

<.0001
<.0001
<.0001

Analysis of Maximum Likelihood Estimates

Parameter
fin
age
race
wexp
mar
paro
prio

Parameter
Estimate

Standard
Error

Chi-Square

Pr > ChiSq

Hazard
Ratio

1
1
1
1
1
1
1

-0.37902
-0.05724
0.31415
-0.15113
-0.43280
-0.08497
0.09114

0.19136
0.02198
0.30802
0.21212
0.38180
0.19575
0.02863

3.9228
6.7798
1.0402
0.5076
1.2850
0.1884
10.1331

0.0476
0.0092
0.3078
0.4762
0.2570
0.6642
0.0015

0.685
0.944
1.369
0.860
0.649
0.919
1.095

The preliminary information is the same as in PROC LIFEREG,
except for the line “Ties Handling: BRESLOW.” This line refers to the
default method for handling ties—two or more observations that have
exactly the same event time. Although Breslow’s method is nearly
universal, we’ll consider three superior alternatives later in the Tied Data
section. The next table, labeled “Model Fit Statistics,” displays measures
of fit that were described in Chapter 4:
 –2 times the log likelihood
 Akaike’s information criterion
 Schwarz’s Bayesian criterion (equivalent to the Bayesian
information criterion [BIC] statistic reported by PROC LIFEREG)
The difference here is that these statistics are based on the partial
likelihood rather than the full likelihood. Notice that the three fit measures
are the same for a model with no covariates. This is not surprising because
the main purpose of Akaike’s information criterion (AIC) and Schwartz’s
Bayesian criterion (SBC) statistics is to penalize models for having more
covariates, allowing one to compare non-nested models with different
numbers of covariates.
The table labeled “Testing Global Null Hypothesis: BETA=0”
displays tests of the null hypothesis that all the coefficients are 0. Three
alternative chi-square statistics are given: a likelihood-ratio test, a score

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test, and a Wald test. The likelihood ratio chi-square is just the difference
in the –2 log L values for the models with and without covariates shown in
the previous table. I already discussed the general properties of these tests
in Chapter 4 (see Hypothesis Tests). Here we see that all three statistics are
a bit over 30 with 7 d.f. leading to very small p-values. (The 7 d.f.
correspond to the seven coefficients in the model.) We conclude that at
least one of the coefficients is not 0.
In the last table, we see the coefficient estimates and associated
statistics. Notice that there is no intercept estimate—a characteristic
feature of partial likelihood estimation. The intercept is part of α(t), the
arbitrary function of time, which drops out of the estimating equations. As
with PROC LIFEREG, the chi-square tests are Wald tests for the null
hypothesis that each coefficient is equal to 0. These statistics are
calculated simply by squaring the ratio of each coefficient to its estimated
standard error. The last column, labeled “Hazard Ratio,” is just exp(β).
Hazard ratios can be interpreted almost exactly like odds ratios in
logistic regression. For indicator (dummy) variables with values of 1 and 0,
you can interpret the hazard ratio as the ratio of the estimated hazard for
those with a value of 1 to the estimated hazard for those with a value of 0
(controlling for other covariates). For example, the estimated hazard ratio
for the variable FIN (financial aid) is .685. This means that the hazard of
arrest for those who received financial aid is only about 69 percent of the
hazard for those who did not receive aid (controlling for other covariates).
For quantitative covariates, a more helpful statistic is obtained by
subtracting 1.0 from the hazard ratio and multiplying by 100. This gives
the estimated percent change in the hazard for each 1-unit increase in the
covariate. For the variable AGE, the hazard ratio is .944, which yields
100(.944 – 1) = – 5.6. Therefore, for each 1-year increase in the age at
release, the hazard of arrest goes down by an estimated 5.6 percent.
Overall, the results are similar to those we saw in Chapter 4 with the
LIFEREG procedure. There are highly significant effects of age and the
number of prior convictions and a marginally significant effect of financial
aid. Comparing the coefficients with those in Output 4.2 for the
exponential model, we find that all the numbers are very close, but the
signs are reversed. The p-values are also similar. The sign reversal is not
surprising because the PROC LIFEREG estimates are in log-survival time
format, while the PROC PHREG estimates are in log-hazard format. The
PROC PHREG estimates are all larger in magnitude than the Weibull
estimates in Output 4.3, but, again, that’s merely a consequence of the
alternative ways of expressing the model. When we convert the Weibull
estimates to log-hazard format by dividing by the scale estimate and
changing the sign (as in The Weibull Model section in Chapter 4), the

CHAPTER 5

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results are remarkably close to the PROC PHREG estimates. Because the
other PROC LIFEREG models (gamma, log-logistic, and log-normal) are not
proportional hazards models, their coefficients cannot be converted to loghazard format. Consequently, there is no point in comparing them with the
PROC PHREG coefficients.
Now let’s look at a somewhat more complicated example, the famous
Stanford Heart Transplant Data, as reported by Crowley and Hu (1977).
The sample consists of 103 cardiac patients who were enrolled in the
transplantation program between 1967 and 1974. After enrollment,
patients waited varying lengths of time until a suitable donor heart was
found. Thirty patients died before receiving a transplant, while another
four patients had still not received transplants at the termination date of
April 1, 1974. Patients were followed until death or until the termination
date. Of the 69 transplant recipients, only 24 were still alive at
termination. At the time of transplantation, all but four of the patients were
tissue typed to determine the degree of similarity with the donor.
The following variables were input into SAS:
DOB
DOA
DOT
DLS
DEAD
SURG
M1
M2
M3

date of birth.
date of acceptance into the program.
date of transplant.
date last seen (dead or censored).
coded 1 if dead at DLS; otherwise, coded 0.
coded 1 if patient had open-heart surgery prior
to DOA; otherwise, coded 0.
number of donor alleles with no match in recipient
(1 through 4).
coded 1 if donor-recipient mismatch on HLA-A2 antigen;
otherwise, 0.
mismatch score.

The variables DOT, M1, M2, and M3 are coded as missing for those
patients who did not receive a transplant. All four date measures are coded
in the form mm/dd/yy, where mm is the month, dd is the day, and yy is the
year. Here are the raw data for the first 10 cases:

CHAPTER 5

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DOB

DOA

DOT

DLS

01/10/37
03/02/16
09/19/13
12/23/27
07/28/47
11/08/13
08/29/17
03/27/23
06/11/21
02/09/26

11/15/67
01/02/68
01/06/68
03/28/68
05/10/68
06/13/68
07/12/68
08/01/68
08/09/68
08/11/68

.
.
01/06/68
05/02/68
.
.
08/31/68
.
.
08/22/68

01/03/68
01/07/68
01/21/68
05/05/68
05/27/68
06/15/68
05/17/70
09/09/68
11/01/68
10/07/68

DEAD
1
1
1
1
1
1
1
1
1
1

SURG
0
0
0
0
0
0
0
0
0
0

M1
.
.
2
3
.
.
4
.
.
2

M2
.
.
0
0
.
.
0
.
.
0

These data were read into SAS with the following DATA step. (Note:
The OPTIONS statement preceding the DATA step corrects a Y2K problem
in the data.)
OPTIONS YEARCUTOFF=1900;
DATA stan;
INFILE 'c: stan.dat';
INPUT dob mmddyy9. doa mmddyy9. dot mmddyy9. dls mmddyy9.
dead surg m1 m2 m3;
surv1=dls-doa;
surv2=dls-dot;
ageaccpt=(doa-dob)/365.25;
agetrans=(dot-dob)/365.25;
wait=dot-doa;
IF dot=. THEN trans=0; ELSE trans=1;
RUN;

Notice that the four date variables are read with the MMDDYY9.
format, which translates the date into the number of days since January 1,
1960. (Dates earlier than that have negative values.) We then create two
survival time variables: days from acceptance until death (SURV1) and
days from transplant until death (SURV2). We also calculate the age (in
years) at acceptance into the program (AGEACCPT), the age at transplant
(AGETRANS), and the number of days from acceptance to transplant
(WAIT). Finally, we create an indicator variable (TRANS) coded 1 for those
who received a transplant and 0 for those who did not.
An obvious question is whether transplantation raised or lowered
the hazard of death. A naive approach to answering this question is to do a
Cox regression of SURV1 on transplant status (TRANS), controlling for
AGEACCPT and SURG:

M3
.
.
1.110
1.660
.
.
1.320
.
.
0.610

CHAPTER 5

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135

PROC PHREG DATA=stan;
MODEL surv1*dead(0)=trans surg ageaccpt;
RUN;

The results in Output 5.2 show very strong effects of both transplant
status and age at acceptance. We see that each additional year of age at the
time of acceptance into the program is associated with a 6 percent increase
in the hazard of death. On the other hand, the hazard for those who
received a transplant is only about 18 percent of the hazard for those who
did not (see the Hazard Ratio column). Or equivalently (taking the
reciprocal), those who did not receive transplants are about 5-1/2 times
more likely to die at any given point in time.

Output 5.2 Results for All Patients, No Time-Dependent Variables
Testing Global Null Hypothesis: BETA=0
Test
Likelihood Ratio
Score
Wald

Chi-Square

Pr > ChiSq

45.4629
52.0469
46.6680

3
3
3

<.0001
<.0001
<.0001

Analysis of Maximum Likelihood Estimates

Parameter
trans
surg
ageaccpt

Parameter
Estimate

Standard
Error

Chi-Square

Pr > ChiSq

Hazard
Ratio

1
1
1

-1.70813
-0.42130
0.05860

0.27860
0.37098
0.01505

37.5902
1.2896
15.1611

<.0001
0.2561
<.0001

0.181
0.656
1.060

While the age effect may be real, the transplant effect is almost surely
an artifact. The main reason why patients did not get transplants is that
they died before a suitable donor could be found. Thus, when we compare
the death rates for those who did and did not get transplants, the rates are
much higher for those who did not. In effect, the covariate is actually a
consequence of the dependent variable: an early death prevents a patient
from getting a transplant. The way around this problem is to treat
transplant status as a time-dependent covariate, but that will have to wait
until the Time-Dependent Covariates section later in this chapter.
We can also ask a different set of questions that do not require any
time-dependent covariates. Restricting the analysis to the 65 patients who
did receive heart transplants, we can ask why some of these patients
survived longer than others:

CHAPTER 5

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136

PROC PHREG DATA=stan;
WHERE trans=1;
MODEL surv2*dead(0)=surg m1 m2 m3 agetrans wait dot;
RUN;

Notice that we now use a different origin—the date of the
transplant—in calculating survival time. (It is possible to use the date of
acceptance as the origin, using the methods in the Left Truncation and
Late Entry into the Risk Set section later in this chapter, but it is probably
not worth the trouble.)

Output 5.3 Results for Transplant Patients, No Time-Dependent Covariates
Testing Global Null Hypothesis: BETA=0
Test
Likelihood Ratio
Score
Wald

Chi-Square

Pr > ChiSq

16.5855
15.9237
14.9076

7
7
7

0.0203
0.0258
0.0372

Analysis of Maximum Likelihood Estimates

Parameter
surg
m1
m2
m3
agetrans
wait
dot

Parameter
Estimate

Standard
Error

Chi-Square

Pr > ChiSq

Hazard
Ratio

1
1
1
1
1
1
1

-0.77029
-0.24857
0.02958
0.64407
0.04927
-0.00197
-0.0001650

0.49718
0.19437
0.44268
0.34276
0.02282
0.00514
0.0002991

2.4004
1.6355
0.0045
3.5309
4.6619
0.1469
0.3044

0.1213
0.2009
0.9467
0.0602
0.0308
0.7015
0.5811

0.463
0.780
1.030
1.904
1.050
0.998
1.000

Results in Output 5.3 show, again, that older patients have higher
risks of dying. Specifically, each additional year of age at the time of the
transplant is associated with a 5 percent increase in the hazard of death.
That does not tell us whether the surgery is riskier for older patients,
however. It merely tells us that older patients are more likely to die. There
is also some evidence of higher death rates for those who have a higher
level of tissue mismatch, as measured by the M3 score. None of the other
variables approaches statistical significance, however. (Note that four cases
are lost in this analysis because of missing data on M1-M3.)

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137

Partial Likelihood: Mathematical and Computational Details

Now that we’ve seen the partial likelihood method in action, let’s
take a closer look at how it does what it does. Using the same notation as
in Chapter 4, we have n independent individuals (i = 1,…, n). For each
individual i, the data consist of three parts: ti, δi and xi, where ti is the time
of the event or the time of censoring, δi is an indicator variable with a
value of 1 if ti is uncensored or a value of 0 if ti is censored, and xi = [xi1 …
xik] is a vector of k covariate values.
An ordinary likelihood function is typically written as a product of
the likelihoods for all the individuals in the sample. On the other hand,
you can write the partial likelihood as a product of the likelihoods for all
the events that are observed. Thus, if J is the number of events, we can
write
J

PL = ∏ L j

(5.4)

j =1

where Lj is the likelihood for the jth event. Next we need to see how the
individual Ljs are constructed. This is best explained by way of an
example. Consider the data in Output 5.4, which is taken from Collett
(2003) with a slight modification (the survival time for observation 8 is
changed from 26 to 25 to eliminate ties). The variable SURV contains the
survival time in months, beginning with the month of surgery, for 45 breast
cancer patients. Twenty-six of the women died (DEAD=1) during the
observation period, so there are 26 terms in the partial likelihood. The
variable X has a value of 1 if the tumor had a positive marker for possible
metastasis; otherwise, the variable has a value of 0. The cases are arranged
in ascending order by survival time, which is convenient for constructing
the partial likelihood.
Output 5.4 Survival Times for Breast Cancer Patients

OBS
1
2
3
4
5
6
7
8
9
10

EVENT
1
2
3
4
5
6
7
8
9
10

SURV
5
8
10
13
18
23
24
25
26
31

DEAD

1
1
1
1
1
1
1
1
1
1

1
1
1
1
1
0
1
1
1
1

(continued)

CHAPTER 5

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138
Output 5.4 (continued)

11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45

11
12
13
14
15
16
17
18
19
20
.
21
.
.
.
.
.
.
.
22
.
23
24
25
.
.
26
.
.
.
.
.
.
.
.

35
40
41
47
48
50
59
61
68
69
70
71
71
76
100
101
105
107
109
113
116
118
143
148
154
162
181
188
198
208
212
212
217
224
225

1
1
1
1
1
1
1
1
1
1
0
1
0
0
0
0
0
0
0
1
0
1
1
1
0
0
1
0
0
0
0
0
0
0
0

1
1
1
0
1
1
1
1
1
0
0
1
0
1
0
0
1
1
1
1
1
1
1
0
1
1
0
1
0
0
0
1
1
0
1

The first death occurred to patient 1 in month 5. To construct the
partial likelihood (L1) for this event, we ask the following question: Given
that a death occurred in month 5, what is the probability that it happened
to patient 1 rather than to one of the other patients? The answer is the
hazard for patient 1 at month 5 divided by the sum of the hazards for all
the patients who were at risk of death in that same month. At month 5, all
45 patients were at risk of death, so the probability is

L1 =

h1 (5)
h1 (5) + h2 (5) +  + h45 (5)

(5.5)

While this expression has considerable intuitive appeal, the derivation is
actually rather involved and will not be presented here.
The second death occurred to patient 2 in month 8. Again we ask,
given that a death occurred in month 8, what is the probability that it

CHAPTER 5

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139

occurred to patient 2 rather than to one of the other patients at risk?
Patient 1 is no longer at risk of death because she already died. So L2 has
the same form as L1, but the hazard for patient 1 is removed from the
denominator:

h2 (8)
h2 (8) + h3 (8) +  + h45 (8)

L2 =

(5.6)

The set of all individuals who are at risk at a given point in time is
often referred to as the risk set. At time 8, the risk set consists of patients 2
through 45, inclusive.
We continue in this way for each successive death, deleting from the
denominator the hazards for all those who have already died. Also deleted
from the denominator are those who have been censored at an earlier point
in time. That’s because they are no longer at risk of an observed event. For
example, the 21st death occurred to patient 22 in month 71. Patient 21 was
censored at month 70, so her hazard does not appear in the denominator of
L21. On the other hand, if an event time is the same as a censoring time, the
convention is to assume that the censored observation was still at risk at
that time. Thus, patient 23 who was censored in month 71 does show up
in the denominator of L21.
The last term in the likelihood corresponds to the 26th death, which
occurred to the 37th patient in month 181:

L26 =

h37 (181)
h37 (181) + h38 (181) +  + h45 (181)

(5.7)

All the hazards in the denominator, except for the first, are for patients
who were censored in months later than 181.
The results to this point have made no assumptions about the form
of the hazard function. Now, we invoke the proportional hazards model of
equation (5.1) and substitute the expression for the hazard into the
expression for L1,

L1 =

λ0 (5)e

βx1

λ0 (5)e βx

+ λ0 (5)e

βx 2

+  + λ0 (5)e

βx 45

(5.8)

where xi is the value of x for the ith patient. This leads to a considerable
simplification because the unspecified function λ0(5) is common to every
term in the expression. Canceling, we get

L1 =

e βx1

e βx1
.
+ e βx 2 +  + e βx 45

(5.9)

CHAPTER 5

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It is this cancellation of the λs that makes it possible to estimate the β
coefficients without having to specify the baseline hazard function. Of
course, the λs also cancel for all the other terms in the partial likelihood.
Earlier I remarked that the partial likelihood depends only on the
order of the event times, not on their exact values. You can easily see this
by considering each of the Lj terms. Although the first death occurred in
month 5, L1 would be exactly the same if it had occurred at any time from
0 up to (but not including) 8, the month of the second event. Similarly, L2
would have been the same if the second death had occurred any time
greater than 5 and less than 10 (the month of the third death).
A general expression for the partial likelihood for data with timeinvariant covariates from a proportional hazards model is



n
eβxi
PL = ∏  n

βx j
i =1
 ∑ Yij e
 j =1

δi








(5.10)

where Yij = 1 if tj ≥ ti; and Yij = 0 if tj < ti. (The Ys are just a convenient
mechanism for excluding from the denominator those individuals who
have already experienced the event and are, thus, not part of the risk set.)
Although this expression has the product taken over all individuals rather
than over all events (as in equation [5.4]), the terms corresponding to
censored observations are effectively excluded because δi =0 for those
cases. This expression is not valid for tied event times, but it does allow
for ties between one event time and one or more censoring times.
Once the partial likelihood is constructed, you can maximize it with
respect to β just like an ordinary likelihood function. As usual, it’s
convenient to maximize the logarithm of the likelihood, which is
n
n

βx 
log PL = ∑ δ i βx i − log ∑ Yij e j  .
i =1
j =1



(5.11)

Most partial likelihood programs use some version of the NewtonRaphson algorithm to maximize this function with respect to β . For
details, see Chapter 4 (in the Maximum Likelihood Estimation:
Mathematics section).
As with PROC LIFEREG, there will occasionally be times when the
Newton-Raphson algorithm does not converge. A message in the OUTPUT
window will say WARNING: The information matrix is not
positive definite and thus the convergence is questionable.

Unfortunately, PROC PHREG’s convergence criterion sometimes makes it
look as though the algorithm has converged when, in fact, true
convergence is not possible. This problem typically arises when one of the

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explanatory variables is an indicator variable (1 or 0) and all the
observations are censored for one of the levels of that variable. In such
cases, the log-likelihood reaches a stable value, but the coefficient of the
offending variable keeps heading off toward plus or minus infinity. The
only indication of a problem is that the variable in question will have a
large coefficient with a much larger standard error. Beginning with SAS
9.2, there is now a simple solution to this problem. Using the FIRTH
option in the MODEL statement requests that the partial likelihood
function be modified to produce penalized partial likelihood estimates.
This method reduces the small-sample bias that often occurs with
conventional likelihood methods and usually clears up convergence
problems (Firth, 1993; Heinze and Schemper, 2001).
To complete the breast cancer example, let’s take a look at the partial
likelihood results in Output 5.5. With only one covariate, the Global Null
Hypothesis statistics provide us with three alternative tests for the effect of
that variable. The Wald and score tests have p-values that exceed the
conventional .05 level, while the likelihood-ratio test has a p-value that is
slightly below .05. This degree of discrepancy is not at all surprising with a
small sample. The estimated hazard ratio of 2.483 tells us that the hazard
of death for those whose tumor had the positive marker was nearly 2.5
times the hazard for those without the positive marker.
Because the covariate is dichotomous, an alternative approach is to
use PROC LIFETEST to test for differences in survival curves. When I did
this, the p-value for the log-rank test (.0607) was identical to the p-value
for the score test in Output 5.5. This is no accident. The log-rank test is the
exact equivalent of the partial likelihood score test for a single,
dichotomous covariate (when there are no ties).

Output 5.5 PROC PHREG Results for Breast Cancer Data

Testing Global Null Hypothesis: BETA=0
Test
Likelihood Ratio
Score
Wald

Chi-Square

Pr > ChiSq

3.8843
3.5194
3.2957

1
1
1

0.0487
0.0607
0.0695

Analysis of Maximum Likelihood Estimates

Parameter
x

Parameter
Estimate

Standard
Error

Chi-Square

Pr > ChiSq

Hazard
Ratio

0.90933

0.50089

3.2957

0.0695

2.483

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TIED DATA
The formula for the partial likelihood in equation (5.10) is valid only
for data in which no two events occur at the same time. It’s quite common
for data to contain tied event times, however, so we need an alternative
formula to handle those situations. Most partial likelihood programs use a
technique called Breslow’s approximation, which works well when ties are
relatively few. But when data are heavily tied, the approximation can be
quite poor (Farewell and Prentice, 1980; Hsieh, 1995). Although PROC
PHREG uses Breslow’s approximation as the default, it also provides a
better approximation proposed by Efron (1977) as well as two exact
methods.
This section explains the background, rationale, and implementation
of these alternative methods for handling ties. Because this issue is
somewhat confusing, I’m going to discuss it at some length. Those who
just want the bottom line can skip to the end of the section where I
summarize the practical implications. Because the formulas get rather
complicated, I won’t go into all the mathematical details. But I will try to
provide some intuitive understanding of why there are different
approaches and the basic logic of each one.
To illustrate the problem and the various solutions, let’s turn again to
the recidivism data. As Output 5.6 shows, these data include a substantial
number of tied survival times (weeks to first arrest). For weeks 1 through 7,
there is only one arrest in each week. For these seven events, the partial
likelihood terms are constructed exactly as described in the Partial
Likelihood: Mathematical and Computational Details section. Five arrests
occurred in week 8, however, so the construction of L8 requires a different
method. Two alternative approaches have been proposed for the
construction of the likelihood for tied event times; these are specified in
PROC PHREG by TIES=EXACT or TIES=DISCRETE as options in the
MODEL statement. This terminology is somewhat misleading because both
methods give exact partial likelihoods; the difference is that the EXACT
method assumes that there is a true but unknown ordering for the tied
event times (that is, time is continuous), while the DISCRETE method
assumes that the events really occurred at exactly the same time.

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143
Output 5.6

Week of First Arrest for Recidivism Data
Cumulative Cumulative
WEEK
Frequency
Percent
Frequency
Percent
------------------------------------------------1
1
0.2
1
0.2
2
1
0.2
2
0.5
3
1
0.2
3
0.7
4
1
0.2
4
0.9
5
1
0.2
5
1.2
6
1
0.2
6
1.4
7
1
0.2
7
1.6
8
5
1.2
12
2.8
9
2
0.5
14
3.2
10
1
0.2
15
3.5
11
2
0.5
17
3.9
12
2
0.5
19
4.4
13
1
0.2
20
4.6
14
3
0.7
23
5.3
15
2
0.5
25
5.8
16
2
0.5
27
6.2
17
3
0.7
30
6.9
18
3
0.7
33
7.6
19
2
0.5
35
8.1
20
5
1.2
40
9.3
21
2
0.5
42
9.7
22
1
0.2
43
10.0
23
1
0.2
44
10.2
24
4
0.9
48
11.1
25
3
0.7
51
11.8
26
3
0.7
54
12.5
27
2
0.5
56
13.0
28
2
0.5
58
13.4
30
2
0.5
60
13.9
31
1
0.2
61
14.1
32
2
0.5
63
14.6
33
2
0.5
65
15.0
34
2
0.5
67
15.5
35
4
0.9
71
16.4
36
3
0.7
74
17.1
37
4
0.9
78
18.1
38
1
0.2
79
18.3
39
2
0.5
81
18.8
40
4
0.9
85
19.7
42
2
0.5
87
20.1
43
4
0.9
91
21.1
44
2
0.5
93
21.5
45
2
0.5
95
22.0
46
4
0.9
99
22.9
47
1
0.2
100
23.1
48
2
0.5
102
23.6
49
5
1.2
107
24.8
50
3
0.7
110
25.5
52
322
74.5
432
100.0

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144

The EXACT Method

Let’s begin with the EXACT method because its underlying model is
probably more plausible for most applications. Since arrests can occur at
any point in time, it’s reasonable to suppose that ties are merely the result
of imprecise measurement of time and that there is a true but unknown
time ordering for the five arrests that occurred in week 8. If we knew that
ordering, we could construct the partial likelihood in the usual way. In the
absence of any knowledge of that ordering, however, we have to consider
all the possibilities. With five events, there are 5! = 120 different possible
orderings. Let’s denote each of those possibilities by Ai, where i = 1, … ,
120. What we want is the probability of the union of those possibilities,
that is, Pr(A1 or A2 or … or A120). Now, a fundamental law of probability
theory is that the probability of the union of a set of mutually exclusive
events is just the sum of the probabilities for each of the events. Therefore,
we can write
120

L8 = ∑ Pr( Ai ) .

(5.12)

i =1

Each of these 120 probabilities is just a standard partial likelihood.
Suppose, for example, that we arbitrarily label the five arrests at time 8
with the numbers 8, 9, 10, 11, and 12, and suppose further that A1 denotes
the ordering {8, 9, 10, 11, 12}. Then

eβx 8
Pr( A1 ) =  βx 8
βx 9
βx 432
 e + e ++ e


eβx 9
 βx
βx
βx 432
9
10
 e + e +  + e

 
eβx12
   βx
βx
βx 432
13
12
  e + e ++ e

.



On the other hand, if A2 denotes the ordering {9, 8, 10, 11, 12}, we
have

eβx 9
Pr( A2 ) =  βx 8
βx 9
βx 432
 e + e ++ e


eβx 8
 βx
βx
βx 432
8
10
 e + e +  + e

 
eβx12
   βx
βx
βx 432
13
12
  e + e ++ e

.



We continue in this way for the other 118 possible orderings. Then L8 is
obtained by adding all the probabilities together.
The situation is much simpler for week 9 because only two arrests
occurred, giving us two possible orderings. For L9, then, we have

eβx13
L9 =  βx13
βx 432
βx14
 e + e ++ e


eβx14
 βx
βx15
βx 432
14
 e + e +  + e


eβx14
 βx
βx 432
βx
13
14
 e + e ++ e


 +



eβx13
 βx
βx
βx 432
13
15
 e + e +  + e





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145

where the numbers 13 and 14 are arbitrarily assigned to the two events.
When we get to week 10, there’s only one event, so we’re back to the
standard partial likelihood formula:

eβx15
L10 =  βx15
βx16
βx 432
 e + e ++ e

.



It’s difficult to write a general formula for the exact likelihood with
tied data because the notation becomes very cumbersome. For one version
of a general formula, see Kalbfleisch and Prentice (2002). Be forewarned
that the formula in the official PROC PHREG documentation bears no
resemblance to that given by Kalbfleisch and Prentice or to the explanation
given here. That’s because it’s based on a re-expression of the formula in
terms of a definite integral, which facilitates computation (DeLong,
Guirguis, and So, 1994).
It should be obvious, by this point, that computation of the exact
likelihood can be a daunting task. With just five tied survival times, we
have seen that one portion of the partial likelihood increased from 1 term
to 120 terms. If 10 events occur at the same time, there are over 3 million
possible orderings to evaluate. Until recently, statisticians abandoned all
hope that such computations might be practical. What makes it possible
now is the development of an integral representation of the likelihood,
which is much easier to evaluate numerically (DeLong, Guirguis, and So,
1994). Even with this innovation, however, computation of the exact
likelihood when large numbers of events occur at the same time can take a
lot of computing time.
Early recognition of these computational difficulties led to the
development of approximations. The most popular of these is widely
attributed to Breslow (1974), but it was first proposed by Peto (1972). This
is the default in PROC PHREG, and it is nearly universal in other
programs. Efron (1977) proposed an alternative approximation that is also
available in PROC PHREG. The results that we saw earlier in Output 5.1
for the recidivism data were obtained with the Breslow approximation.
To use the EXACT method, we specify
PROC PHREG DATA=recid;
MODEL week*arrest(0)=fin age race wexp mar paro prio
/ TIES=EXACT;
RUN;

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146

This PROC step produces the results in Output 5.7. Comparing this with
Output 5.1, it’s apparent that the Breslow approximation works well in this
case. The coefficients are generally the same to at least two (and sometimes
three) decimal places. The test statistics all yield the same conclusions.

Output 5.7

Recidivism Results Using the EXACT Method
Testing Global Null Hypothesis: BETA=0
Test
Likelihood Ratio
Score
Wald

Chi-Square

Pr > ChiSq

33.2663
33.5289
32.1190

7
7
7

<.0001
<.0001
<.0001

Analysis of Maximum Likelihood Estimates

Parameter
fin
age
race
wexp
mar
paro
prio

Parameter
Estimate

Standard
Error

Chi-Square

Pr > ChiSq

Hazard
Ratio

1
1
1
1
1
1
1

-0.37942
-0.05743
0.31393
-0.14981
-0.43372
-0.08486
0.09152

0.19138
0.02200
0.30800
0.21223
0.38187
0.19576
0.02865

3.9305
6.8152
1.0389
0.4983
1.2900
0.1879
10.2067

0.0474
0.0090
0.3081
0.4803
0.2560
0.6646
0.0014

0.684
0.944
1.369
0.861
0.648
0.919
1.096

Output 5.8 shows the results from using Efron’s approximation
(invoked by using TIES=EFRON). If Breslow’s approximation is good, this
one is superb. Nearly all the numbers are the same to four decimal places.
In all cases where I’ve tried the two approximations, Efron’s
approximation gave results that were much closer to the exact results than
Breslow’s approximation. This improvement comes with only a trivial
increase in computation time.

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147

Output 5.8

Recidivism Results Using Efron’s Approximation
Testing Global Null Hypothesis: BETA=0
Test
Likelihood Ratio
Score
Wald

Chi-Square

Pr > ChiSq

33.2659
33.5287
32.1192

7
7
7

<.0001
<.0001
<.0001

Analysis of Maximum Likelihood Estimates

Parameter
fin
age
race
wexp
mar
paro
prio

Parameter
Estimate

Standard
Error

Chi-Square

Pr > ChiSq

Hazard
Ratio

1
1
1
1
1
1
1

-0.37942
-0.05743
0.31392
-0.14981
-0.43372
-0.08486
0.09152

0.19138
0.02200
0.30799
0.21223
0.38187
0.19576
0.02865

3.9304
6.8152
1.0389
0.4983
1.2900
0.1879
10.2067

0.0474
0.0090
0.3081
0.4803
0.2560
0.6646
0.0014

0.684
0.944
1.369
0.861
0.648
0.919
1.096

If the approximations are so good, why do we need the
computationally intensive EXACT method? Farewell and Prentice (1980)
showed that the Breslow approximation deteriorates as the number of ties
at a particular point in time becomes a large proportion of the number of
cases at risk. For the recidivism data in Output 5.6, the number of tied
survival times at any given time point is never larger than 2 percent of the
number at risk, so it’s not surprising that the approximations work well.
Now let’s look at an example where the conditions are less favorable.
The data consist of 100 simulated job durations, measured from the year of
entry into the job until the year that the employee quit. Durations after the
fifth year are censored. If the employee was fired before the fifth year, the
duration is censored at the end of the last full year in which the employee
was working. We know only the year in which the employee quit, so the
survival times have values of 1, 2, 3, 4, or 5.

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148

Here’s a simple life table for these data:

Duration

Number
Quit

Number
Censored

1
2
3
4
5

22
18
16
8
4

7
3
4
1
17

Number
At Risk
100
71
50
30
21

Number Quit/
At Risk
.22
.25
.32
.27
.19

The number at risk at each duration is equal to the total number of
cases (100) minus the number who quit or were censored at previous
durations. Looking at the last column, we see that the ratio of the number
quitting to the number at risk is substantial at each of the five points in
time. Three covariates were measured at the beginning of the job: years of
schooling (ED), salary in thousands of dollars (SALARY), and prestige of
the occupation (PRESTIGE) measured on a scale from 1 to 100.
Output 5.9 displays selected results from using PROC PHREG with
the three different methods for handling ties. Breslow’s method yields
coefficient estimates that are about one-third smaller in magnitude than
those using the EXACT method, while the p-values (for testing the
hypothesis that each coefficient is 0) are substantially higher. In fact, the pvalue for the SALARY variable is above the .05 level for Breslow’s method,
but it is only .01 for the EXACT method. Efron’s method produces
coefficients that are about midway between the other two methods, but the
p-values are much closer to those of the EXACT method. Clearly, the
Breslow approximation is unacceptable for this application. Efron’s
approximation is not bad for drawing qualitative conclusions, but there is
an appreciable loss of accuracy in estimating the magnitudes of the
coefficients.

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149

Output 5.9 Results for Job Duration Data: Three Methods for Handling Ties

Ties Handling: BRESLOW
Variable DF

Parameter
Estimate

ED
PRESTIGE
SALARY

0.116453
-0.064278
-0.014957

1
1
1

Standard
Wald
Pr >
Error Chi-Square Chi-Square
0.05918
0.00959
0.00792

3.87257
44.93725
3.56573

0.0491
0.0001
0.0590

Hazard
Ratio
1.124
0.938
0.985

Ties Handling: EFRON
Variable DF

Parameter
Estimate

ED
PRESTIGE
SALARY

0.144044
-0.079807
-0.020159

1
1
1

Standard
Wald
Pr >
Error Chi-Square Chi-Square
0.05954
0.00996
0.00830

5.85271
64.20009
5.90363

0.0156
0.0001
0.0151

Hazard
Ratio
1.155
0.923
0.980

Ties Handling: EXACT
Variable DF

Parameter
Estimate

ED
PRESTIGE
SALARY

0.164332
-0.092019
-0.022545

1
1
1

Standard
Wald
Pr >
Error Chi-Square Chi-Square
0.06380
0.01240
0.00884

6.63419
55.10969
6.50490

0.0100
0.0001
0.0108

Hazard
Ratio
1.179
0.912
0.978

The DISCRETE Method

The DISCRETE option in PROC PHREG is also an exact method but
one based on a fundamentally different model. In fact, it is not a
proportional hazards model at all. The model does fall within the
framework of Cox regression, however, because it was proposed by Cox in
his original 1972 paper and because the estimation method is a form of
partial likelihood. Unlike the EXACT model, which assumes that ties are
merely the result of imprecise measurement of time, the DISCRETE model
assumes that time is really discrete. When two or more events appear to
happen at the same time, there is no underlying ordering—they really
happen at the same time.
While most applications of survival analysis involve events that can
occur at any moment on the time continuum, there are definitely some
events that are best treated as if time were discrete. If the event of interest
is a change in the political party occupying the U.S. presidency, that can
only occur once every four years. Or suppose the aim is to predict how

CHAPTER 5

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150

many months it takes before a new homeowner misses a mortgage
payment. Because payments are only due at monthly intervals, a discretetime model is the natural way to go.
Cox’s model for discrete-time data can be described as follows. The
time variable t can only take on integer values. Let Pit be the conditional
probability that individual i has an event at time t, given that an event has
not already occurred to that individual. This probability is sometimes
called the discrete-time hazard. The model says that Pit is related to the
covariates by a logistic regression equation:

 P
log it
 1 − Pit


 = α t + β1 xi1 +  + β k xik .


The expression on the left side of the equation is the logit or log-odds of
Pit. On the right side, we have a linear function of the covariates, plus a
term αt that plays the same role as α(t) in expression (5.2) for the
proportional hazards model. αt is just a set of constants—one for each time
point—that can vary arbitrarily from one time point to another.
This model can be described as a proportional odds model. The odds
that individual i has an event at time t (given that i did not already have an
event) is Oit =Pit /(1 - Pit ). The model implies that the ratio of the odds for
any two individuals Oit/Ojt does not depend on time (although it may vary
with the covariates).
How can we estimate this model? In Chapter 7, “Analysis of Tied or
Discrete Data with PROC LOGISTIC,” we will see how to estimate it using
standard maximum likelihood methods that yield estimates of both the β
coefficients and the αts. Using partial likelihood, however, we can treat the
αts as nuisance parameters and estimate only the βs. If there are J unique
times at which events occur, there will be J terms in the partial likelihood
function:
J

PL = ∏ L j
j =1

where Lj is the partial likelihood of the jth event. Thus, for the job duration
data, there are only five terms in the partial likelihood function. But each
of those five terms is colossal. Here’s why.
At time 1, 22 people had events out of 100 people who were at risk.
To get L1, we ask the question: given that 22 events occurred, what is the
probability that they occurred to these particular 22 people rather than to
some different set of 22 people from among the 100 at risk? How many
different ways are there of selecting 22 people from among a set of 100? A
lot! Specifically, 7.3321 × 1021. Let’s call that number Q, and let q be a
running index from 1 to Q, with q = 1 denoting the set that actually

CHAPTER 5

Estimating Cox Regression Models with PROC PHREG

experienced the events. For a given set q, let ψq be the product of the odds
for all the individuals in that set. Thus, if the individuals who actually
experienced events are labeled i = 1 to 22, we have
22

ψ 1 = ∏ Oi1 .
i =1

We can then write

L1 =

ψ1
.
ψ 1 +ψ 2  +ψ Q

This looks like a simple expression, but there are trillions of terms being
summed in the denominator. Fortunately, there is a recursive algorithm
that makes it practical, even with substantial numbers of ties (Gail et al.,
1981). Still, doing this with a very large data set with many ties can take a
lot of computer time.
Does the discrete-time model make sense for these data? For most
jobs it’s possible to quit at any point in time, suggesting that the model
might not be appropriate. Remember, however, that these are simulated
data. Because the simulation is actually based on a discrete-time model, it
makes perfectly good sense in this case. Output 5.10 displays the results.
Comparing these with the results for the EXACT method in Output 5.9, we
see that the chi-square statistics and the p-values are similar. However, the
coefficients for the DISCRETE method are about one-third larger for ED
and PRESTIGE and about 15 percent larger for SALARY. This discrepancy
is due largely to the fact that completely different models are being
estimated, a hazard model and a logit model. The logit coefficients will
usually be larger. For the logit model, 100(eβ – 1) gives the percent change
in the odds that an event will occur for a one-unit increase in the
covariate. Thus, each additional year of schooling increases the odds of
quitting a job by 100(e.219–1) = 24 percent.

Output 5.10

Job Duration Results Using the DISCRETE Method

Ties Handling: DISCRETE
Analysis of Maximum Likelihood Estimates
Variable DF

Parameter
Estimate

ED
PRESTIGE
SALARY

0.219378
-0.120474
-0.026108

1
1
1

Standard
Wald
Pr >
Error Chi-Square Chi-Square
0.08480
0.01776
0.01020

6.69295
46.02220
6.55603

0.0097
0.0001
0.0105

Hazard
Ratio
1.245
0.886
0.974

151

CHAPTER 5

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152

Comparison of Methods

Though the job duration coefficients differ for the two exact
methods, they are at least in the same ballpark. More generally, it has been
shown that if ties result from grouping continuous time data into intervals,
the logit model converges to the proportional hazards model as the interval
length gets smaller (Thompson, 1977). When there are no ties, the partial
likelihoods for all four methods (the two exact methods and the two
approximations) reduce to the same formula.
The examples that we’ve seen so far have been small enough, both in
number of observations and numbers of ties, that the computing times for
the two exact methods were quite small (much less than a second). Before
concluding, let’s see what happens with larger data sets. I took the 100
observations in the job duration data set and duplicated them to produce
data sets of size 1000 to 6000 in increments of 1000. I then estimated the
models on a Dell laptop running at 1.8GHz with 1 Gb of memory. Here are
the CPU times, in seconds, for the two exact methods and six sample sizes:
EXACT
DISCRETE

1000
3.6
1.6

2000
16
6

3000 4000
43
93
13
23

5000
171
33

6000
316
51

The good news is that these computing times are more than an order
of magnitude smaller than the times that I reported in the 1995 edition of
this book. For 1000 observations, what previously took the EXACT method
more than 2 minutes now takes less than 4 seconds. The upshot is that
these methods are now practical for fairly large data sets, at least if you are
willing to be patient. For the DISCRETE method, computing time goes up
approximately with the square of the sample size. Computing time for the
EXACT method goes up even faster. Incidentally, both approximate
methods (BRESLOW and EFRON) took less than a quarter of a second for
6000 observations.
What we’ve learned about the handling of ties can be summarized in
six points:
 When there are no ties, all four options in PROC PHREG give
identical results.
 When there are few ties, it makes little difference which method
is used. But because computing times will also be comparable,
you might as well use one of the exact methods.
 When the number of ties is large, relative to the number at risk,
the approximate methods tend to yield coefficients that are
biased toward 0.

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153

Both the EXACT and DISCRETE methods produce exact results
(that is, true partial likelihood estimates), but the EXACT method
assumes that ties arise from grouping continuous, untied data,
while the DISCRETE method assumes that events really occur at
the same, discrete times. The choice should be based on
substantive grounds, although qualitative results will usually be
similar.
 Both of the exact methods need a substantial amount of
computer time for large data sets containing many ties, especially
the EXACT method.
 If the exact methods are too time-consuming, use the Efron
approximation, at least for model exploration. It’s nearly always
better than the Breslow method, with virtually no increase in
computer time.


TIME-DEPENDENT COVARIATES
Time-dependent covariates are those that may change in value over
the course of observation. While it’s simple to modify Cox’s model to allow
for time-dependent covariates, the computation of the resulting partial
likelihood is more time-consuming, and the practical issues surrounding
the implementation of the procedure can be quite complex. It’s easy to
make mistakes without realizing it, so be sure you know what you’re
doing.
To modify the model in equation (5.2) to include time-dependent
covariates, all we need to do is write (t) after the xs that are timedependent. For a model with one time-invariant covariate and one
time-dependent covariate, we have
log hi (t ) = α (t ) + β1 xi1 + β 2 xi 2 (t ).

This says that the hazard at time t depends on the value of x1 and on the
value of x2 at time t. What may not be clear is that x2(t) can be defined
using any information about the individual before time t, thereby allowing
for lagged or cumulative values of some variables. For example, if we want
a model in which the hazard of arrest depends on employment status, we
could specify employment as
 whether the person is currently employed in week t
 whether the person was employed in the previous month (t-1)
 the number of weeks of employment in the preceding 3 months
 the number of bouts of unemployment in the preceding 12
months.

CHAPTER 5

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154

The use of lagged covariates is often essential for resolving issues of causal
ordering (more on that later).
For handling time-dependent covariates, PROC PHREG has two very
different ways of setting up the data and specifying the model. In the
programming statements method, there is one record per individual, and
the information about time-dependent covariates is contained in multiple
variables on that record. The time-dependent covariates are then defined
in programming statements that are part of the PROC PHREG step. In the
counting process method, on the other hand, there may be multiple
records for each individual, with each record corresponding to an interval
of time during which all the covariates remain constant. Once this special
data set has been constructed, time-dependent covariates are treated just
like time-invariant covariates. However, a different syntax (the counting
process syntax) is needed to specify the dependent variable.
If done properly, these two methods produce exactly the same
results, so the choice between them is purely a matter of computational
convenience. Both methods may require a substantial amount of careful
programming. However, with the counting process syntax, all the
programming is up front in the construction of the data set. That makes it
straightforward to check the data set for any programming errors. The
method also lends itself to a division of labor in which a skilled
programmer produces the required data set. The data analyst can then
easily specify whatever models are desired in PROC PHREG.
With the programming statements method, on the other hand, the
program to generate the time-dependent covariates must be a part of every
PROC PHREG run. Furthermore, because the temporary data set produced
by the program is not accessible, there is no easy way to check for
programming mistakes. All this would seem to favor the counting process
method. Nonetheless, as we shall see, the counting process method often
requires considerably more program code. We will illustrate these two
methods with several examples, first applying the programming statements
method and then using the counting process method.
Heart Transplant Example

We begin with another look at the Stanford Heart Transplant Data. In
the Partial Likelihood: Examples section, we attempted to determine
whether a transplant raised or lowered the risk of death by examining the
effect of a time-invariant covariate TRANS that was equal to 1 if the patient
ever had a transplant and 0 otherwise. I claimed that those results were
completely misleading because patients who died quickly had less time
available to get transplants. Now we’ll do it right by defining a time-

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dependent covariate PLANT equal to 1 if the patient has already had a
transplant at day t; otherwise, PLANT is equal to 0.
Here’s how it’s done with the programming statements method:
PROC PHREG DATA=stan;
MODEL surv1*dead(0)=plant surg ageaccpt / TIES=EFRON;
IF wait>=surv1 OR wait=. THEN plant=0; ELSE plant=1;
RUN;

Recall that SURV1 is the time in days from acceptance into the
program until death or termination of observation (censoring). Recall also
that SURG =1 if the patient had previous heart surgery; otherwise,
SURG=0. AGEACCPT is the patient’s age in years at the time of
acceptance, and WAIT is the time in days from acceptance until transplant
surgery, coded as missing for those who did not receive transplants. Note
that I have used the EFRON method for handling ties. Although there are
not many ties in these data, the discussion in the last section concluded
that the EFRON method is a better default than the usual BRESLOW
method.
The IF statement defines the new time-varying covariate PLANT.
Note that programming statements must follow the MODEL statement. At
first glance, this IF statement may be puzzling. For patients who were not
transplanted, the IF condition will be true (and PLANT will be assigned a
value of 0) because their WAIT value will be missing. On the other hand,
for those who did receive transplants, it appears that the IF condition will
always be false because their waiting time to transplant cannot be greater
than their survival time, giving us a fixed covariate rather than a timedependent covariate. Now it’s true that waiting time is never greater than
survival time for transplanted patients (with one patient dying on the
operating table). But, unlike an IF statement in the DATA step, which only
operates on a single case at a time, this IF statement compares waiting
times for patients who were at risk of a death with survival times for
patients who experienced events. Thus, SURV1 in this statement is
typically not the patient’s own survival time, but the survival time of some
other patient who died. This fact will become clearer in the next
subsection when we examine the construction of the partial likelihood.
Results in Output 5.11 indicate that transplantation has no effect on
the hazard of death. The effect of age at acceptance is somewhat smaller
than it was in Output 5.2, although still statistically significant. However,
the effect of prior heart surgery is larger and now significant at the .05
level.

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Output 5.11 Results for Transplant Data with a Time-Dependent Covariate

Parameter
plant
surg
ageaccpt

Parameter
Estimate

Standard
Error

Chi-Square

Pr > ChiSq

Hazard
Ratio

1
1
1

0.01610
-0.77332
0.03054

0.30859
0.35967
0.01389

0.0027
4.6229
4.8312

0.9584
0.0315
0.0279

1.016
0.461
1.031

Now let’s redo the analysis using the counting process method. To
do this, we must construct a data set with multiple records per person, one
record for each period of time during which all the covariates remain
constant. For persons who did not receive transplants, only one record is
needed. Persons who received transplants need two records: one for the
interval of time between acceptance and transplantation (with PLANT=0)
and a second record for the interval between transplantation and either
death or censoring (PLANT=1). The first interval is coded as censored. The
second interval may be either censored or uncensored, depending on
whether the person died during the observation period. Time-invariant
variables are replicated for the two records. For each record, we also need
two time variables, one for the start of the interval and one for the end of
the interval. Both time variables are measured in days since acceptance
into the study.
Here is the DATA step needed to produce the required data set:
DATA stanlong;
SET stan;
plant=0;
start=0;
IF trans=0 THEN DO;
dead2=dead;
stop=surv1;
IF stop=0 THEN stop=.1;
OUTPUT;
END;
ELSE DO;
stop=wait;
IF stop=0 THEN stop=.1;
dead2=0;
OUTPUT;
plant=1;
start=wait;
IF stop=.1 THEN start=.1;
stop=surv1;
dead2=dead;
OUTPUT;
END;
RUN;

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Clearly this is a lot more programming than was required for the
programming statements method. Here is a brief explanation. The first two
assignment statements initialize the time-dependent covariate (PLANT)
and the starting time at 0. The IF-DO statement begins a block of
definitions for those who did not receive transplants. A new censoring
indicator, DEAD2, is specified to have the same value as the old indicator
DEAD, and the stop time is set to be the same as the original time variable
SURV1. The OUTPUT statement writes the record to the new data set. By
default, all other variables in the original data set are included in this
record.
A peculiarity of this data set is that there were two individuals who
died on their day of acceptance into the study, so both their start time and
their stop time would be 0. Unfortunately, the counting process syntax
deletes records that have the same start and stop time. That’s why there is
an IF statement that changes a stop time of 0 to .1.
The ELSE DO statement begins a block of variable definitions for
those who received transplants. First, a record is created for the interval
from acceptance to transplant. The stop time is set to be the waiting time to
transplant. Again, because there was one person who was transplanted on
the day of acceptance, it is necessary to change the stop time from 0 to .1.
The censoring indicator DEAD2 is set to 0 since this interval did not end
in a death. And the OUTPUT statement writes the record to the new data
set.
Next, a record is created for the interval from transplant to death or
censoring. PLANT is now set to 1. The START time is set to the time of
transplant (WAIT). The STOP time is reset to the original death or
censoring time, and the censoring indicator is reset to its original values.
For the one person transplanted on the day of acceptance, the IF statement
aligns the start time of the second interval with the stop time of the first
interval.
The new data set has 172 records, one each for the 34 people who
did not receive transplants and two records for each of the 69 people who
did receive transplants. Output 5.12 displays the last 15 records in the new
data set, representing eight people. All but one (record 164) had a
transplant and, therefore, the rest have two records. Only one person died
(record 163).

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Output 5.12 Last 15 Records for Counting Process Data Set

Obs

start

stop

158
159
160
161
162
163
164
165
166
167
168
169
170
171
172

0.0
35.0
0.0
57.0
0.0
36.0
0.0
0.0
40.0
0.0
22.0
0.0
50.0
0.0
24.0

35.0
1141.0
57.0
1321.0
36.0
1386.0
1400.0
40.0
1407.0
22.0
1571.0
50.0
1586.0
24.0
1799.0

dead2

plant

surg

ageaccpt

0
0
0
0
0
1
0
0
0
0
0
0
0
0
0

0
1
0
1
0
1
0
0
1
0
1
0
1
0
1

1
1
1
1
0
0
0
1
1
0
0
0
0
0
0

36.6543
36.6543
45.3032
45.3032
54.0123
54.0123
30.5352
48.4819
48.4819
40.5530
40.5530
48.9035
48.9035
33.2238
33.2238

To estimate the model, we use the counting process syntax,
specifying both the starting time and the stopping time for each record:
PROC PHREG DATA=stanlong;
MODEL (start,stop)*dead2(0)=plant surg ageaccpt /TIES=EFRON;
RUN;

Results are identical to those in Output 5.11.
Although this syntax is similar to the PROC LIFEREG syntax for
interval-censored data, the intent is completely different. In the PROC
LIFEREG syntax, the event is known to occur sometime within the
specified interval. In the counting process syntax, on the other hand,
(START, STOP) represents an interval of time during which the individual
was continuously at risk of the event, and events can only occur at the end
of the interval.
Construction of the Partial Likelihood with Time-Dependent Covariates

With time-dependent covariates, the partial likelihood function has
the same form that we saw previously in equation (5.4) and equation
(5.10). The only thing that changes is that the covariates are now indexed
by time. Consider, for example, the 12 selected cases from the Stanford
Heart Transplant Data that are shown in Output 5.13. These are all the
cases that had death or censoring times between 16 and 38 days, inclusive.

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Output 5.13 Selected Cases from the Stanford Heart Transplant Data

OBS

SURV1

DEAD

WAIT

19
20
21
22
23
24
25
26
27
28
29
30

16
17
20
20
27
29
30
31
34
35
36
38

1
1
1
1
1
1
0
1
1
1
1
1

4
.
.
.
17
4
.
.
.
.
.
35

On day 16, one death occurred (to case 19). Because 18 people had
already died or been censored by day 16, there were 103–18 = 85 people
left in the risk set on that day. Let’s suppose that we have a single
covariate, the time-dependent version of transplant status. The partial
likelihood for day 16 is therefore

e βx19 (16 )
.
e βx19 (16 ) + e βx 20 (16 ) + e βx 21 (16 ) +  + e βx103 (16 )
To calculate this quantity, PROC PHREG must compute the value of x on
day 16 for each of the 85 people at risk. For cases 19 and 24, a transplant
occurred on day 4. Since this was before day 16, we have x19(16) = 1 and
x24(16) = 1. Waiting time is missing for cases 20−22 and cases 25−29,
indicating that they never received a transplant. Therefore, x20(16) =
x21(16) = x22(16) = x25(16) = x26(16) = x27(16) = x28(16) = x29(16) = 0. Case
23 had a transplant on day 17, so on day 16, the patient was still without a
transplant and x23(16) = 0. Similarly, we have x30(16) = 0 because case 30
didn’t get a transplant until day 35.
For the programming statements method, the calculation of the
appropriate x values is accomplished by the IF statement shown earlier in
the heart transplant example. At each unique event time, PROC PHREG
calculates a term in the partial likelihood function (like the one above) by
applying the IF statement to all the cases in the risk set at that time. Again,
the value of SURV1 in the IF statement is the event time that PROC PHREG
is currently operating on, not the survival time for each individual at risk.
For the counting process method, to compute the partial likelihood
for a death occurring on day 16, PHREG simply searches through all the
records in the data set to find those with START and STOP times that
bracket day 16. Note that the intervals defined by those times are open on
the left and closed on the right. That means that, if the START time is 16

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and the STOP time is 24, day 24 would be included in the interval but day
16 would not be included.
This property of the counting process method has important
implications for the next term in the partial likelihood function, which
corresponds to the death that occurred to person 20 on day 17:

e βx 20 (17 )
.
e βx 20 (17 ) + e βx 21 (17 ) +  + e βx103 (17 )
Of course person 19 no longer shows up in this formula because that
patient left the risk set at death. The values of x are all the same as they
were for day 16, even for patient 23, who had a transplant on day 17. This
patient has two records, one running from 0 to 17 and the other from 17 to
27. The first record includes day 17, but the second record does not. The
first record has PLANT=0, so x23(17) = 0.
For the data in Output 5.13, there are eight additional terms in the
partial likelihood function. For the first five of these, the values of x for
cases remaining in the risk set are the same as they were on day 17. On day
36, however, the value of x for case 30 switches from 0 to 1.
Covariates Representing Alternative Time Origins

When I discussed the choice of time origin in Chapter 2, I mentioned
that you can include alternative origins as covariates, sometimes as timedependent covariates. Let’s see how this might work with the Stanford
Heart Transplant Data. In the analysis just completed, the origin was the
date of acceptance into the program, with the time of death or censoring
computed from that point. It is certainly plausible, however, that the
hazard of death also depends on age or calendar time. We have already
included age at acceptance into the program as a time-constant covariate
and found that patients who were older at acceptance had a higher hazard
of death. But if that’s the case, we might also expect that the hazard will
continue to increase with age after acceptance into the program. A natural
way to allow for this possibility is to specify a model with current age as a
time-dependent covariate. Here’s how to do that with PROC PHREG:
PROC PHREG DATA=stan;
MODEL surv1*dead(0)=plant surg age / TIES=EFRON;
IF wait>surv1 OR wait=. THEN plant=0; ELSE plant=1;
age=ageaccpt+surv1;
RUN;

In this program, current age is defined as age at acceptance plus the
time to the current event. While this is certainly correct, a surprising thing
happens: the results are exactly the same as in Output 5.11. Here’s why.
We can write the time-dependent version of the model as

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log h(t ) = α (t ) + β1 x1 + β 2 x2 (t ) + β 3 x3 (t )

(5.13)

where x1 is the surgery indicator, x2 is the transplant status, and x3 is the
current age. We also know that x3(t) = x3(0) + t, where x3(0) is age at the
time of acceptance. Substituting into equation (5.13), we have
log h(t ) = α * (t ) + β1 x1 + β 2 x2 (t ) + β 3 x3 (0)

where α*(t) = α(t) + β3t. Thus, we have converted a model with a timedependent version of age to one with a time-invariant version of age. In the
process, the arbitrary function of time changes, but that’s of no
consequence because it drops out of the estimating equations anyway. The
same trick works with calendar time: instead of specifying a model in
which the hazard depends on current calendar time, we can estimate a
model with calendar time at the point of acceptance and get exactly the
same results.
The trick does not work, however, if the model says that the log of
the hazard is a nonlinear function of the alternative time origin. For
example, suppose we want to estimate the model

log h(t ) = α (t ) + β1 x1 + β 2 x2 (t ) + β 3 log x3 (t )
where x3(t) is again age at time t. Substitution with x3(t) = x3(0) + t gets us
nowhere in this case because the β3 coefficient does not distribute across
the two components of x3(t). You must estimate this model with log x3(t) as
a time-dependent covariate:
PROC PHREG DATA=stan;
MODEL surv1*dead(0)=plant surg logage / TIES=EFRON;
IF wait>surv1 OR wait=. THEN plant=0; ELSE plant=1;
logage=LOG(ageaccpt+surv1);
RUN;

In sum, if you are willing to forego nonlinear functions of time, you
can include any alternative time origin as a time-invariant covariate,
measured at the origin that is actually used in calculating event times.
Time-Dependent Covariates Measured at Regular Intervals

As we saw earlier, calculation of the partial likelihood requires that
the values of the covariates be known for every individual who was at risk
at each event time. In practice, because we never know in advance when
events will occur, we need to know the values of all the time-dependent
covariates at every point in time. This requirement was met for the heart
transplant data: death times were measured in days, and for each day, we

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could construct a variable indicating whether a given patient had already
had a heart transplant.
Often, however, the information about the covariates is only
collected at regular intervals of time that may be longer (or shorter) than
the time units used to measure event times. For example, in a study of time
to death among AIDS patients, there may be monthly follow-ups in which
vital signs and blood measurements are taken. If deaths are reported in
days, there is only about a 1-in-30 chance that the time-dependent
measurements will be available for a given patient on a particular death
day. In such cases, it is necessary to use some ad-hoc method for assigning
covariate values to death days. In a moment, I will discuss several issues
related to such ad-hoc approaches. First, let’s look at an example in which
the time intervals for covariate measurement correspond exactly to the
intervals in which event times are measured.
For the recidivism example that we studied earlier, additional
information was available on the employment status of the released
convicts over the 1-year follow-up period. Specifically, for each of the 52
weeks of follow-up, there was a dummy variable coded 1 if the person was
employed full-time during that week; otherwise, the variable was coded 0.
The data are read as follows:
DATA RECID;
INFILE 'c:\recid.dat';
INPUT week arrest fin age race wexp mar paro prio educ emp1-emp52;
RUN;

The employment status information is contained in the variables EMP1 to
EMP52. For the programming statements method, we can work directly
with the RECID data set. The PROC PHREG statements are
PROC PHREG DATA=recid;
MODEL week*arrest(0)=fin age race wexp mar paro prio employed
/ TIES=EFRON;
ARRAY emp(*) emp1-emp52;
DO i=1 TO 52;
IF week=i THEN employed=emp[i];
END;
RUN;

The aim here is to pick out the employment indicator that
corresponds to the week in which an event occurred and assign that value
to the variable EMPLOYED for everyone who is in the risk set for that
week. The ARRAY statement makes it possible to treat the 52 distinct
dummy variables as a single subscripted array, thereby greatly facilitating
the subsequent operations. Note that in the IF statement, the “subscript” to
EMP must be contained in brackets rather than parentheses.

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The only problem with this code is that the program has to cycle
through 52 IF statements to pick out the right value of the employment
variable. A more efficient program that directly retrieves the correct value
is as follows:
PROC PHREG DATA=recid;
MODEL week*arrest(0)=fin age race wexp mar paro prio employed
/ TIES=EFRON;
ARRAY emp(*) emp1-emp52;
employed=emp[week];
RUN;

Output 5.14 shows the results (for either version). For the
time-invariant variables, the coefficients and test statistics are pretty much
the same as in Output 5.8. Judging by the chi-square test, however, the new
variable EMPLOYED has by far the strongest effect of any variable in the
model. The hazard ratio of .265 tells us that the hazard of arrest for those
who were employed full time is a little more than one-fourth the hazard for
those who were not employed full time.

Output 5.14 Recidivism Results with a Time-Dependent Covariate

Parameter
fin
age
race
wexp
mar
paro
prio
employed

Parameter
Estimate

Standard
Error

Chi-Square

Pr > ChiSq

Hazard
Ratio

1
1
1
1
1
1
1
1

-0.35672
-0.04633
0.33867
-0.02557
-0.29374
-0.06420
0.08515
-1.32823

0.19113
0.02174
0.30960
0.21142
0.38303
0.19468
0.02896
0.25071

3.4835
4.5442
1.1966
0.0146
0.5881
0.1088
8.6455
28.0679

0.0620
0.0330
0.2740
0.9037
0.4432
0.7416
0.0033
<.0001

0.700
0.955
1.403
0.975
0.745
0.938
1.089
0.265

Unfortunately, these results are undermined by the possibility that
arrests affect employment status rather than vice versa. If someone is
arrested and incarcerated near the beginning of a particular week, the
probability of working full time during the remainder of that week is likely
to drop precipitously. This potential reverse causation is a problem that is
quite common with time-dependent covariates, especially when event
times or covariate times are not measured precisely.
One way to reduce ambiguity in the causal ordering is to lag the
covariate values. Instead of predicting arrests in a given week based on
employment status in the same week, we can use employment status in the
prior week. This requires only minor modifications in the SAS code:

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PROC PHREG;
WHERE week>1;
MODEL week*arrest(0)=fin age race wexp mar paro prio employed
/ TIES=EFRON;
ARRAY emp(*) emp1-emp52;
employed=emp[week-1];
RUN;

One change is to add a WHERE statement to eliminate cases (only
one case, in fact) with an arrest in the first week after release. This change
is necessary because there were no values of employment status before the
first week. The other change is to use WEEK–1 as the subscript for EMP
rather than WEEK. With these changes, the coefficient for EMPLOYED
drops substantially, from –1.33 to –.79, which implies that the hazard of
arrest for those who were employed is about 45 percent of the risk of those
who were not employed. While this is a weaker effect than we found using
unlagged values of employment status, it is still highly significant with a
chi-square value of 13.1. The effects of the other variables remain virtually
unchanged.
As this example points out, there are often many different ways of
specifying the effect of a time-dependent covariate. Let’s consider a couple
of the alternatives. Instead of a single lagged version of the employment
status indicator, we can have both a 1-week and a 2-week lag, as shown in
the following program:
PROC PHREG;
WHERE week>2;
MODEL week*arrest(0)=fin age race wexp mar paro prio employ1
employ2 / TIES=EFRON;
ARRAY emp(*) emp1-emp52;
employ1=emp[week-1];
employ2=emp[week-2];
RUN;

Note that because of the 2-week lag, it is necessary to eliminate cases
with events in either week 1 or week 2. When I tried this variation, I found
that neither EMPLOY1 nor EMPLOY2 was significant (probably because
they are highly correlated—a joint test that both coefficients are 0 yielded a
p-value of .0018). However, the 1-week lag was much stronger than the 2week lag. So it looks as though we’re better off sticking with the single 1week lag.
Another possibility is that the hazard of arrest may depend on the
cumulative employment experience after release rather than the
employment status in the preceding week. Consider the following SAS
code:

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DATA RECIDCUM;
SET recid;
ARRAY emp(*) emp1-emp52;
ARRAY cum(*) cum1-cum52;
cum1=emp1;
DO i=2 TO 52;
cum(i)=(cum(i-1)*(i-1) + emp(i))/i;
END;
PROC PHREG DATA=recidcum;
WHERE week>1;
MODEL week*arrest(0)=fin age race wexp mar paro prio employ
/ TIES=EFRON;
ARRAY cumemp(*) cum1-cum52;
EMPLOY=cumemp[week-1];
RUN;

The DATA step defines a new set of variables CUM1−CUM52 that are
the cumulative proportions of weeks worked for each of the 52 weeks. The
DO loop creates the cumulative number of weeks employed and converts it
into a proportion. The PROC PHREG statements have the same structure as
before, except that the cumulative employment (lagged by 1 week) has
been substituted for the lagged employment indicator. This run produces a
marginally significant effect of cumulative employment experience. When
I also included the 1-week-lagged employment indicator, the effect of the
cumulated variable faded to insignificance, while the lagged indicator
continued to be a significant predictor. Again, it appears that the 1-week
lag is a better specification.
You can create the cumulated variable in the PROC PHREG step
rather than the DATA step, but that would increase computing time. In
general, whatever programming can be done in the DATA step should be
done there because the computations only have to be done once. In the
PROC PHREG step, on the other hand, the same computations may have to
be repeated many times.
Now let’s repeat our analysis of the RECID data set using the
counting process method. Here’s the code for producing the data set:
DATA RECIDCOUNT;
ARRAY emp(*) emp1-emp52;
SET recid;
arrest2=0;
DO stop=1 TO week;
start=stop-1;
IF stop=week THEN arrest2=arrest;
employed=emp(stop);
IF week>=1 THEN emplag=emp(stop-1); ELSE emplag=.;
OUTPUT;
END;
RUN;

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This DATA step produced 19,377 person-weeks. Once again, we
need an ARRAY to hold the 52 values of the employment indicator. Next,
ARREST2, our new censoring indicator is initialized at 0 because
individuals can only be arrested in their last person-week. The DO loop,
running from week 1 until the week of arrest or censoring, creates a new
person-week for each cycle through the loop. For each person-week, a
START time and a STOP time are defined. For the last person-week
(STOP=WEEK), the IF statement resets ARREST2 to be the same as
ARREST, the original censoring indicator. Next, the variable EMPLOYED is
set equal to the EMP indicator variable corresponding to that week. An
employment indicator lagged by 1 week is also created. Note that here in a
DATA step, we can put parentheses around the array subscript rather than
the brackets that are required in the PROC PHREG step (although brackets
would work also).
Because most periods of employment or unemployment extended
over multiple weeks, we could have gotten by with a lot fewer records in
the new data set. That is, we could have created one record for each
contiguous period of employment or unemployment. But that would
require a considerably more complicated program to produce the data set.
In many applications, it’s easier to just create a new record for each
discrete point in time. This will not change the results. You must break the
observation period whenever any covariate changes in value, but you can
always break it into smaller units without affecting the outcome.
Here is the PROC PHREG step, which produces exactly the same
table as the one shown in Output 5.14:
PROC PHREG DATA=recidcount;
MODEL (start,stop)*arrest2(0)=fin age race wexp mar paro prio employed
/TIES=EFRON;
RUN;

To do the analysis with a 1-week lag of employment status, just substitute
EMPLAG for EMPLOYED.

Ad-Hoc Estimates of Time-Dependent Covariates

It often happens that time-dependent covariates are measured at
regular intervals, but the intervals don’t correspond to the units in which
event times are measured. For example, we may know the exact day of
death for a sample of cancer patients but have only monthly measurements
of, say, albumin level in the blood. For partial likelihood estimation with
such data, we really need daily albumin measurements, so we must
somehow impute these from the monthly data. There are often several

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possible ways to do this, and, unfortunately, none has any formal
justification. On the other hand, it’s undoubtedly better to use some
common-sense method for imputing the missing values rather than
discarding the data for the time-dependent covariates.
Let’s consider some possible methods and some rough rules of thumb.
For the case of monthly albumin measurements and day of death, an
obvious method is to use the closest preceding albumin level to impute the
level at any given death time. For data over a 1-year period, the SAS code
for the programming statements method might look like this:
DATA blood;
INFILE 'c:\blood.dat';
INPUT deathday status alb1-alb12;
PROC PHREG;
MODEL deathday*status(0)=albumin;
ARRAY alb(*) alb1-alb12;
deathmon=CEIL(deathday/30.4);
albumin=alb[deathmon];
RUN;

Assume that ALB1 is measured at the beginning of the first month,
and so on for the next 11 measurements. Dividing DEATHDAY by 30.4
converts days into months (including fractions of a month). The CEIL
function then returns the smallest integer larger than its argument. Thus,
day 40 would be converted to 1.32, which then becomes 2 (that is, the
second month). This value is then used as a subscript in the ALB array to
retrieve the albumin level recorded at the beginning of the second month.
(There may be some slippage here because months vary in length.
However, if we know the exact day at which each albumin measurement
was taken, we can avoid this difficulty by using the methods for irregular
intervals described in the next section.)
Here’s how to accomplish the same thing with the counting process
syntax:
DATA bloodcount;
SET blood;
ARRAY alb(*) alb1-alb12;
status2=0;
deathmon=CEIL(deathday/30.4);
DO j=1 TO deathmon;
start=(j-1)*30.4;
stop=start+30.4;
albumin=alb(j);
IF j=deathmon THEN DO;
status2=status;
stop=deathday-start;
END;
OUTPUT;
END;

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PROC PHREG DATA=bloodcount;
MODEL (start,stop)*status2(0)=albumin;
RUN;

It may be possible to get better imputations of daily albumin levels
by using information about the blood levels in earlier months. If we
believe, for example, that albumin levels are likely to worsen (or improve)
steadily, it might be sensible to calculate a linear extrapolation based on
the most recent 2 months. Here is the programming statements code:
PROC PHREG DATA=blood;
MODEL deathday*status(0)=albumin;
ARRAY alb(*) alb1-alb12;
deathmon=deathday/30.4;
j=CEIL(deathmon);
IF j=1 THEN albumin=alb(1);
ELSE albumin=alb[j]+(alb[j]-alb[j-1])*(deathmon-j+1);
RUN;

To accomplish the same thing with the counting process method, the
data set must consist of person months rather than person days. Here is the
code:
DATA bloodcount;
SET blood;
ARRAY alb(*) alb1-alb12;
status2=0;
DO i=1 TO deathday;
start=i-1
stop=i;
j=CEIL(i/30.4);
albumin=alb(j)+(alb(j)-alb(j-1))*(i/30.4-j+1);
IF i=deathday THEN status2=status;
OUTPUT;
END;
PROC PHREG DATA=bloodcount;
MODEL (start,stop)*status2(0)=albumin;
RUN;

In these examples, we made no use of information about albumin
levels that were recorded after the death date. Obviously, we had no other
option for patients who died. Remember, though, that for every death date,
PROC PHREG retrieves (or constructs) the covariates for all individuals
who were at risk of death on that date, whether or not they died. For those
who did not die on that death date, we could have used the average of the
albumin level recorded before the death date and the level recorded after
the death date. I don’t recommend this, however. Using different
imputation rules for those who died and those who didn’t die is just
asking for artifacts to creep into your results. Even in cases where the

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occurrence of the event does not stop the measurement of the timedependent covariate, it’s a dangerous practice to use information recorded
after the event to construct a variable used as a predictor of the event. This
would be sensible only if you are completely confident that the event
could not have caused any changes in the time-dependent covariate. For
example, in modeling whether people will purchase a house, it might be
reasonable to use an average of the local mortgage rates before and after the
purchase.
Time-Dependent Covariates That Change at Irregular Intervals

In the Stanford Heart Transplant example, we had a time-dependent
covariate—previous receipt of a transplant—that changed
at unpredictable times. No more than one such change could occur for any
of the patients. Now we consider the more general situation in which the
time-dependent covariate may change at multiple, irregularly spaced
points in time.
We’ll do this by way of an example. The survival data (hypothetical)
in Output 5.15 are for 29 males, aged 50 to 60, who were diagnosed with
alcoholic cirrhosis. At diagnosis, they were measured for blood
coagulation time (PT). The men were then remeasured at clinic visits that
occurred at irregular intervals until they either died or the study was
terminated. The maximum number of clinic visits for any patient was 10.
The length of the intervals between visits ranged between 3 and 33
months, with a mean interval length of 9.5 months and a standard
deviation of 6.0. In Output 5.15, SURV is the time of death or time of
censoring, calculated in months since diagnosis. DEAD is coded 1 for a
death and 0 if censored. TIME2−TIME10 contain the number of months
since diagnosis for each clinic visit. PT1−PT10 contain the measured
values of the PT variable at each clinic visit. Variables are recorded as
missing if there was no clinic visit.

CHAPTER 5

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Output 5.15 Survival Data for 29 Males with Alcoholic Cirrhosis

S
U
R
V

D
E
A
D

T
I
M
E
2

90
80
36
68
62
47
84
57
7
49
55
43
42
11
36
36
2
23
10
29
16
15
5
15
1
13
39
20
45

0
0
0
0
0
0
0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1
1
1
1
1
1

7
6
17
15
22
5
8
6
4
16
3
6
6
9
16
9
.
7
6
21
6
7
3
6
.
4
22
12
18

T
I
M
E
3

T
I
M
E
4

T
I
M
E
5

T
I
M
E
6

20
36
28
20
40
12
27
21
7
.
9
11
12
.
22
26
.
13
.
27
12
12
.
.
.
10
35
.
24

26
42
34
26
46
24
31
27
.
.
15
18
18
.
28
32
.
.
.
.
.
.
.
.
.
.
.
.
38

35
54
.
32
.
35
43
34
.
.
21
24
23
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

44
67
.
51
.
46
76
39
.
.
33
42
29
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

T
I
M
E
7

T
I
M
E
8

T
I
M
E
9

T
I
M
E
1
0

P
T
1

P
T
2

P
T
3

P
T
4

P
T
5

P
T
6

P
T
7

P
T
8

P
T
9

P
T
1
0

50 56 80 83 23.9 20.8 23.6 23.6 24.0 22.5 24.6 25.1 29.4 27.9
78 . . . 29.6 15.1 15.4 16.3 13.9 14.6 16.1
.
.
.
. . . . 25.9 24.4 24.8 24.3
.
.
.
.
.
.
. . . . 26.8 27.9 26.5 26.5 26.8 26.6
.
.
.
.
. . . . 23.0 25.2 27.1 27.8
.
.
.
.
.
.
. . . . 25.8 26.0 25.2 24.9 26.3 26.6
.
.
.
.
. . . . 14.2 11.5 12.9 12.6 12.5 18.6
.
.
.
.
45 51 . . 27.6 27.5 28.0 27.8 29.1 28.2 28.3 28.4
.
.
. . . . 25.0 25.1 24.7
.
.
.
.
.
.
.
. . . . 25.5 27.4
.
.
.
.
.
.
.
.
42 49 . . 14.8 16.7 16.9 17.7 13.8 13.8 13.7 14.2
.
.
. . . . 20.6 19.9 20.3 20.2 19.7 27.1
.
.
.
.
35 . . . 27.6 27.0 28.1 28.8 29.0 28.4 28.8
.
.
.
. . . . 25.3 27.8
.
.
.
.
.
.
.
.
. . . . 22.5 22.3 25.2 26.4
.
.
.
.
.
.
. . . . 26.9 26.9 24.2 26.2
.
.
.
.
.
.
. . . . 19.2
.
.
.
.
.
.
.
.
.
. . . . 21.8 20.3 23.8
.
.
.
.
.
.
.
. . . . 21.6 22.3
.
.
.
.
.
.
.
.
. . . . 18.7 20.2 22.5
.
.
.
.
.
.
.
. . . . 28.4 28.7 28.7
.
.
.
.
.
.
.
. . . . 17.8 17.7 17.4
.
.
.
.
.
.
.
. . . . 20.7 22.6
.
.
.
.
.
.
.
.
. . . . 28.0 28.8
.
.
.
.
.
.
.
.
. . . . 31.6
.
.
.
.
.
.
.
.
.
. . . . 26.0 22.7 25.4
.
.
.
.
.
.
.
. . . . 25.5 29.0 29.2
.
.
.
.
.
.
.
. . . . 21.3 21.0
.
.
.
.
.
.
.
.
. . . . 23.9 28.7 29.5 30.2
.
.
.
.
.
.

Let’s estimate a model in which the hazard of death at time t
depends on the value of PT at time t. Because we don’t have measures of
PT at all death times, we’ll use the closest preceding measurement. The
SAS code for accomplishing this with the programming statements method
is as follows:
PROC PHREG DATA=alco;
MODEL surv*dead(0)=pt;
time1=0;
ARRAY time(*) time1-time10;
ARRAY p(*) pt1-pt10;
DO j=1 TO 10;

CHAPTER 5

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171

IF surv > time[j] AND time[j] NE . THEN pt=p[j];
END;
RUN;

For a given death time, the DO loop cycles through all 10 possible
clinic visits. If the death time is greater than the time of jth visit, the value
of PT is reassigned to be the value observed at visit j. PROC PHREG keeps
doing this until it either
 encounters a missing value of the TIME variable (no clinic visit)
 encounters a TIME value that is greater than the death time
 goes through all 10 possible visits.
Hence, PROC PHREG always stops at the most recent clinic visit and
assigns the value of PT recorded at that visit.
When I ran this model, I got a coefficient for PT of .084 with a
nonsignificant likelihood-ratio chi-square value of 1.93, suggesting that the
blood measurement is of little predictive value. But suppose the hazard
depends on the change in PT relative to the patient’s initial measurement
rather than upon the absolute level. You can implement this idea by
changing the IF statement above to read as follows:
IF surv > time[j] AND time[j] NE . THEN pt=p[j]-pt1;

With this minor change in specification, the coefficient of PT increased in
magnitude to .365 with a chi-square of 7.35, which is significant at beyond
the .01 level.
Here is the code for producing the same results with the counting
process method:
DATA alcocount;
SET alco;
time1=0;
time11=.;
ARRAY t(*) time1-time11;
ARRAY p(*) pt1-pt10;
dead2=0;
DO j=1 TO 10 WHILE (t(j) NE .);
start=t(j);
pt=p(j);
stop=t(j+1);
IF t(j+1)=. THEN DO;
stop=surv;
dead2=dead;
END;
OUTPUT;
END;
PROC PHREG DATA=alcocount;
MODEL (start,stop)*dead2(0)=pt;
RUN;

CHAPTER 5

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172

To make the predictor measure the change from TIME1, simply modify the
definition of PT to be
pt=p(j)-pt1;

In general, the only additional requirement for handling covariates
with irregular intervals between measurements is data on the times of
measurement. The variables containing the measurement times must
always be in the same metric as the event-time variable.

COX MODELS WITH NONPROPORTIONAL HAZARDS
Earlier, I mentioned that the Cox model can be easily extended to
allow for nonproportional hazards. In fact, we’ve just finished a lengthy
discussion of one class of nonproportional models. Whenever you
introduce time-dependent covariates into a Cox regression model, it’s no
longer accurate to call it a proportional hazards (PH) model. Why? Because
the time-dependent covariates will change at different rates for different
individuals, so the ratios of their hazards cannot remain constant. As
we’ve seen, however, that creates no real problem for the partial likelihood
estimation method.
But suppose you don’t have any time-dependent covariates. How do
you know whether your data satisfy the PH assumption, and what happens
if the assumption is violated? Although these are legitimate questions, I
personally believe that concern about the PH assumption is often
excessive. Every model embodies many assumptions, some more
questionable or consequential than others. The reason people focus so
much attention on the PH assumption is that the model is named for that
property. At the same time, they often ignore such critical questions as:
Are all the relevant covariates included? Is the censoring mechanism
noninformative? Is measurement error in the covariates acceptably low? As
in ordinary linear regression, measurement error in the covariates tends to
attenuate coefficients (Nakamura, 1992).
To put this issue in perspective, you need to understand that
violations of the PH assumption are equivalent to interactions between one
or more covariates and time. That is, the PH model assumes that the effect
of each covariate is the same at all points in time. If the effect of a variable
varies with time, the PH assumption is violated for that variable. It’s
unlikely that the PH assumption is ever exactly satisfied, but that’s true of
nearly all statistical assumptions. If we estimate a PH model when the
assumption is violated for some variable (thereby suppressing the

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173

interaction), then the coefficient that we estimate for that variable is a sort
of average effect over the range of times observed in the data. Is this so
terrible? In fact, researchers suppress interactions all the time when they
estimate regression models. In models with even a moderately large
number of variables, no one tests for all the possible 2-way interactions—
there are just too many of them.
In some cases, however, we may have reason to believe that the
interactions with time are so strong that it would be misleading to
suppress them. In other situations, we might have a strong theoretical
interest in those interactions, even when they are not very strong. For
those cases, we need to examine methods for testing and modeling
nonproportional hazards.
We begin with methods for testing for nonproportional hazards using
residuals. Then we examine two methods that extend the Cox model to
allow for nonproportional hazards. One method explicitly incorporates the
interactions into the model. The other, called stratification, subsumes the
interactions into the arbitrary function of time.
Testing the Proportionality Assumption with the ASSESS Statement

A variety of different kinds of residuals have been defined for Cox
regression models (Collett, 2003). One kind, martingale residuals, can be
used to test for nonproportionality, using the method proposed by Lin,
Wei, and Ying (1993). This method has been incorporated into the ASSESS
statement in PROC PHREG. Unfortunately, this statement will not work if
the model contains time-dependent covariates, either with the
programming statements method or the counting process method.
Here’s how to use the ASSESS statement for the recidivism data:
ODS GRAPHICS ON;
PROC PHREG DATA=recid;
MODEL week*arrest(0)=fin age race wexp mar paro prio
/ TIES=EFRON;
ASSESS PH / RESAMPLE;
RUN;
ODS GRAPHICS OFF;

In the ASSESS statement, the PH option requests an assessment of
the proportional hazards assumption. For each covariate, this statement
produces a graphical display of the empirical score process, which is
based on the martingale residuals. Output 5.16 displays the graph for AGE.
The solid line is the observed empirical score process. The dashed lines
are empirical score processes based on 20 random simulations that
embody the proportional hazards assumption. If the observed process
deviates markedly from the simulated processes, it is evidence against the

CHAPTER 5

Estimating Cox Regression Models with PROC PHREG

174

proportional hazards assumption. The observed process for AGE does
indeed appear to be somewhat more extreme than the simulated processes.

Output 5.16

Testing the Proportional Hazards Assumption with the ASSESS Statement

In the lower right corner of Output 5.16, we get a more quantitative
assessment in the form of a p-value. For 1000 simulated paths, only 1.5
percent of them had extreme points (suprema) that exceeded the most
extreme point of the observed path (yielding a Kolmogorov-Smirnov type
test). The p-value was produced by the RESAMPLE option, which
generated the 1,000 simulated paths. That option also produces a table
summarizing the results for each covariate, as seen in Output 5.17. The
only variable with clear evidence for a violation of the PH assumption is
AGE. However, this test does not tell us anything about the nature of the
violation. Note also that if you only want the table but not the graphs, you
can just run the code above without the ODS statements.

CHAPTER 5

Estimating Cox Regression Models with PROC PHREG

175

Output 5.17

Testing the Proportional Hazards Assumption with the ASSESS Statement
Supremum Test for Proportional Hazards Assumption

Variable
fin
age
race
wexp
mar
paro
prio

Maximum
Absolute
Value

Replications

Seed

Pr >
MaxAbsVal

0.5408
1.8192
0.9435
1.3008
0.9349
0.5383
0.6104

1000
1000
1000
1000
1000
1000
1000

292778001
292778001
292778001
292778001
292778001
292778001
292778001

0.8240
0.0150
0.2210
0.0940
0.2350
0.8070
0.7240

You can also test for nonproportional hazards using Schoenfeld
residuals. Unlike the ASSESS statement, this method works even if you
have time-varying covariates defined by the counting process method (but
not with the programming statements method). It also takes much less
computing time, which may be important with large samples.
Schoenfeld residuals have an unusual property: instead of a single
residual for each individual, there is a separate residual for each covariate
for each individual. They are not defined for censored observations (they
are missing in the output data set). Here’s how they are calculated.
Suppose that individual i is arrested at time ti, and at that time there were
330 people at risk of arrest, indexed by j = 1, …, 330. For each of those 330
people, the estimated Cox model implies a certain probability of being
arrested at that time, denoted by pj. Imagine randomly selecting one of
these 330 people, with probability pj. For each covariate xk, we can
calculate its expected value for a randomly selected person (from that risk
set) as
330

xk = ∑ xkj p j .
j =1

The Schoenfeld residual is then defined as the covariate value for the
person who actually died, xik, minus the expected value.

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The main use of these residuals is to detect possible departures from
the proportional hazards assumption. If the assumption is satisfied, the
Schoenfeld residuals should be independent of time. Testing the
proportional hazards assumption is equivalent, therefore, to testing
whether the Schoenfeld residuals are correlated with time or with some
function of time. For the recidivism data, I produced a data set containing
the Schoenfeld residuals using these statements:
PROC PHREG DATA=recid;
MODEL week*arrest(0)=fin age prio race wexp mar paro prio /
TIES=EFRON;
OUTPUT OUT=b RESSCH=schfin schage schprio schrace schwexp schmar
schparo schprio;
RUN;

Note that the keyword RESSCH is followed by eight arbitrarily
chosen variable names corresponding to the eight covariates in the model.
These variables contain the Schoenfeld residuals in the output data set B.
The next step is to modify the data set to incorporate two familiar
functions of time, the log and the square, followed by PROC CORR to
compute the correlations of the residuals with time and its two functions.
Other functions of time might also be considered.
DATA c;
SET b;
lweek=LOG(week);
week2=week**2;
PROC CORR;
VAR week lweek week2 schfin schage schprio schrace schwexp schmar
schparo schprio;
RUN;

The results, shown in Output 5.18, are consistent with what we
found using the ASSESS statement. There is a fairly clear-cut problem
with AGE (with p-values less than .05 for all three functions) and a
marginal problem with WEXP. Notice that the sample size is only 114
because the Schoenfeld residuals are not defined for the censored cases. In
the next subsection, we will see how to estimate a Cox model that allows
the effect of AGE to change with time. This will also give us a more
definitive test of the proportionality assumption for this variable.

CHAPTER 5

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177
Output 5.18

Testing the Proportional Hazards Assumption with Schoenfeld Residuals
Pearson Correlation Coefficients
Prob > |r| under H0: Rho=0
Number of Observations
week

lweek

week2

schfin
Schoenfeld Residual for fin

-0.00874
0.9265
114

0.03484
0.7129
114

-0.01688
0.8585
114

schage
Schoenfeld Residual for age

-0.21563
0.0212
114

-0.23849
0.0106
114

-0.18679
0.0466
114

schprio
Schoenfeld Residual for prio

-0.06043
0.5230
114

0.01215
0.8979
114

-0.08285
0.3809
114

schrace
Schoenfeld Residual for race

-0.13550
0.1506
114

-0.11724
0.2141
114

-0.13527
0.1513
114

schwexp
Schoenfeld Residual for wexp

0.18432
0.0496
114

0.13489
0.1524
114

0.20662
0.0274
114

schmar
Schoenfeld Residual for mar

0.09382
0.3208
114

0.09654
0.3069
114

0.08061
0.3939
114

schparo
Schoenfeld Residual for paro

0.01596
0.8661
114

0.04421
0.6404
114

0.00346
0.9709
114

schprio
Schoenfeld Residual for prio

-0.06043
0.5230
114

0.01215
0.8979
114

-0.08285
0.3809
114

INTERACTIONS WITH TIME AS TIME-DEPENDENT COVARIATES
A common way of representing interaction between two variables in
a linear regression model is to include a new variable that is the product of
the two variables in question. To represent the interaction between a
covariate x and time in a Cox model, we can write

log h(t ) = α (t ) + β1 x + β 2 xt .
Factoring out the x, we can rewrite this as

log h(t ) = α (t ) + ( β1 + β 2t ) x .

CHAPTER 5

178

Estimating Cox Regression Models with PROC PHREG

In this equation, the effect of x is β1 + β2t. If β2 is positive, then the
effect of x increases linearly with time; if it’s negative, the effect decreases
linearly with time. β1 can be interpreted as the effect of x at time 0, the
origin of the process.
You can easily estimate this model by defining a time-dependent
covariate z=xt. Here’s an example. Using both the ASSESS statement and
Schoenfeld residuals, we just saw evidence that the variable AGE violates
the proportional hazards assumption, implying that the effect of AGE
varies with time since release. To incorporate that interaction using the
programming statements method, we can run the following SAS code,
which yields the results in Output 5.19:
PROC PHREG DATA=recid;
MODEL week*arrest(0)=fin age race wexp mar paro prio ageweek
/ TIES=EFRON;
ageweek=age*week;
RUN;

Output 5.19 A Model for Recidivism with Time by Age Interaction

Parameter
fin
age
race
wexp
mar
paro
prio
ageweek

Parameter
Estimate

Standard
Error

Chi-Square

Pr > ChiSq

Hazard
Ratio

1
1
1
1
1
1
1
1

-0.37823
0.03692
0.32290
-0.12224
-0.41162
-0.09293
0.09354
-0.00369

0.19129
0.03917
0.30804
0.21285
0.38212
0.19583
0.02869
0.00146

3.9096
0.8883
1.0989
0.3298
1.1604
0.2252
10.6294
6.4241

0.0480
0.3459
0.2945
0.5658
0.2814
0.6351
0.0011
0.0113

0.685
1.038
1.381
0.885
0.663
0.911
1.098
0.996

The interaction term is significant at about the .01 level, which
confirms what we found using the ASSESS statement and the Schoenfeld
residuals. But now we can go further and say something specific about
how the effect of AGE varies with time. The main effect of AGE is the
effect of that variable when WEEK is 0 (that is, when people are first
released from prison). The hazard ratio tells us that each additional year of
AGE increases the hazard of arrest by about 3.8 percent, although that
number is not significantly different from 0. To get the effect of age for any
given week, we can calculate .03692 – .00369*WEEK. For example, at 10
weeks from release, the effect is approximately 0. At 20 weeks, it’s –.0369.
At 30 weeks, it’s –.0738. And at 40 weeks, it’s –.1107. We can test whether
the effect of AGE is significantly different from 0 at any given week by

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using a TEST statement. For example, to test the effect at week 30, the
TEST statement would be
TEST age+30*ageweek=0;

This returns a chi-square value of 9.06, with a p-value of .0026.
Some textbooks recommend forming the product of the covariate
and the logarithm of time, possibly to avoid numerical overflows. But
there is no theoretical reason to prefer this specification, and the algorithm
used by PROC PHREG is robust to numerical problems. When I used the
logarithm of time with the recidivism data, the chi-square for the
interaction was higher but not by much.
As an alternative to treating time as continuous in the interaction, I
defined it as dichotomous with the following programming statement:
ageweek=age*(week>26);

In this case, the main effect of AGE (−.004 with a p-value of .88) may be
interpreted as the effect of AGE during the first 6 months of the year. To get
the effect of AGE during the second 6x months, we add the interaction to
the main effect, which yields a value of –.145 with a p-value of .0004. So it
appears that AGE has essentially no effect on the hazard of arrest during
the first 6 months but a big negative effect during the second 6 months.
To sum up, we now have a definitive way to test for violations of the
PH assumption: For any suspected covariate, simply add to the model a
time-dependent covariate representing the interaction of the original
covariate and time. If the interaction covariate does not have a significant
coefficient, then we may conclude that the PH assumption is not violated
for that variable. On the other hand, if the interaction variable does have a
significant coefficient, we have evidence for nonproportionality. Of course,
we also have a model that incorporates the nonproportionality. So, in this
case, the method of diagnosis is also the cure.

NONPROPORTIONALITY VIA STRATIFICATION
Another approach to nonproportionality is stratification, a technique
that is most useful when the covariate that interacts with time is both
categorical and not of direct interest. Consider the myelomatosis data
described in Chapter 2 and analyzed in Chapter 3. For that data, the
primary interest was in the effect of the treatment indicator TREAT. There
was also a covariate called RENAL, an indicator of whether renal
functioning was impaired at the time of randomization to treatment. In

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Chapter 3, we found that RENAL was strongly related to survival time, but
the effect of TREAT was not significant. Suppose we believe (or suspect)
that the effect of RENAL varies with time since randomization.
Alternatively, we can say that the shape of the hazard function is different
for those with and without impaired renal functioning. Letting x represent
the treatment indicator and z the renal functioning indicator, we can
represent this idea by postulating separate models for the two renal
functioning groups:
Impaired
log hi (t ) = α 0 (t ) + βxi

Not Impaired
log hi (t ) = α1 (t ) + βxi

Notice that the coefficient of x is the same in both equations, but the
arbitrary function of time is allowed to differ. We can combine the two
equations into a single equation by writing

log hi (t ) = α z (t ) + βxi .
One can easily estimate this model by the method of partial
likelihood using these steps:
1. Construct separate partial likelihood functions for each of the
two renal functioning groups.
2. Multiply those two functions together.
3. Choose values of β that maximize this function.
PROC PHREG does this automatically if you include a STRATA statement:
PROC PHREG DATA=myel;
MODEL dur*stat(0)=treat / TIES=EXACT;
STRATA renal;
RUN;

Output 5.20 shows the results. PROC PHREG first reports some
simple statistics for each of the strata; then it reports the usual output for
the regression model. In contrast to the PROC LIFETEST results of Chapter
3, we now find a statistically significant effect of the treatment. Notice,
however, that there is no estimated coefficient for RENAL. Whatever
effects this variable may have, they are entirely absorbed into the two
arbitrary functions of time.

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Output 5.20 Myelomatosis Data with Stratified Cox Regression

Summary of the Number of Event and Censored Values

Stratum

renal

Total

Event

Censored

Percent
Censored

1
0
18
10
8
44.44
2
1
7
7
0
0.00
------------------------------------------------------------------Total
25
17
8
32.00

Testing Global Null Hypothesis: BETA=0
Test
Likelihood Ratio
Score
Wald

Chi-Square

Pr > ChiSq

6.0736
5.7908
4.9254

1
1
1

0.0137
0.0161
0.0265

Analysis of Maximum Likelihood Estimates

Parameter
treat

Parameter
Estimate

Standard
Error

Chi-Square

Pr > ChiSq

Hazard
Ratio

1.46398

0.65965

4.9254

0.0265

4.323

For comparison purposes, I also estimated three other PROC PHREG
models:
 a model with TREAT as the only covariate and no stratification.
The coefficient was .56, with a p-value (based on the Wald chisquare test) of .27, which is approximately the same as the pvalue for the log-rank test reported in Chapter 3.
 a model with both TREAT and RENAL as covariates. The
coefficient of TREAT was 1.22, with a p-value of .04.
 a model that included TREAT, RENAL, and the (time-varying)
product of RENAL and DUR. The coefficient of TREAT was 1.34,
with a p-value of .05. The interaction term was not significant.

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Based on these results, it is clearly essential to control for RENAL in
order to obtain good estimates of the treatment effect, but whether you do
it by stratification or by including RENAL as a covariate doesn’t seem to
make much difference. Note that you cannot stratify by a variable and also
include it as a covariate. These are alternative ways of controlling for the
variable.
Compared with the explicit interaction method of the previous
section, the method of stratification has two main advantages:
 The interaction method requires that you choose a particular
form for the interaction, but stratification allows for any change
in the effect of a variable over time. For example, including the
product of RENAL and DUR as a covariate forces the effect of
RENAL to either increase linearly or decrease linearly with time.
Stratification by RENAL allows for reversals (possibly more than
one) in the relationship between time and the effect of RENAL.
 Stratification takes less computing time. This can be important
in working with large samples.
But there are also important disadvantages of stratification:
 No estimates are obtained for the effect of the stratifying
variable. As a result, stratification only makes sense for nuisance
variables whose effects have little or no intrinsic interest.
 There is no way to test for either the main effect of the stratifying
variable or its interaction with time. In particular, it is not
legitimate to compare the log-likelihoods for models with and
without a stratifying variable.
 If the form of the interaction with time is correctly specified, the
explicit interaction method should yield more efficient estimates
of the coefficients of the other covariates. Again, there is a tradeoff between robustness and efficiency.
Now for some complications. In the example we’ve just been
considering, the stratifying variable had only two values. If a variable with
more than two values is listed in the STRATA statement, PROC PHREG
creates a separate stratum for each value. The stratifying variable can be
either character or numeric, with no restrictions on the possible values. If
more than one variable is listed in the STRATA statement, PROC PHREG
creates a separate stratum for every observed combination of values.
Obviously you must exercise some care in using this option because you
can easily end up with strata that have only one observation. For
stratifying variables that are numeric, it is also easy to specify a limited
number of strata that correspond to cut points on the variable. The syntax
for accomplishing this is identical to the description in Chapter 3 for the
STRATA statement in PROC LIFETEST.

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In my view, the most useful application of stratification is for
samples involving some kind of clustering or multi-level grouping.
Examples include animals within litters, children within schools, patients
within hospitals, and so on. In all of these situations, it is reasonable to
expect that observations within a cluster will not be truly independent,
thereby violating one of the standard assumptions used in constructing the
likelihood function. The result is likely to be standard error estimates that
are biased downward and test statistics that are biased upward. A highly
effective solution is to treat each cluster as a distinct stratum. You can
accomplish this by giving every cluster a unique identification number
and then stratifying on the variable containing that number. This method
assumes that observations are conditionally independent within clusters
and that the coefficients of the covariates are the same across clusters. (The
conditional independence assumption would be violated if, for example,
the death of one patient had a deleterious effect on the risk of death for
other patients in the same hospital.) In Chapter 8, we will see an
alternative method for dealing with clustering.

LEFT TRUNCATION AND LATE ENTRY

INTO THE

RISK SET

In standard treatments of partial likelihood, it is assumed that every
individual is at risk of an event at time 0 and continues to be at risk until
either the event of interest occurs or the observation is censored. At that
point the individual is removed from the risk set, never to return. It doesn’t
have to be this way, however. If there are periods of time when it is known
that a particular individual is not at risk for some reason, the partial
likelihood method allows for the individual to be removed from the risk
set and then returned at some later point in time. Unfortunately, many
partial likelihood programs are not set up to handle this possibility.
The most common situation is left truncation or late entry to the risk
set. This situation often arises without the investigator even realizing it.
Here are three examples:
 Patients are recruited into a study at varying lengths of time after
diagnosis with some disease. The investigator wishes to specify a
model in which the hazard of death depends on the time since
diagnosis. But by the design of the study, it is impossible for the
patients to die between diagnosis and recruitment. If they had
died, they would not have been available for recruitment.
 At a specific point in time, all employees in a firm are
interviewed and asked how long they have been working for the
firm. Then, they are followed forward to determine how long

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they stay with the firm. The investigator estimates a model in
which the hazard of termination depends on the length of
employment with the firm. But again, it is impossible for these
employees to have terminated before the initial interview. If they
had terminated, they would not have been present for that
interview.
 High school girls are recruited into a study of teen pregnancy.
The desired model expresses the hazard of pregnancy as a
function of age. But girls were not included in the study if they
had become pregnant before the study began. Thus, the girls in
the study were not observationally at risk of pregnancy at ages
before the initial interview. Obviously, they were at risk during
those ages, but the design of the study makes it impossible to
observe pregnancies that occurred before the initial interview.
In all three cases, the solution is to remove the individual from the
risk set between the point of origin and the time of the initial contact,
which you can easily accomplish using the counting process syntax in
PROC PHREG.
To illustrate these methods, let’s specify a model for the Stanford
Heart Transplant Data in which the hazard is expressed as a function of
patient age rather than time since acceptance into the program. To do this,
we must first convert the death time variable into patient age (in years):
DATA stan2;
SET stan;
agels=(dls-dob)/365.25;
RUN;

Recall that DLS is date last seen and DOB is date of birth. Then
AGELS is age at which the patient was last seen, which may be either a
death time or a censoring time. Our problem is that, by design, patients
were not at risk of an observable death before they were accepted into the
program.
Using the counting process syntax, we can specify a model in the
following way:
PROC PHREG DATA=stan2;
MODEL (ageaccpt,agels)*dead(0)=surg ageaccpt / TIES=EFRON;
RUN;

Here, AGEACCPT is the time of entry into the risk set and AGELS is the
time of departure. Note that AGEACCPT can also be included as a
covariate.
It is also possible to combine the counting process syntax with the
programming statements method for defining a time-dependent covariate.

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The following program defines PLANT as a time-dependent indicator of
transplant status, with results shown in Output 5.21:
PROC PHREG DATA=stan2;
MODEL (ageaccpt,agels)*dead(0)=plant surg ageaccpt / TIES=EFRON;
IF agetrans>=agels OR agetrans=. THEN plant=0;
ELSE plant=1;
RUN;

Compare the results in Output 5.21 with those in Output 5.11, which had
the same covariates but specified the principal time axis as time since
acceptance. For PLANT and SURG, the results are very similar. But there is
a dramatic increase in the effect of age at acceptance, which is a direct
consequence of specifying age as the time axis. With age held constant
(every risk set consists of people who are all the same age), age at
acceptance is an exact linear function of time since acceptance. In the
earlier analysis, any effects of time since acceptance were part of the
baseline hazard function and were, therefore, not visible in the estimated
coefficients.

Output 5.21 Results for Transplant Data with Age as the Principal Time Axis

Parameter
plant
surg
ageaccpt

Parameter
Estimate

Standard
Error

Chi-Square

Pr > ChiSq

Hazard
Ratio

1
1
1

-0.28042
-0.85312
0.75857

0.43761
0.46529
0.28338

0.4106
3.3618
7.1655

0.5217
0.0667
0.0074

0.755
0.426
2.135

So far, I have explained how to handle situations in which
individuals who are not in the risk set at time 0 enter the risk set at some
later point in time and then remain in the risk set until an event occurs.
Now let’s consider situations in which individuals temporarily leave the
risk set for some reason and then return at a later point in time. Suppose,
for example, that a sample of children is followed for a period of 5 years to
detect any occurrences of parental abuse. If some children are placed in
foster homes for temporary periods, it may be desirable to remove them
from the risk set for those periods.
To accomplish this, a separate record is created for each interval
during which the individual was continuously at risk. Thus, for a child
who had one temporary period in a foster home, the first record
corresponds to the interval from time 0 (however defined) to the time of
entry into the foster home. This record is coded as censored because the
event of interest did not occur. The second record is for the interval from

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the time of exit from the foster home to the last time of observation, coded
as censored if no event occurred at the termination of observation. The
MODEL statement then uses the counting process syntax:
MODEL (start, finish)*CENSOR(0)=X1 X2;

Here, start is the time at which the interval begins (measured from the
origin) and finish is the time at which the interval ends. The extension to
multiple time outs should be straightforward.

ESTIMATING SURVIVOR FUNCTIONS
As we have seen, the form of the dependence of the hazard on time
is left unspecified in the proportional hazards model. Furthermore, the
partial likelihood method discards that portion of the likelihood function
that contains information about the dependence of the hazard on time.
Nevertheless, it is still possible to get nonparametric estimates of the
survivor function based on a fitted proportional hazards model.
When there are no time-dependent covariates, the Cox model can be
written as

Si (t ) = [S 0 (t )]

exp( βx i )

(5.14)

where Si(t) is the probability that individual i with covariate values xi
survives to time t. S0(t) is called the baseline survivor function (that is, the
survivor function for an individual whose covariate values are all 0). After
β is estimated by partial likelihood, S0(t) can be estimated by a
nonparametric maximum likelihood method. With that estimate in hand,
you can generate the estimated survivor function for any set of covariate
values by substitution into equation (5.14).
In PROC PHREG, this is accomplished with the BASELINE statement
and the PLOTS option. The easiest task is to get the survivor function for
x i = x , the vector of sample means. For the recidivism data set, the SAS
code for accomplishing this task is as follows:
ODS GRAPHICS ON;
PROC PHREG DATA=recid PLOTS=S;
MODEL week*arrest(0)=fin age prio
/ TIES=EFRON;
BASELINE OUT=a SURVIVAL=s LOWER=lcl UPPER=ucl;
RUN;
ODS GRAPHICS OFF;

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In the BASELINE statement, the OUT=A option puts the survival
estimates in a temporary data set named A. SURVIVAL=S stores the
survival probabilities in a variable named S. The UPPER and LOWER
options store the upper and lower 95% confidence limits in variables UCL
and LCL, respectively.
Data set A is printed in Output 5.22. There are 50 records,
corresponding to the 49 unique weeks in which arrests were observed to
occur, plus an initial record for time 0. Each record gives the mean value
for each of the three covariates. The S column gives the estimated survival
probabilities. The last two columns are the upper and lower 95%
confidence limits. The survivor function reported here is very similar to
one that would be produced by PROC LIFETEST using the Kaplan-Meier
method, but it is not identical. This one allows for heterogeneity in the
hazard/survivor functions while the Kaplan-Meier estimator does not.

Output 5.22

Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25

fin
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5

Survivor Function Estimates for Recidivism Data at Sample Means
age
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972

prio
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380

week
0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24

s
1.00000
0.99801
0.99602
0.99403
0.99204
0.99005
0.98805
0.98605
0.97610
0.97209
0.97009
0.96606
0.96197
0.95993
0.95381
0.94970
0.94558
0.93942
0.93321
0.92904
0.91865
0.91448
0.91240
0.91031
0.90198

lcl
.
0.99411
0.99050
0.98728
0.98425
0.98134
0.97851
0.97573
0.96258
0.95747
0.95494
0.94991
0.94487
0.94237
0.93494
0.92999
0.92507
0.91778
0.91047
0.90560
0.89356
0.88876
0.88636
0.88396
0.87445

ucl
.
1.00000
1.00000
1.00000
0.99989
0.99884
0.99769
0.99648
0.98981
0.98694
0.98547
0.98248
0.97939
0.97783
0.97307
0.96983
0.96654
0.96158
0.95651
0.95309
0.94445
0.94096
0.93920
0.93744
0.93037

(continued)

CHAPTER 5

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Output 5.22 (continued)

26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50

0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5
0.5

24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972
24.5972

2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380
2.98380

25
26
27
28
30
31
32
33
34
35
36
37
38
39
40
42
43
44
45
46
47
48
49
50
52

0.89573
0.88938
0.88514
0.88090
0.87665
0.87451
0.87024
0.86596
0.86165
0.85302
0.84653
0.83787
0.83569
0.83133
0.82262
0.81823
0.80948
0.80505
0.80061
0.79171
0.78947
0.78500
0.77384
0.76712
0.75812

0.86735
0.86017
0.85539
0.85062
0.84584
0.84345
0.83867
0.83389
0.82909
0.81953
0.81235
0.80281
0.80042
0.79563
0.78610
0.78130
0.77177
0.76696
0.76214
0.75251
0.75010
0.74528
0.73328
0.72608
0.71644

0.92504
0.91959
0.91593
0.91227
0.90857
0.90672
0.90300
0.89925
0.89547
0.88788
0.88214
0.87446
0.87252
0.86863
0.86085
0.85691
0.84903
0.84503
0.84102
0.83294
0.83091
0.82684
0.81665
0.81049
0.80221

The PLOTS=S option produces a graph of the adjusted survivor
function, evaluated at the means of the covariates, displayed in Output
5.23. The PLOTS option only works with ODS graphics, which is why the
PROC PHREG step is preceded and followed by ODS statements. The
“Reference Setting” in the graph title refers to the mean values of the
covariates. However, if the MODEL statement contains any CLASS
variables, the survivor function is evaluated at the reference levels of those
variables rather than at their means. In a moment, we will see how to
evaluate and graph the survivor function at other specified covariate
values.

CHAPTER 5

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Output 5.23 Graph of the Adjusted Survivor Function for Recidivism Data

By combining the BASELINE statement with stratification, we can
also produce graphs that compare survivor functions for two or more
groups, while adjusting for covariates. Suppose we take financial aid (FIN)
as the stratifying variable for the recidivism data. That might seem selfdefeating because FIN is the variable of greatest interest and stratifying on
it implies that no tests or estimates of its effect are produced. But after
stratifying, we can graph the baseline survivor functions for the two
financial aid groups using the following code:
ODS GRAPHICS ON;
PROC PHREG DATA=recid PLOTS(OVERLAY=ROW)=S;
MODEL week*arrest(0)=fin age prio
/ TIES=EFRON;
STRATA fin;
RUN;
ODS GRAPHICS OFF;

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The OVERLAY option is necessary to get the two survival curves on the
same graph. Note that the BASELINE statement used in the previous
program is not necessary if you only want the graphs. The resulting figure
in Output 5.24 shows the survivor functions for each of the two financial
aid groups, evaluated at the means of the covariates.

Output 5.24

Survivor Plots for the Two Financial Aid Groups

Another major use of the baseline survivor function is to obtain
predictions about survival time for particular sets of covariate values.
These covariate values need not appear in the data set being analyzed. For
the recidivism data, for example, we may want to say something about
arrest times for 20-year-olds with four prior convictions who did not
receive financial aid. The mechanics of doing this are a bit awkward. You
must first create a new data set containing the values of the covariates for
which you want predictions and then pass the name of that data set to
PROC PHREG:
DATA covals;
INPUT fin age prio;
DATALINES;
0 40 3
;

CHAPTER 5

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PROC PHREG DATA=recid;
MODEL week*arrest(0)=fin age prio / TIES=EFRON;
BASELINE OUT=a COVARIATES=covals SURVIVAL=s LOWER=lcl
UPPER=ucl;
PROC PRINT DATA=a;
RUN;

The advantage of doing it this way is that predictions can easily be
generated for many different sets of covariate values just by including
more input lines in the data set COVALS. Each input line produces a
complete set of survivor estimates, but all estimates are output to a single
data set. The LOWER= and UPPER= options give 95% confidence intervals
around the survival probability.
Output 5.25 displays a portion of the data set generated by the
BASELINE statement above. In generating predictions, it’s typical to focus
on a single summary measure rather than the entire distribution. The
median survival time is easily obtained by finding the smallest value of t
such that S(t) ≤ .50. That won’t work for the recidivism data, however,
because the data are censored long before a .50 probability is reached. For
these data, it’s probably more useful to pick a fixed point in time and
calculate survival probabilities at that time under varying conditions. For
the covariate values in Output 5.25, the 6-month (26-week) survival
probability is .82, with a 95% confidence interval of .77 to .88.

Output 5.25

Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16

Portion of Survivor Function Estimate for Recidivism Data
fin
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

age
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20
20

prio
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4
4

week

lcl

ucl

0
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15

1.00000
0.99644
0.99289
0.98935
0.98582
0.98229
0.97875
0.97522
0.95769
0.95068
0.94718
0.94017
0.93308
0.92954
0.91898
0.91191

.
0.98948
0.98305
0.97730
0.97189
0.96671
0.96168
0.95675
0.93345
0.92444
0.92000
0.91117
0.90232
0.89794
0.88497
0.87636

.
1.00000
1.00000
1.00000
0.99994
0.99811
0.99613
0.99404
0.98257
0.97767
0.97517
0.97009
0.96489
0.96226
0.95429
0.94890

(continued)

CHAPTER 5

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192
Output 5.25 (continued)

17
18
19
20
21
22
23
24
25
26
27
28
29
30

0
0
0
0
0
0
0
0
0
0
0
0
0
0

20
20
20
20
20
20
20
20
20
20
20
20
20
20

4
4
4
4
4
4
4
4
4
4
4
4
4
4

16
17
18
19
20
21
22
23
24
25
26
27
28
30

0.90486
0.89435
0.88381
0.87677
0.85932
0.85237
0.84891
0.84543
0.83166
0.82140
0.81102
0.80413
0.79726
0.79039

0.86783
0.85524
0.84267
0.83433
0.81381
0.80570
0.80166
0.79762
0.78169
0.76989
0.75799
0.75011
0.74227
0.73445

0.94346
0.93525
0.92695
0.92137
0.90738
0.90175
0.89894
0.89611
0.88482
0.87634
0.86776
0.86204
0.85631
0.85058

Note that the BASELINE statement will not produce any output
when there are time-dependent covariates defined by the programming
statements method. The BASELINE statement will produce output when
time-dependent covariates are handled with the counting process method,
although the log window will contain the following warning: NOTE:
Since the counting process style of response was specified
in the MODEL statement, the SURVIVAL= statistics in the
BASELINE statement should be used with caution. For most

applications, however, this should not be a concern.

TESTING LINEAR HYPOTHESES WITH CONTRAST
STATEMENTS

TEST

In Chapter 4, I discussed the CLASS statement in PROC LIFEREG,
and I showed how to do non-standard hypothesis tests using the estimated
covariance matrix of the coefficients. PROC PHREG also has a CLASS
statement (in SAS 9.2 and later), but it has many more options than the
one in PROC LIFEREG. PROC PHREG also has a TEST statement and a
CONTRAST statement that make it easy to test any linear hypothesis about
more than one coefficient. As we shall see, these two statements are
somewhat redundant. Almost anything you can do with one can be done
with the other. Furthermore, if you are already familiar with these
statements from other SAS procedures (for example, REG, GLM,
LOGISTIC, GENMOD), you won’t find much new in this section.
As an example, let’s treat the variable EDUC as a CLASS variable in a
model for the recidivism data:

CHAPTER 5

Estimating Cox Regression Models with PROC PHREG

193

PROC PHREG DATA=recid;
CLASS educ;
MODEL week*arrest(0)=fin age prio educ/ TIES=EFRON;
TEST educ3=educ5;
CONTRAST 'ed3 vs. ed5' educ 0 1 0 -1 ;
RUN;

The EDUC variable has five different integer values: 2, 3, 4, 5, and 6.
The CLASS statement creates a set of dummy variables to represent four
out of the five categories. (In this context, they are sometimes referred to as
design variables.) By default, the omitted (reference) category is the one
with the highest value, in this case 6. You can override this default by
specifying, for example,
CLASS educ(REF='2');

which makes level 2 the reference category.
The TEST statement tests the null hypotheses that the coefficient for
level 3 is the same as the coefficient for level 5. Note that you refer to the
coefficients by appending the variable value to the variable name. The
CONTRAST statement accomplishes exactly the same task but with a very
different syntax. The text enclosed in single quotes is a required label
describing the test. It can be any text that you choose. The set of numbers
following the variable name encodes the following instructions: Take the
coefficient for the second dummy variable (EDUC3) and multiply it by 1.
Take the coefficient for the fourth dummy variable (EDUC5) and multiply
it by −1. Add the two numbers together and test whether the sum is equal
to 0. Of course, this is equivalent to testing whether the two coefficients
are equal to each other.
Output 5.26 displays the results. When a model contains a CLASS
variable, PROC PHREG reports a Type 3 table. For variables that are not
CLASS variables, this table is completely redundant with the usual
regression table below. For the CLASS variable, however, it gives us the
very useful test that all the coefficients associated with this variable are 0.
What makes this test particularly attractive is that it does not depend on
the choice of the reference category. In this case, with a chi-square of 4.53
and 4 degrees of freedom, the test is far from statistically significant.
In the regression table, we see coefficients and hazard ratios for
levels 2 through 5 of EDUC. Each of these is a comparison with the omitted
level 6. For example, the hazard ratio for level 3 is 2.738. This means that
the hazard of arrest for someone at level 3 is almost three times the hazard
for someone at level 6. However, the difference is not statistically
significant, nor are any of the other differences with level 6. Below the
regression table, we see the output from the TEST statement, which shows

CHAPTER 5

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194

that the difference between level 3 and level 5 is not statistically
significant. Below that is the same result from the CONTRAST statement.
You can include as many TEST or CONTRAST statements as you
like. Also, a single CONTRAST or TEST statement can test more than one
hypothesis at a time—just separate the hypotheses by commas. You’ll get a
single test statistic and p-value for a simultaneous test that all the
hypotheses are true.

Output 5.26 Results from Using the CLASS Statement for the Recidivism Data

Type 3 Tests

Effect
fin
age
prio
educ

Wald
Chi-Square

Pr > ChiSq

1
1
1
4

3.6040
8.1609
8.6708
4.5257

0.0576
0.0043
0.0032
0.3395

Analysis of Maximum Likelihood Estimates

Parameter
fin
age
prio
educ
educ
educ
educ

2
3
4
5

Parameter
Estimate

1
1
1
1
1
1
1

-0.36314
-0.05986
0.08355
0.46192
1.00730
0.77558
0.28079

Standard
Error Chi-Square Pr > ChiSq
0.19128
0.02095
0.02837
1.12059
1.00996
1.01952
1.09597

3.6040
8.1609
8.6708
0.1699
0.9948
0.5787
0.0656

Hazard
Ratio Label

0.0576
0.0043
0.0032
0.6802
0.3186
0.4468
0.7978

Linear Hypotheses Testing Results

Label
Test 1

Wald
Chi-Square

Pr > ChiSq

2.4159

0.1201

Contrast Test Results

Contrast
ed3 vs. ed5

Wald
Chi-Square

Pr > ChiSq

2.4159

0.1201

0.695
0.942
1.087
1.587
2.738
2.172
1.324

educ
educ
educ
educ

2
3
4
5

CHAPTER 5

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195

CUSTOMIZED HAZARD RATIOS
As we have seen, for each predictor variable PROC PHREG reports a
hazard ratio (HR), which is just the exponentiated value of its coefficient.
And if we further calculate 100(HR-1), we get the percentage change in the
hazard of the event for a 1-unit increase in that particular variable. Two
problems can arise, however. First, a 1-unit change may be either too small
or too large to be usefully interpreted. If a person’s income is measured in
dollars, for example, a $1 increase will most likely produce a very tiny
percentage change in the hazard. On the other hand, if a lab measurement
varies only between 0 and 0.1, a 1-unit increase (from 0 to 1) will be 10
times the observed range of the variable. Although you can do hand
calculations to get the hazard ratio for different units (or modify the units
of the variables before estimating the model), the HAZARDRATIO
statement (available in SAS 9.2 and later) makes things much easier.
Let’s apply these options to the recidivism data. In Output 5.1, we
saw a hazard ratio of .944 for AGE, which we interpreted to mean that each
1-year increase in age was associated with a 5.6 percent reduction in the
hazard of arrest. Suppose, instead, that we want the hazard ratios for a 5year increase and a 10-year increase in age. We can get that with the
following code:
PROC PHREG DATA=my.recid;
MODEL week*arrest(0)=fin age race wexp mar paro prio;
HAZARDRATIO age / UNITS=5 10;
RUN;

In addition to the usual output, we now get Output 5.27, which tells us
that a 5-year increase in age is associated with a 25 percent reduction in
the hazard of arrest, while a 10-year increase in age is associated with a 44
percent reduction.

Output 5.27 Results from Using the HAZARDRATIO Statement for the Recidivism Data

Hazard Ratios for age

Description

Point
Estimate

age Unit=5
age Unit=10

0.751
0.564

95% Wald Confidence
Limits
0.606
0.367

0.932
0.868

CHAPTER 5

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The HAZARDRATIO statement also solves another problem. More
recent versions of PROC PHREG allow for interactions to be directly
specified in the MODEL statement. But if you include an interaction,
PROC PHREG does not report hazard ratios for either the interaction itself
or the main effects associated with the interaction. For example, suppose
that we fit a model with an interaction between FIN and AGE, which
produces the results in Output 5.28:
PROC PHREG DATA=my.recid;
MODEL week*arrest(0)=fin age race wexp mar paro prio fin*age;
RUN;

Output 5.28 Model with Interaction between FIN and AGE

Parameter DF
fin
age
race
wexp
mar
paro
prio
fin*age

1
1
1
1
1
1
1
1

Parameter
Estimate
1.62266
-0.02263
0.32112
-0.15996
-0.45026
-0.08408
0.09380
-0.08832

Standard
Error Chi-Square Pr > ChiSq
1.02829
0.02595
0.30827
0.21231
0.38264
0.19601
0.02846
0.04504

2.4902
0.7600
1.0851
0.5677
1.3846
0.1840
10.8618
3.8449

0.1146
0.3833
0.2976
0.4512
0.2393
0.6680
0.0010
0.0499

Hazard
Ratio Label
.
.
1.379
0.852
0.637
0.919
1.098
.
fin * age

The interaction is just barely significant at the .05 level. Like all twoway interactions, this one can be interpreted in two different ways: the
effect of AGE depends on the level of FIN and the effect of FIN depends on
the level of AGE. The latter makes more sense because AGE is
predetermined and FIN is an experimental treatment. So it is natural to ask
how the effect of the treatment depends on the characteristics of the
subjects. Output 5.28 is not helpful because it doesn’t give us hazard ratios
for either the interaction or the two main effects. To answer this question,
we can use the HAZARDRATIO statement to give us hazard ratios for FIN
at various selected ages. Here’s how:
HAZARDRATIO fin / at (age=20 25 30 35 40) CL=PL;

The CL=PL option is not essential here. It requests that confidence
intervals be computed using the profile likelihood method, which is
somewhat more accurate than the conventional Wald method, especially
in small samples.

CHAPTER 5

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197
Output 5.29 Hazard Ratios for FIN by AGE

Hazard Ratios for fin

Point
Estimate

Description
fin
fin
fin
fin
fin

Unit=1
Unit=1
Unit=1
Unit=1
Unit=1

At
At
At
At
At

age=20
age=25
age=30
age=35
age=40

0.866
0.557
0.358
0.230
0.148

95% Profile
Likelihood
Confidence Limits
0.559
0.351
0.157
0.065
0.027

1.333
0.854
0.743
0.701
0.676

Results are shown in Output 5.29. At age 20, the hazard ratio is not
significantly different from 1 (as indicated by the 95% confidence interval,
which includes 1). For later ages, all the odds ratios are significantly
different from 1. The differences are striking. At age 25, receipt of financial
aid reduces the hazard of arrest by 44 percent. At age 40, receipt of
financial aid reduces the hazard by 85 percent.

BAYESIAN ESTIMATION AND TESTING
Beginning with SAS 9.2, PROC PHREG can do a Bayesian analysis of
the Cox regression model. We discussed Bayesian analysis for PROC
LIFEREG in the last section of Chapter 4. Because the theory, syntax, and
output for Bayesian analysis are virtually identical for these two
procedures, I will not go into all the details here.
As with PROC LIFEREG, requesting a Bayesian analysis in PROC
PHREG can be as simple as including the BAYES statement after the
MODEL statement. Here’s how to do it for the recidivism data:
PROC PHREG DATA=my.recid;
MODEL week*arrest(0)=fin age race wexp mar paro prio;
BAYES;
RUN;

The default is to use non-informative priors for the coefficients and to
produce 10,000 random draws from the posterior distribution using a
Gibbs sampler algorithm. Output 5.30 displays some key summary
statistics for the posterior distribution of the regression coefficients.

CHAPTER 5

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Output 5.30 Results from Using the BAYES Statement for the Recidivism Data

Bayesian Analysis
Posterior Summaries

Parameter
fin
age
race
wexp
mar
paro
prio

Mean

Standard
Deviation

10000
10000
10000
10000
10000
10000
10000

-0.3823
-0.0593
0.3520
-0.1542
-0.4798
-0.0868
0.0889

0.1927
0.0220
0.3133
0.2113
0.3959
0.1947
0.0286

Percentiles
25%
50%
-0.5111
-0.0738
0.1371
-0.2988
-0.7338
-0.2180
0.0700

-0.3806
-0.0586
0.3434
-0.1534
-0.4593
-0.0887
0.0897

75%

-0.2494
-0.0443
0.5526
-0.00844
-0.2022
0.0435
0.1084

Posterior Intervals
Parameter

Alpha

Equal-Tail Interval

fin
age
race
wexp
mar
paro
prio

0.050
0.050
0.050
0.050
0.050
0.050
0.050

-0.7655
-0.1047
-0.2310
-0.5708
-1.3054
-0.4669
0.0315

-0.00737
-0.0186
1.0003
0.2548
0.2385
0.3009
0.1435

HPD Interval
-0.7572
-0.1028
-0.2634
-0.5734
-1.2621
-0.4469
0.0310

-0.00339
-0.0175
0.9563
0.2499
0.2575
0.3194
0.1425

As expected, the results from the Bayesian analysis do not differ
much from the conventional analysis in Output 5.1. Either the means or
medians (50th percentiles) of the posterior could be used as point
estimates, and both are about the same as the conventional partial
likelihood estimates. The standard deviations in Output 5.30 are about the
same as the standard errors in Output 5.1. The big difference is in
computing time, which increases by a factor of nearly 1,000 when the
BAYES statement is used. There are a lot of additional tables in the default
output that are not shown here, including fit statistics, autocorrelations,
and Geweke diagnostics. And if you want to see graphs of the posterior
distribution, you will need ODS statements before and after the code
above, as shown in the last section of Chapter 4.
One surprising thing you can do with the BAYES statement is
estimate a piecewise exponential model, which we discussed in Chapter 4

CHAPTER 5

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199

and estimated with PROC LIFEREG. For the piecewise exponential model,
a special case of the Cox model, we divide the time scale into intervals
with cut points a0, a1, … , aJ, where a0 = 0, and aJ = ∞. We assume that the
hazard is constant within each interval but can vary across intervals. The
hazard for individual i is assumed to have the form

log hi (t ) = α j + βx i

for aj-1 ≤ t < aj

Thus, the intercept in the equation is allowed to vary in an unrestricted
fashion from one interval to another.
You can do a Bayesian analysis of this model by putting the
PIECEWISE option in the BAYES statement. On the other hand, if you only
want maximum likelihood estimates, you can suppress the Gibbs sampler
iterations with the statement
BAYES PIECEWISE NBI=0 NMC=1;

which gives the maximum likelihood estimates in Output 5.31. The default
is to construct eight intervals with approximately equal numbers of events
in each interval, as shown in the table. However, you can directly specify
either the number of intervals or the cut points for the intervals.

Output 5.31 Maximum Likelihood Estimates of Piecewise Exponential Model

Constant Hazard Time Intervals
Interval
[Lower,
Upper)
0
11.5
18.5
24.5
30.5
36.5
42.5
47.5

11.5
18.5
24.5
30.5
36.5
42.5
47.5
Infty

Event

17
16
15
12
14
13
13
332

17
16
15
12
14
13
13
14

Log Hazard
Parameter
Alpha1
Alpha2
Alpha3
Alpha4
Alpha5
Alpha6
Alpha7
Alpha8

(continued)

CHAPTER 5

Estimating Cox Regression Models with PROC PHREG

200
Output 5.31 (continued)

Maximum Likelihood Estimates

Parameter
Alpha1
Alpha2
Alpha3
Alpha4
Alpha5
Alpha6
Alpha7
Alpha8
fin
age
race
wexp
mar
paro
prio

Estimate

Standard
Error

1
1
1
1
1
1
1
1
1
1
1
1
1
1
1

-4.5804
-4.0752
-3.9299
-4.1037
-3.9063
-3.9233
-3.6959
-3.4703
-0.3801
-0.0575
0.3131
-0.1483
-0.4327
-0.0839
0.0917

0.6295
0.6316
0.6346
0.6457
0.6376
0.6433
0.6462
0.6432
0.1914
0.0220
0.3080
0.2123
0.3820
0.1957
0.0287

95% Confidence Limits
-5.8142
-5.3131
-5.1737
-5.3692
-5.1560
-5.1841
-4.9626
-4.7310
-0.7552
-0.1006
-0.2904
-0.5645
-1.1813
-0.4675
0.0355

-3.3466
-2.8373
-2.6861
-2.8381
-2.6567
-2.6625
-2.4293
-2.2095
-0.00507
-0.0144
0.9167
0.2678
0.3159
0.2998
0.1479

CONCLUSION
It’s no accident that Cox regression has become the overwhelmingly
favored method for doing regression analysis of survival data. It makes no
assumptions about the shape of the distribution of survival times. It allows
for time-dependent covariates. It is appropriate for both discrete-time and
continuous-time data. It easily handles left truncation. It can stratify on
categorical control variables. And, finally, it can be extended to
nonproportional hazards. The principal disadvantage is that you lose the
ability to test hypotheses about the shape of the hazard function. As we’ll
see in Chapter 8, however, the hazard function is often so confounded with
unobserved heterogeneity that it’s difficult to draw any substantive
conclusion from the shape of the observed hazard function. So the loss
may not be as great as it seems.
PROC PHREG goes further than most Cox regression programs in
realizing the full potential of this method. In this chapter, I have
particularly stressed those features of PROC PHREG that most distinguish
it from other Cox regression programs: its extremely general capabilities for
time-dependent covariates, its exact methods for tied data, and its ability
to handle left truncation. If you already have some familiarity with PROC
PHREG, you may have noticed that I said nothing about its several variable

CHAPTER 5

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201

selection methods. I confess that I have never used them. While I am not
totally opposed to such automated model-building methods, I think they
should be reserved for those cases in which there are a large number of
potential covariates, little theoretical guidance for choosing among them,
and a goal that emphasizes prediction rather than hypothesis testing. If
you find yourself in this situation, you may want to consider the best
subset selection method that uses the SELECTION=SCORE and BEST=
options. This method is computationally efficient because no parameters
are estimated. Yet, it gives you lots of useful information.
We still haven’t finished with PROC PHREG. Chapter 6, “Competing
Risks,” shows you how to use it for competing risks models in which there
is more than one kind of event. Chapter 8, “Heterogeneity, Repeated
Events, and Other Topics,” considers PROC PHREG’s capabilities for
handling repeated events.

CHAPTER 5

202

Estimating Cox Regression Models with PROC PHREG

CHAPTER 6
Competing Risks
Introduction 203
Type-Specific Hazards 204
Time in Power for Leaders of Countries: Example 207
Estimates and Tests without Covariates 208
Covariate Effects via Cox Models 213
Accelerated Failure Time Models 220
Alternative Approaches to Multiple Event Types 227
Conclusion 232

INTRODUCTION
In the previous chapters, all the events in each analysis were treated
as if they were identical: all deaths were the same, all job terminations
were the same, and all arrests were the same. In many cases, this is a
perfectly acceptable way to proceed. But more often than not, it is
essential—or at least desirable—to distinguish different kinds of events
and treat them differently in the analysis. To evaluate the efficacy of heart
transplants, you will certainly want to treat deaths due to heart failure
differently from deaths due to accident or cancer. Job terminations that
occur when an employee quits are likely to have quite different
determinants than those that occur when the employee is fired. And
financial aid to released convicts will more plausibly reduce arrests for
theft or burglary than for rape or assault.
In this chapter, we consider the method of competing risks for
handling these kinds of situations. What is most characteristic of
competing risk situations is that the occurrence of one type of event
removes the individual from risk of all the other event types. People who
die of heart disease are no longer at risk of dying of cancer. Employees who
quit can no longer be fired.
Because competing risk analysis requires no new SAS procedures,
this chapter will be short. With only minor changes in the SAS statements,
you can estimate competing risks models with the LIFETEST, LIFEREG, or
PHREG procedures. All that’s necessary is to change the code that specifies
which observations are censored and which are not. After considering a bit
of the theory of competing risks, most of the chapter will be devoted to

CHAPTER 6

Competing Risks

204

examples. At the end, however, I also discuss some alternative to
competing risks analysis that may be more appropriate for some situations.
Despite the practical importance of methods for handling multiple
kinds of events, many textbooks give minimal coverage to this topic. For
example Collett’s (2003) otherwise excellent survey of survival analysis
makes only a brief mention of competing risks. In this chapter, I have
relied heavily on Kalbfleisch and Prentice (2002) and Cox and Oakes
(1984).

TYPE-SPECIFIC HAZARDS
The classification of events into different types is often somewhat
arbitrary and may vary according to the specific goals of the analysis. I’ll
have more to say about that later. For now, let’s suppose that the events
that we are interested in are deaths, and we have classified them into five
types according to cause: heart disease, cancer, stroke, accident, and a
residual category that we’ll call other. Let’s assign the numbers 1 through
5, respectively, to these death types. For each type of death, we are going
to define a separate hazard function that we’ll call a type-specific or causespecific hazard.
As before, let Ti be a random variable denoting the time of death for
person i. Now let Ji be a random variable denoting the type of death that
occurred to person i. Thus, J5=2 means that person 5 died of cancer. We
now define hij(t), the hazard for death type j at time t for person i, as
follows:

hij (t ) = lim

∆t → 0

Pr(t < Ti < t + ∆t , J i = j | Ti ≥ t )
,
∆t

j = 1,,5

(6.1)

Comparing this with the definition of the hazard in equation (2.2), we see
that the only difference is the appearance of Ji =j. Thus, the conditional
probability in equation (6.1) is the probability that death occurs between t
and t+∆t, and the death is of type j, given that the person had not already
died by time t. The overall hazard of death is just the sum of all the typespecific hazards:
hi (t ) = ∑ hij (t ) .

(6.2)

You can interpret type-specific hazards in much the same way as
ordinary hazards. Their metric is the number of events per unit interval of
time, except now the events are of a specific type. The reciprocal of the

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205

hazard is the expected length of time until an event of that type occurs,
assuming that the hazard stays constant.
Based on the type-specific hazards, we can also define type-specific
survival functions:

 t
S j (t ) = exp− ∫ h j (u )du  .

 0

(6.3)

The interpretation of these functions is somewhat controversial, however.
Lawless (2003) claims that they don’t refer to any well-defined random
variable and that they are useful only as a way of examining hypotheses
about the hazard. On the other hand, we can define the hypothetical
variable Tij as the time at which the jth event type either occurred to the
ith person or would have occurred if other event types had not preceded it.
In other words, we suppose that a person who dies of cancer at time T2
would have later died of heart disease at time T1 if the cancer death had
not occurred. For a given set of Tijs, we only observe the one that is
smallest. If we further assume that the Tijs are independent across event
types, we can say that
Sij (t ) = Pr(Tij > t ) .

(6.4)

That is, the type-specific survival function gives the probability that an
event of type j occurs later than time t. You can see why people might want
to forgo this interpretation, but I personally find it meaningful.
Now that we have the type-specific hazards, we can proceed to
formulate models for their dependence on covariates. Any of the models
that we have considered so far—both proportional hazards models and
accelerated failure time models—are possible candidates. For example, we
can specify general proportional hazards models for all five death types:
log hij (t ) = α j (t ) + β j x i (t ),

j = 1,  ,5.

(6.5)

where xi(t) is a vector of covariates, some of which may vary with time.
Note that the coefficient vector β is subscripted to indicate that the effects
of the covariates may be different for different death types. In particular,
some coefficients may be set to 0, thereby excluding the covariate for a
specific death type. The α(t) function is also subscripted to allow the
dependence of the hazard on time to vary across death types. If β j and αj(t)
are the same for all j, then the model reduces to the proportional hazards
model of Chapter 5 where no distinction is made among event types.
Although it is a bit unusual, there is nothing to prevent you from
choosing, say, a log-normal model for heart disease, a gamma model for
cancer, and a proportional hazards model for strokes. What makes this
possible is that the models may be estimated separately for each event

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206

type, with no loss of statistical precision. This is, perhaps, the most
important principle of competing risks analysis. More technically, the
likelihood function for all event types taken together can be factored into a
separate likelihood function for each event type. The likelihood function
for each event type treats all other events as if the individual were
censored at the time when the other event occurred.
There are a few computer programs that estimate models for
competing risks simultaneously, but their only advantage is to reduce the
number of statements needed to specify the models. This advantage must
be balanced against the disadvantage of having to specify the same
functional form and the same set of covariates for all event types.
The only time you need to estimate models for two or more events
simultaneously is when there are parameter restrictions that cross event
types. For example, Kalbfleisch and Prentice (2002) consider a model that
imposes the restriction αj(t) = αj + α0(t) on equation (6.3). This restriction
says that the baseline hazard function for each event type is proportional
to a common baseline hazard function. Later in the chapter, we’ll see how
to test this restriction (see Estimates and Tests without Covariates).
A further implication is that you don’t need to estimate models for
all event types unless you really want to. If you’re only interested in the
effects of covariates on deaths from heart disease, then you can simply
estimate a single model for heart disease, treating all other death types as
censoring. Besides reducing the amount of computation, this fact also
makes it unnecessary to do an exhaustive classification of death types. You
only need to distinguish the event type of interest from all other types of
event.
Since we are treating events other than those of immediate interest
as a form of censoring, it’s natural to ask what assumptions must be made
about the censoring mechanism. In Chapter 2, “Basic Concepts of Survival
Analysis,” we saw that censoring must be noninformative if the estimates
are to be unbiased. We must make exactly the same assumption in the case
of competing risks. That is, we must assume that, conditional on the
covariates, those who are at particularly high (or low) risk of one event
type are no more (or less) likely to experience other kinds of events. Thus,
if we know that someone died of heart disease at age 50, that should give
us no information (beyond what we know from the covariates) about his
risk of dying of cancer at that age. Noninformativeness is implied by the
somewhat stronger assumption that the hypothetical Tijs (discussed above)
are independent across j. Unfortunately, as we saw in Chapter 2 with
censoring, it is impossible to test whether competing events are actually
noninformative. I’ll discuss this issue further in the conclusion to this
chapter.

CHAPTER 6

TIME

Competing Risks

POWER

FOR

LEADERS

COUNTRIES: EXAMPLE

The main data set that I will use to illustrate competing risks
analysis was constructed by Bienen and van de Walle (1991). They
identified a fairly exhaustive set of primary leaders of all countries
worldwide over the past 100 years or so. For each leader, they determined
the number of years in power and the manner by which he or she lost
power. Each period of leadership is called a spell. The mode of exit for a
leader from a position of power was classified into three categories:
 death from natural causes
 constitutional means
 nonconstitutional means (including assassination).
Clearly, it is unreasonable to treat all three types as equivalent. While
constitutional and nonconstitutional exits might have similar
determinants, death from natural causes is clearly a distinct phenomenon.
The data set included the following variables:
YEARS
LOST
MANNER
START
MILITARY
AGE
CONFLICT
LOGINC
GROWTH
POP
LAND
LITERACY
REGION

Number of years in power, integer valued. Leaders in power
less than 1 year were coded 0.
0=still in power in 1987; 1=exit by constitutional means;
2=death by natural causes; and 3=nonconstitutional exit.
How the leader reached power: 0=constitutional means;
1=nonconstitutional means.
Year of entry into power.
Background of the leader: 1=military; 0=civilian.
Age of the leader, in years, at the time of entry into power.
Level of ethnic conflict: 1=medium or high; 0=low.
Natural logarithm of GNP per capita (dollar equivalent) in
1973.
Average annual rate of per capita GNP growth between
1965−1983.
Population, in millions (year not indicated).
Land area, in thousands of square kilometers.
Literacy rate (year not indicated).
0=Middle East; 1=Africa; 2=Asia; 3=Latin America; and
4=North America, Europe, and Australia.

207

CHAPTER 6

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208

In the analysis that follows, I restrict the leadership spells to
 countries outside of Europe, North America, and Australia
 spells that began in 1960 or later
 only the first leadership spell for those leaders with multiple
spells.
This leaves a total of 472 spells, of which 115 were still in progress
at the time observation was terminated in 1987. Of the remaining spells, 27
ended when the leader died of natural causes, 165 were terminated by
constitutional procedures, and 165 were terminated by nonconstitutional
means. The restriction to starting years greater than 1960 was made so that
the variables describing countries (income, growth, population, and
literacy) would be reasonably concurrent with the exposure to risk.

ESTIMATES AND TESTS WITHOUT COVARIATES
The simplest question we might ask about the multiple event types
is whether the type-specific hazard functions are the same for all event
types (that is, hj(t) = h(t) for all j). For the LEADER data set, just looking at
the frequencies of the three event types suggests that deaths due to natural
causes (n=27) are much less likely to occur than the other two types,
which have identical frequencies (n=165). In fact, we can easily obtain a
formal test of the null hypothesis from these three frequencies. If the null
hypothesis of equal hazards is correct, the expected frequencies of the
three event types should be equal in any time interval. Thus, we have an
expected frequency of 119=(472−115)/3 for each of the three types.
Calculating Pearson’s chi-square test (by hand) yields a value of 88.9 with
2 d.f., which is far above the .05 critical value. So we can certainly
conclude that the hazard for naturally occurring deaths is lower than for
the other two exit modes.
Although the hazards are not equal, it’s still possible that they might
be proportional (in a different sense than that of Chapter 5); that is, if the
hazard for death changes with time, the hazards for constitutional and
nonconstitutional exits may also change by a proportionate amount. We
can write this hypothesis as

h j (t ) = ω j h(t ), j = 1, 2, 3,
where the ωjs are constants of proportionality.

(6.6)

CHAPTER 6

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209

We can obtain a graphic examination of this hypothesis by using
PROC LIFETEST to estimate log-log survivor functions for each of the
three event types. If the hazards are proportional, the log-log survivor
functions should be strictly parallel. We can get graphs of those functions
by first creating a separate data set for each event type, with a variable
EVENT equal to 1 if that event occurred and 0 otherwise. We then
concatenate the three data sets into a single data set called COMBINE.
Finally, we run PROC LIFETEST, stratifying on a variable TYPE that
distinguishes the three data sets, and requesting the log-log survivor
function with the PLOTS=LLS option. Here is the code:
DATA const;
SET leaders;
event=(lost=1);
type=1;
DATA nat;
SET leaders;
event=(lost=2);
type=2;
DATA noncon;
SET leaders;
event=(lost=3);
type=3;
DATA combine;
SET const nat noncon;
PROC LIFETEST DATA=COMBINE PLOTS=LLS;
TIME years*event(0);
STRATA type;
RUN;

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210

Output 6.1

Log-Log Survival Plot for Three Types of Exits, LEADERS Data Set

Output 6.1 shows the log-log survivor curves for the three event
types. The curves for constitutional and nonconstitutional exits are
virtually indistinguishable. Not surprisingly, the curve for natural deaths is
much lower than the other two curves. There is also some tendency for the
natural death curve to move closer to the other two in later years, which is
evidence against the proportionality hypothesis.
We can also examine smoothed hazard plots using the kernel
smoothing option for PROC LIFETEST that was described in Chapter 3:
ODS GRAPHICS ON;
PROC LIFETEST DATA=combine PLOTS=H(BW=10);
TIME years*event(0);
STRATA type;
RUN;
ODS GRAPHICS OFF;

In Output 6.2, we see that the smoothed hazard functions for
constitutional and nonconstitutional exits are very similar, except after the
15th year. The sharp increase in the hazard for constitutional exits after
this year could arguably be disregarded because the standard errors also

CHAPTER 6

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211

become large at later points in time. On the other hand, it could represent
the fact that many governments have constitutional limits on the number
of years a leader can spend in power. As might be expected, the hazard for
natural death is much lower but gradually increases with time (except for
the very end where, again, the estimates are not very reliable).
Output 6.2 Smoothed Hazard Functions for Three Types of Exits

Cox and Oakes (1984) proposed a parametric test of the proportional
hazards hypothesis in equation (6.6). Consider the model

log h j (t ) = α 0 (t ) + α j + β j t ,

j = 1, 2,

(6.7)

If βj = β for all j, then the proportional hazards hypothesis is satisfied.
Otherwise, this model says that the log-hazards for any two event types
diverge linearly with time. Cox and Oakes showed that if there are two
event types, equation (6.7) implies a binary logistic regression model for
the type of event, with the time of the event as an independent variable.
Under the proportional hazards hypothesis, the coefficient for time will be
0.

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212

For more than two event types, equation (6.7) implies a multinomial
logit model for event type. Here’s how to estimate that model with PROC
LOGISTIC:
PROC LOGISTIC DATA=leaders;
WHERE lost NE 0;
MODEL lost=years / LINK=GLOGIT;
RUN;

Notice that leadership spells that are still in progress (LOST=0)
when the observation is terminated are excluded from the analysis. The
LINK=GLOGIT option specifies an unordered multinomial logit model
(rather than the default cumulative logit model). Output 6.3 shows
selected portions of the output.

Output 6.3 Test of Proportionality with PROC LOGISTIC

Type 3 Analysis of Effects

Effect

Wald
Chi-Square

Pr > ChiSq

years

17.8915

0.0001

Analysis of Maximum Likelihood Estimates

Parameter

lost

Intercept
Intercept
years
years

1
2
1
2

Estimate

Standard
Error

Wald
Chi-Square

Pr > ChiSq

1
1
1
1

0.0134
-2.5393
-0.00391
0.1394

0.1432
0.3141
0.0267
0.0359

0.0088
65.3689
0.0215
15.0990

0.9253
<.0001
0.8834
0.0001

Looking first at the TYPE 3 table, we see that the effect of YEARS is
highly significant, indicating a rejection of the proportionality hypothesis.
The coefficients in the lower half of the output tell us which hazard
functions are proportional and which are not. The first coefficient for
YEARS is the β coefficient for the contrast between type 1 (constitutional
exit) and type 3 (nonconstitutional exit). The chi-square statistic is
minuscule, indicating that proportionality cannot be rejected for these two
hazard types. Moreover, the fact that the coefficient is also near 0 suggests
that the hazard functions for these two event types are nearly identical. On

CHAPTER 6

Competing Risks

213

the other hand, the second YEARS coefficient is highly significant. This
coefficient is a contrast between the type 2 hazard (death from natural
causes) and the type 3 hazard (nonconstitutional exit). Its value tells us
that the hazard for natural death increases much more rapidly with time
than the hazard for nonconstitutional exit; specifically, their ratio increases
by about 15 percent each year (calculated as 100(exp(.1394)–1))).

COVARIATE EFFECTS VIA COX MODELS
We have just seen evidence that the hazard for natural death is lower
than that for the other two exit types but that it increases more rapidly
with time. On the other hand, the hazard functions for constitutional and
nonconstitutional exits are nearly indistinguishable. We now look at
whether the effects of covariates are the same or different across event
types by fitting a Cox model to each type. The SAS program for doing this
is as follows:
PROC PHREG DATA=leaders;
MODEL years*lost(0)=manner start military age
loginc growth pop land literacy;
STRATA region;
PROC PHREG DATA=leaders;
MODEL years*lost(0,1,2)=manner start military
loginc growth pop land literacy;
STRATA region;
PROC PHREG DATA=leaders;
MODEL years*lost(0,1,3)=manner start military
loginc growth pop land literacy;
STRATA region;
PROC PHREG DATA=leaders;
MODEL years*lost(0,2,3)=manner start military
loginc growth pop land literacy;
STRATA REGION;
RUN;

conflict

age conflict

The first model treats all event types the same. I’ve included that
model for comparison, as well as for some hypothesis tests that we’ll look
at shortly. The second model focuses on type 3 by treating types 1 and 2 as
censoring. That’s followed by a model for type 2, treating types 1 and 3 as
censoring, and a model for type 1, treating types 2 and 3 as censoring.
Notice that I have stratified by region. Alternatively, I could have created
three dummy variables to represent the four regions and included them in
the model. However, stratification is less restrictive because it allows for
interactions between region and time, as we saw in Chapter 5.

CHAPTER 6

Competing Risks

214

In Output 6.4 we see what happens when no distinction is made
among the different kinds of exits. Four variables are statistically
significant by conventional criteria. The strongest effect is age at entry into
power, with each additional year of age associated with a 2 percent
increase in the risk of leaving power. Recall from Chapter 5, moreover, that
if age at origin is a covariate in a Cox model, its coefficient may actually be
interpreted as the effect of age as a time-dependent covariate. We also see
that
 leaders who attained power by nonconstitutional means
(MANNER=1) have a 44 percent greater risk of leaving power
 leaders in countries with higher GNP per capita (LOGINC) have a
lower risk of exit
 the risk of exit declined by about 1.6 percent per year (START)
since 1960.
The coefficient for the logarithm of GNP (–.1710) needs a little
explanation. If both the hazard and a covariate in a regression model are
logged, we can interpret the coefficient (without transformation) as the
percent change in the hazard for a 1 percent increase in the covariate (in
its original metric). Thus, we conclude that a 1 percent increase in per
capita GNP yields a .17 percent decrease in the hazard of exit.

Output 6.4

PROC PHREG Analysis for Exits by Any Means
Testing Global Null Hypothesis: BETA=0
Test
Likelihood Ratio
Score
Wald

Chi-Square

Pr > ChiSq

29.9069
29.5835
29.3945

10
10
10

0.0009
0.0010
0.0011

Analysis of Maximum Likelihood Estimates

Parameter

Parameter
Estimate

Standard
Error

Chi-Square

Pr > ChiSq

Hazard
Ratio

manner
start
military
age
conflict

1
1
1
1
1

0.36403
-0.01591
-0.22510
0.02016
0.12052

0.15412
0.00815
0.16251
0.00555
0.13175

5.5787
3.8129
1.9186
13.1731
0.8368

0.0182
0.0509
0.1660
0.0003
0.3603

1.439
0.984
0.798
1.020
1.128

(continued)

CHAPTER 6

Competing Risks

215
Output 6.4 (continued)

loginc
growth
pop
land
literacy

1
1
1
1
1

-0.17103
-0.00140
-0.0000210
0.0000107
0.0008773

0.08244
0.02131
0.0006375
0.0000478
0.00321

4.3044
0.0043
0.0011
0.0502
0.0747

0.0380
0.9478
0.9737
0.8228
0.7847

0.843
0.999
1.000
1.000
1.001

The picture changes somewhat when we focus on the hazard for
nonconstitutional exits in Output 6.5. Leaders who acquired power by
nonconstitutional means are 2.4 times as likely as other leaders to exit by
nonconstitutional means. Income also has a stronger effect: a 1 percent
increase in per capita GNP now yields a .42 percent decrease in the risk of
nonconstitutional exit. Age is no longer important, but we still see a
reduction in the hazard over time since 1960. In addition, we find that
leaders in countries with ethnic conflict have a 62 percent greater chance
of nonconstitutional exit.
Constitutional exits show a different pattern (Output 6.6). Manner of
acquiring power is not important, nor is income or conflict. Age shows up
again, however, and we also see an effect of literacy. Each 1-percentagepoint increase in the literacy rate is associated with a 1.4 percent increase
in the risk of a constitutional exit.

Output 6.5 PROC PHREG Analysis for Nonconstitutional Exits

Parameter
manner
start
military
age
conflict
loginc
growth
pop
land
literacy

Parameter
Estimate

Standard
Error

Chi-Square

Pr > ChiSq

Hazard
Ratio

1
1
1
1
1
1
1
1
1
1

0.88049
-0.03189
-0.38832
0.00827
0.48053
-0.42010
-0.04585
-0.00101
0.0000185
-0.00572

0.21958
0.01220
0.22651
0.00841
0.20398
0.14129
0.03130
0.00157
0.0000809
0.00454

16.0796
6.8278
2.9392
0.9670
5.5498
8.8412
2.1466
0.4170
0.0525
1.5901

<.0001
0.0090
0.0865
0.3254
0.0185
0.0029
0.1429
0.5184
0.8188
0.2073

2.412
0.969
0.678
1.008
1.617
0.657
0.955
0.999
1.000
0.994

CHAPTER 6

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216

Output 6.6

PROC PHREG Analysis for Constitutional Exits

Parameter
manner
start
military
age
conflict
loginc
growth
pop
land
literacy

DF
1
1
1
1
1
1
1
1
1
1

Parameter
Estimate
-0.27001
0.00276
-0.02793
0.02267
-0.03383
-0.12894
0.03254
0.0004148
-0.0000233
0.01376

Standard
Error Chi-Square
0.25623
0.01200
0.25913
0.00865
0.20309
0.11904
0.03452
0.0008444
0.0000699
0.00559

1.1105
0.0530
0.0116
6.8729
0.0278
1.1733
0.8887
0.2413
0.1108
6.0532

Pr > ChiSq
0.2920
0.8178
0.9142
0.0088
0.8677
0.2787
0.3458
0.6233
0.7393
0.0139

Hazard
Ratio
0.763
1.003
0.972
1.023
0.967
0.879
1.033
1.000
1.000
1.014

For deaths from natural causes (Output 6.7), it is not terribly
surprising that age is the only significant variable. Each 1-year increase in
age is associated with a 7.8 percent increase in the hazard of death.
Output 6.7 Deaths from Natural Causes

Parameter
manner
start
military
age
conflict
loginc
growth
pop
land
literacy

Parameter
Estimate

Standard
Error

Chi-Square

Pr > ChiSq

Hazard
Ratio

1
1
1
1
1
1
1
1
1
1

0.30550
-0.05731
-0.30438
0.07834
-0.55599
0.19416
0.09201
0.0009613
0.0000348
-0.01268

0.70173
0.03526
0.78263
0.02035
0.50965
0.28346
0.08551
0.00220
0.0001800
0.01364

0.1895
2.6410
0.1513
14.8145
1.1901
0.4692
1.1579
0.1910
0.0373
0.8644

0.6633
0.1041
0.6973
0.0001
0.2753
0.4934
0.2819
0.6621
0.8468
0.3525

1.357
0.944
0.738
1.081
0.574
1.214
1.096
1.001
1.000
0.987

We see, then, that the coefficients can differ greatly across different
event types. But perhaps these differences are merely the result of random
variation. What we need is a test of the null hypothesis that β j = β for all
j, where β j is the vector of coefficients for event type j. A test statistic is
readily constructed from output given by PROC PHREG. For each model,

CHAPTER 6

Competing Risks

PROC PHREG reports –2 × log-likelihood (for the model with covariates).
For the four models we just estimated, the values are
All types combined
Nonconstitutional
Constitutional
Natural death

2620.54
1216.76
1158.92
156.24

To construct the test, we sum the values for the three specific death
types, yielding a total of 2531.92. We then subtract that from the value for
all types combined, for a difference of 88.62. This is a likelihood ratio chisquare statistic for the null hypothesis. How many degrees of freedom?
Well, when we estimated separate models for the three event types, we got
a total of 30 coefficients. When we collapsed them all together, we only
estimated 10. The difference of 20 is the degrees of freedom. Because the
chi-square statistic is significant at well beyond the .01 level, we may
reject the hypothesis that the coefficients are all equal across event types.
That result is not terribly surprising because deaths from natural
causes are unlikely to have the same determinants as constitutional and
nonconstitutional exits. A more interesting question is whether the
covariates for the latter two event types have identical coefficients. We can
obtain a test statistic for that null hypothesis by estimating a model that
combines constitutional and nonconstitutional exits and that treats natural
deaths as censoring. For that model, –2 × log-likelihood=2443.71. If we
sum the values for the two separate models (given above), we get 2375.68.
The difference of 68.03 has 10 degrees of freedom (the difference in the
number of estimated parameters) for a p-value less than .0001. So again,
we may conclude that different models are required for constitutional and
nonconstitutional exits.
These tests (for all three events or for just two of them) are only valid
when using the Breslow method for handling ties or when there are no tied
event times. A more complicated procedure is necessary if there are ties,
and one of the other three tie-handling methods is used. Suppose, for
example, that we re-estimate all the models using the TIES=EFRON option
in the MODEL statement. We get the following values for the –2 log L
statistics:
All types combined
Nonconstitutional
Constitutional
Natural death

2561.08
1202.04
1139.41
155.38

As before, we can add up the –2 log L statistics for the three separate event
types, yielding a total of 2496.83. But we can’t validly compare this

217

CHAPTER 6

Competing Risks

218

number with –2 log L for all the types combined. That’s because when we
estimate three models for the different outcomes, we are allowing the
baseline hazard function to be different for each outcome. When we
estimate the model for all the types combined, however, we are only
allowing for a single baseline hazard function. The usual likelihood ratio
chi-square has no way to adjust for this difference.
The solution is to use the COMBINE data set that was created in the
Estimation and Testing without Covariates section above. This data set
was constructed by first creating a separate data set for each event type,
with a variable EVENT equal to 1 if that event occurred and 0 otherwise.
We then concatenated the three data sets into a single data set (COMBINE),
which included the variable TYPE with values of 1, 2, or 3 to distinguish
the three kinds of deaths. We now estimate a Cox model for this data set
while stratifying on the TYPE variable. We are already stratifying by
REGION so we can simply add TYPE to the STRATA statement:
PROC PHREG DATA=combine;
MODEL years*event(0)=manner start military age conflict
loginc growth pop land literacy / TIES=EFRON;
STRATA region type;
RUN;

This produces a –2 log L of 2590.68. From this we subtract the summed –2
log Ls for the three separate models (2496.83) to produce a likelihood ratio
chi-square of 93.85. With 20 degrees of freedom, the chi-square statistic is
highly significant, leading again to the conclusion that the regression
coefficients differ across the three event types.
If we had concluded, instead, that corresponding coefficients were
equal across all event types, a natural next step would be to test whether
they were all equal to 0. No special computations are needed for that test,
however. Just estimate the model without distinguishing among the event
types, and examine the global statistics for the null hypothesis that all
coefficients are equal to 0. In Output 6.4, all three chi-square statistics
(Wald, score, and likelihood ratio) have values of about 30 with 10 d.f.,
giving strong evidence against that null hypothesis. Note that this is a
conditional test, the condition being that the coefficients are equal. To test
the hypothesis that corresponding coefficients are equal and that they are
equal to 0, simply add this conditional chi-square statistic to the chisquare statistic for testing equality and then add the degrees of freedom as
well. Thus, for this example, the likelihood ratio statistic is
29.90+88.62=118.52 with 40 d.f.
We can also construct test statistics for hypotheses about coefficients
for specific covariates, using only the coefficients and their standard errors

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(Lagakos, 1978). The coefficient for CONFLICT for nonconstitutional exits
was .4805, with a standard error of .2040. By contrast, the coefficient for
CONFLICT for constitutional exits was –.0338, with a standard error of
.2031. The first coefficient is significantly different from 0; the second is
not. But is there a significant difference between them? A 1-degree-offreedom Wald chi-square statistic for testing the null hypothesis that
β1 = β 2 is easily calculated by the following formula:

(b1 − b2 )2
[s.e.(b1 )]2 + [s.e.(b2 )]2

(6.8)

where b1 is the estimate of β1 and s.e.(.) means estimated standard error.
For our particular question, we have

(− .03383 − .48053)2
(.2031) 2 + (.2040) 2

= 3.19 .

Because that does not exceed the .05 critical value, we have insufficient
statistical justification for concluding that the two coefficients are
different.
This particular statistic, in one form or another, is widely used to test
for differences in parameter estimates across two independent groups.
Here, however, we do not have independent groups because the same set
of 472 leaders is used to estimate models for both constitutional and
nonconstitutional exits. This suggests that we may need a covariance term
in the denominator of the statistic. In fact, we do not. What justifies this
statistic is the fact that the likelihood function factors into a distinct
likelihood for each event type. It follows that the parameter estimates for
each event type are asymptotically independent of the parameter estimates
for all other event types. This only holds for mutually exclusive event
types, however. We cannot use this statistic to test for the difference
between a coefficient for constitutional exits and the corresponding
coefficient for all types of exits.
Because we found no reason to reject the hypothesis that the two
coefficients for CONFLICT are equal, we may want to go further and test
whether they are 0. This is easily accomplished by taking the reported chisquare statistics for CONFLICT, for both constitutional and
nonconstitutional exits, and summing them: .87+5.55=6.42 with 2 d.f. This
is just barely significant at the .05 level.
The tests for a single covariate can be generalized to more than two
event types. For a given covariate, let bj be its coefficient for event type j;
2
2
2
2
let sj be the squared, estimated standard error of bj; and let Xj =bj /sj be the
reported Wald chi-square statistic for testing that βj=0. To test the

CHAPTER 6

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220

hypothesis that all the coefficients for the chosen covariate are 0, we sum
the Wald chi-square statistics as follows:

Q = ∑ X 2j

(6.9)

which has degrees of freedom equal to the number of event types. To test
the hypothesis that all coefficients are equal to each other, we calculate

 bj 
∑ 2 
 j s 
j 
Q−
1
∑j s 2
j

(6.10)

which has degrees of freedom equal to one less than the number of event
types.

ACCELERATED FAILURE TIME MODELS
Competing risks analysis with accelerated failure time models is
basically the same as with Cox models. As before, the key point is to treat
all events as censoring except the one that you’re focusing on. However,
there are some complications that arise in constructing tests of equality of
coefficients across event types. There are also some special characteristics
of the LEADER data set that require slightly different treatment with
accelerated failure time models.
One of those characteristics is the presence of 0s in the YEARS
variable. Recall that any leaders who served less than 1 year were assigned
a time value of 0. Of the 472 leaders, 106 fell in this category. This poses
no problem for PROC PHREG, which is only concerned with the rank
order of the time variable. On the other hand, PROC LIFEREG excludes
any observations with times of 0 or less because it must take the logarithm
of the event time as the first step. But we certainly don’t want to exclude
22 percent of the cases. One approach is to assign some arbitrarily chosen
number between 0 and 1. A more elegant solution is to treat such cases as
if they were left censored at time 1. We discussed how to do this in some
detail in the section Left Censoring and Interval Censoring in Chapter 4,
“Estimating Parametric Regression Models with PROC LIFEREG.” For the
LEADER data set, we need a short DATA step to create LOWER and UPPER
variables that are appropriately coded. Here’s how to prepare the data set
for modeling the constitutional exits.

CHAPTER 6

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221

DATA leaders2;
SET leaders;
lower=years;
upper=years;
IF years=0 THEN DO;
lower=.;
upper=1;
END;
IF lost IN (0,1,2) THEN upper=.;
RUN;

For uncensored observations, LOWER and UPPER have the same value.
For observations with YEARS=0, we set LOWER=. and UPPER=1. For
right-censored observations (including those with events other than the
one of interest), UPPER=. and LOWER=YEARS. If both UPPER and
LOWER are missing (which happens for individuals with YEARS=0 who
have events other than the one of interest), the observation is excluded. To
fit a model for the other two types of exit, we simply change the last IF
statement so that the numbers in parentheses include all outcomes to be
treated as right censored.
To fit an exponential model to this data set, I used the following
PROC LIFEREG program:
PROC LIFEREG DATA=leaders2;
CLASS region;
MODEL (lower,upper)= manner start military age conflict
loginc literacy region / D=EXPONENTIAL;
RUN;

Notice that REGION is declared to be a CLASS variable. In the PROC
PHREG analysis, I simply stratified on this variable. But stratification is
not an option in PROC LIFEREG. I also excluded from the models all
variables that were not statistically significant in any of the PROC PHREG
models (namely, POP, GROWTH, and LAND).
For each of the three event types, I estimated all five of the
accelerated failure time models discussed in Chapter 4. The models and
their log-likelihoods are shown below:

Exponential
Weibull
Log-normal
Gamma
Log-logistic

Nonconstitutional
-383.39
-372.51
-377.04
-372.47
-374.95

Constitutional
-337.30
-336.46
-338.09
-336.14
-335.88

Natural Death
-87.17
-82.48
-83.60
(-81.36)
-82.78

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222

(In fitting the gamma model to natural deaths, PROC LIFEREG reported a
possible convergence failure after 22 iterations. I’ve given the loglikelihood at the last iteration.)
For the nonconstitutional exits, we can reject the hypothesis that the
hazard is constant over time. That’s because the chi-square statistic for
comparing the exponential (constant hazard) model with the generalized
gamma model is 21.84, which was calculated by taking the difference in
the log-likelihoods for the two models and multiplying by 2. With 2 d.f.,
this has a p-value of .00002. The log-normal model must also be rejected
(p=.003). The Weibull model, on the other hand, is only trivially different
from the gamma model.
Output 6.8 shows the parameter estimates for the Weibull model.
Consider first the scale estimate of 1.41, which can tell us whether the
hazard is increasing or decreasing. Using the transformation 1/1.41–1=
–.29, we get the coefficient of log t in the equivalent proportional hazards
model. This result indicates that the hazard of a nonconstitutional exit
decreases with time since entry into power. The p-values for the covariates
are quite similar to those in Output 6.5 for the Cox model, but naturally the
coefficients are all reversed in sign. To directly compare the magnitudes of
the coefficients with those of the Cox model, each must be divided by the
scale estimate of 1.41. After this adjustment, the values are remarkably
similar. We also see major differences among the regions, something that
was hidden in the Cox analysis. Specifically, expected time until a
nonconstitutional exit is more than seven times greater (exp(2.036)) in
Asia (REGION=2) than it is in Latin America (REGION=3), and it is nearly
four times greater (exp(1.36)) in Africa (REGION=1).
Output 6.8 Weibull Model for Nonconstitutional Exits

Parameter

DF Estimate

Intercept
manner
start
military
age
conflict
loginc
literacy
region
region

1 -40.6136
1 -1.3806
1
0.0410
1
0.6510
1 -0.0124
1 -0.7199
1
0.6750
1
0.0072
1
0.9037
1
1.3636

0
1

Standard
Error

95% Confidence
Limits

17.1902 -74.3057
0.3143 -1.9965
0.0178
0.0062
0.3135
0.0366
0.0114 -0.0348
0.2843 -1.2771
0.2073
0.2688
0.0063 -0.0052
0.4457
0.0301
0.3926
0.5941

-6.9215
-0.7646
0.0758
1.2655
0.0100
-0.1626
1.0812
0.0196
1.7773
2.1332

ChiSquare Pr > ChiSq
5.58
19.30
5.33
4.31
1.18
6.41
10.61
1.28
4.11
12.06

0.0181
<.0001
0.0210
0.0378
0.2775
0.0113
0.0011
0.2572
0.0426
0.0005

(continued)

CHAPTER 6

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223
Output 6.8 (continued)

region
2
region
3
Scale
Weibull Shape

1
0
1
1

2.0362
0.0000
1.4064
0.7111

0.4692
.
0.1121
0.0567

1.1166
.
1.2030
0.6082

2.9558
.
1.6441
0.8313

18.83
.

<.0001
.

For constitutional exits, the picture is rather different. All the models
have similar log-likelihoods, and even the exponential model is not
significantly worse than the Weibull or gamma models. Invoking
parsimony, we might as well stick with the exponential model, which is
displayed in Output 6.9. The Lagrange multiplier test at the bottom of the
output shows, again, that the exponential model cannot be rejected. As
with the nonconstitutional exits, the results are quite similar to those of
the Cox model in Output 6.6. Apart from the region differences, the only
significant variables are AGE and LITERACY. Except for the reversal of
sign, the coefficients are directly comparable to those of the Cox model.
Again, we see that Latin America has the shortest expected time until a
constitutional exit, but the longest expected time is now in Africa rather
than Asia.
Output 6.9 Exponential Model for Constitutional Exits

Parameter
Intercept
manner
start
military
age
conflict
loginc
literacy
region
region
region
region
Scale
Weibull Shape

DF Estimate

0
1
2
3

1
1
1
1
1
1
1
1
1
1
1
0
0
0

12.4138
0.3112
-0.0092
0.0284
-0.0317
0.1573
0.1366
-0.0113
0.5413
1.7032
0.5336
0.0000
1.0000
1.0000

Standard
Error

95% Confidence
Limits

11.7600 -10.6353
0.2624 -0.2030
0.0120 -0.0328
0.2526 -0.4667
0.0083 -0.0480
0.1983 -0.2314
0.1135 -0.0858
0.0056 -0.0223
0.3326 -0.1106
0.3702
0.9776
0.2180
0.1062
.
.
0.0000
1.0000
0.0000
1.0000

35.4630
0.8255
0.0144
0.5236
-0.0154
0.5460
0.3590
-0.0003
1.1932
2.4288
0.9609
.
1.0000
1.0000

Lagrange Multiplier Statistics
Parameter
Scale

Chi-Square

Pr > ChiSq

1.4704

0.2253

ChiSquare Pr > ChiSq
1.11
1.41
0.58
0.01
14.52
0.63
1.45
4.09
2.65
21.17
5.99
.

0.2912
0.2356
0.4453
0.9104
0.0001
0.4277
0.2287
0.0432
0.1036
<.0001
0.0144
.

CHAPTER 6

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224

For deaths due to natural causes, the only model that is clearly
rejectable is the exponential model. Of the remaining models, the Weibull
model again appears to be the best choice because its log-likelihood is only
trivially different from that of the gamma model. The Weibull estimates are
reported in Output 6.10. As in the Cox model, there is a strong effect of age
at entry, with older leaders having shorter times until death.

Output 6.10 Weibull Estimates for Natural Deaths

Parameter
Intercept
manner
start
military
age
conflict
loginc
literacy
region
region
region
region
Scale
Weibull Shape

DF Estimate

0
1
2
3

1 -23.1476
1 -0.2216
1
0.0302
1
0.2210
1 -0.0451
1
0.0736
1 -0.1514
1
0.0030
1
0.3597
1
0.7672
1
0.6378
0
0.0000
1
0.5994
1
1.6684

Standard
Error

95% Confidence
Limits

20.0574 -62.4595
0.3909 -0.9878
0.0209 -0.0107
0.4158 -0.5939
0.0108 -0.0662
0.2804 -0.4761
0.1563 -0.4577
0.0073 -0.0112
0.4472 -0.5168
0.4491 -0.1131
0.3745 -0.0963
.
.
0.0885
0.4487
0.2465
1.2489

16.1642
0.5445
0.0712
1.0360
-0.0240
0.6232
0.1549
0.0172
1.2361
1.6475
1.3718
.
0.8007
2.2287

ChiSquare Pr > ChiSq
1.33
0.32
2.09
0.28
17.58
0.07
0.94
0.17
0.65
2.92
2.90
.

0.2485
0.5707
0.1480
0.5950
<.0001
0.7930
0.3328
0.6769
0.4212
0.0876
0.0886
.

The next logical step is to construct tests of hypotheses about
equality of coefficients across different event types. For tests about
individual covariates, the chi-square statistics in equations (6.8) – (6.10)
will do just fine. Unfortunately, we cannot use the method employed for
Cox models to construct a global test of whether all the coefficients for one
event type are equal to the corresponding coefficients for another event
type. For the accelerated failure time models, fitting a model to all events
without distinction involves a different likelihood than constraining
parameters to be equal across the separate likelihoods. As a result, if you
try to calculate the statistics described in the previous section, you’re
likely to get negative chi-square statistics.
In general, there’s no easy way around this problem. For Weibull
models (actually for any parametric proportional hazards model), the
likelihood test can be corrected by a function of the number of events of
each type at each point in time (Narendranathan and Stewart, 1991). Here,
I consider an alternative test for Weibull models that I think is more easily

CHAPTER 6

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225

performed. It also yields a test for each covariate as a by-product. It is, in
fact, a simple generalization of the test for proportional hazards discussed
earlier in this chapter (see Time in Power for Leaders of Countries:
Example).
Suppose we believe that constitutional and nonconstitutional exits
are both governed by Weibull models but with different parameters. We
can write the two models as

log h j (t ) = α j log t + β 0 j + β1 j x1 +  + β kj xk

(6.11)

with j=1 for constitutional exits and j=2 for nonconstitutional exits. Now
consider the following question: Given that an exit (other than a natural
death) occurs at time t, what determines whether it is a constitutional or a
nonconstitutional exit? Equation (6.11) implies that this question is
answered by a logistic regression model:

log

Pr( J = 1 | T = t )
= (α1 − α 2 ) log t + ( β 01 − β 02 ) + ( β11 − β12 ) x1 +  + ( β k1 − β k 2 ) xk
Pr( J = 2 | T = t )

This model can be easily estimated with PROC LOGISTIC:
DATA leaders3;
SET leaders;
lyears=LOG(years+.5);
PROC LOGISTIC DATA=leaders3;
WHERE lost=1 OR lost=3;
CLASS region / PARAM=GLM;
MODEL lost=lyears manner age start military conflict loginc
literacy region;
RUN;

The DATA step is needed so that the log of time can be included in
the model (0.5 was added to avoid problems with times of 0). The WHERE
statement eliminates the censored cases and those who died of natural
causes.
Output 6.11 shows the results. We can interpret each of the
coefficients in this table as an estimate of the difference between
corresponding coefficients in the two Weibull models. We see that there
are highly significant differences in the coefficients for MANNER and
LITERACY but only marginally significant differences between the
coefficients for AGE, START, and LYEARS. The test for LYEARS is
equivalent to a test of whether the scale parameters are the same in the
accelerated failure time version of the model.
The null hypothesis that all the corresponding coefficients are equal
in the two Weibull models is equivalent to the hypothesis that all the
coefficients in the implied logit model are 0. That hypothesis, of course, is

CHAPTER 6

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226

tested by the three chi-square statistics in the panel labeled “Testing Global
Null Hypothesis: BETA=0”. Although the three statistics vary quite a bit in
magnitude, they all indicate that the hypothesis should be rejected.

Output 6.11

Logit Model Comparing Constitutional and Nonconstitutional Exits
Testing Global Null Hypothesis: BETA=0
Test

Chi-Square

Pr > ChiSq

129.1322
108.7634
76.2327

11
11
11

<.0001
<.0001
<.0001

Likelihood Ratio
Score
Wald

Analysis of Maximum Likelihood Estimates

Parameter

Estimate

Standard
Error

Wald
Chi-Square

Pr > ChiSq

Intercept
lyears
manner
age
start
military
conflict
loginc
literacy
region
region
region
region

1
1
1
1
1
1
1
1
1
1
1
1
0

-42.9260
0.2749
-1.3252
0.0277
0.0407
0.0698
-0.2433
0.0581
0.0334
0.1111
-0.6684
0.4737
0

20.7274
0.1424
0.3801
0.0144
0.0214
0.3949
0.3425
0.2502
0.00895
0.5062
0.5034
0.4568
.

4.2890
3.7239
12.1579
3.7136
3.6004
0.0313
0.5044
0.0539
13.9201
0.0481
1.7626
1.0754
.

0.0384
0.0536
0.0005
0.0540
0.0578
0.8596
0.4776
0.8163
0.0002
0.8263
0.1843
0.2997
.

0
1
2
3

Odds Ratio Estimates

Effect
lyears
manner
age
start
military

Point
Estimate
1.316
0.266
1.028
1.041
1.072

95% Wald
Confidence Limits
0.996
0.126
1.000
0.999
0.495

1.740
0.560
1.057
1.086
2.325

(continued)

CHAPTER 6

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227
Output 6.11 (continued)

conflict
loginc
literacy
region
0 vs 3
region
1 vs 3
region
2 vs 3

0.784
1.060
1.034
1.117
0.513
1.606

0.401
0.649
1.016
0.414
0.191
0.656

1.534
1.731
1.052
3.014
1.375
3.932

ALTERNATIVE APPROACHES TO MULTIPLE EVENT TYPES
Conditional Processes

The competing risks approach presumes that each event type has its
own hazard model that governs both the occurrence and timing of events
of that type. The appropriate imagery is one of independent causal
mechanisms operating in parallel. Whichever type of event happens first,
the individual is then no longer at risk of the other types.
This is clearly a defensible way of thinking about deaths due to
natural causes, on the one hand, and forcible removal from power, on the
other. It may not be so sensible, however, for the distinction between
constitutional exits and nonconstitutional exits. We might imagine that a
leader will stay in power as long as his popularity with key groups stays
sufficiently high. When that popularity drops below a certain point,
pressures will build for his removal. How he is removed is another
question, and the answer depends on such things as the constitutional
mechanisms that are available and the cultural traditions of the country. In
this way of thinking about things, we have one mechanism governing the
timing of events and another distinct mechanism determining the type of
event, given that an event occurs.
For an even more blatant example, consider the event buying a
personal computer, and suppose we subdivide this event into two types:
buying a Macintosh computer and buying a Windows based computer.
Now it would be absurd to suppose that there are two parallel processes
here and that we merely observe whichever produces an event first.
Rather, we have one process that governs the decision to buy a computer at
all and another that governs which type is purchased. These kinds of
situations arise most commonly when the different event types are
alternative means for achieving some goal.
If we adopt this point of view, a natural way to proceed is to estimate
one model for the timing of events (without distinguishing among event

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228

types) and a second model for the type of event (restricting the analysis to
those individuals who experienced an event). For the timing of events, any
of the models that we have considered so far are possible contenders. For
the type of event, a binomial or multinomial logit model is a natural
choice, although there are certainly alternatives. A major attraction of this
approach is that there is no need to assume that the different kinds of
events are uninformative for one another.
We have already estimated a logit model for constitutional versus
nonconstitutional exits, with the results displayed in Output 6.11. There
we interpreted the coefficients as differences in coefficients in the
underlying hazard models. Now I am suggesting that we interpret these
coefficients directly as determinants of whether a loss of power was by
constitutional or nonconstitutional means, given that an exit from power
occurred. We see, for example, that those who obtained power by
nonconstitutional means are about four times as likely to lose power in the
same way (based on the odds ratio of .266 for MANNER). On the other
hand, each additional percentage point of literacy increases the odds that
the exit will be constitutional rather than nonconstitutional by about 3
percent. Although the effects are only marginally significant, there are
consistently positive effects of time on constitutional rather than
nonconstitutional changes: older leaders, more recently installed leaders,
and leaders who have been in power longer are all more likely to exit via
constitutional mechanisms.
We still need a model for timing of exits from power, treating deaths
from natural causes as censoring. Output 6.12 shows the results of
estimating a Cox model, with the following statements:
PROC PHREG DATA=leaders;
MODEL years*lost(0,2)=manner age start military conflict
loginc literacy / TIES=EXACT;
STRATA region;
RUN;

Because nearly half of the exits occurred at 0 or 1 year, I used the
TIES=EXACT option.

CHAPTER 6

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229
Output 6.12 Cox Model for Timing of Constitutional and Nonconstitutional Exits

Parameter DF
manner
age
start
military
conflict
loginc
literacy

1
1
1
1
1
1
1

Parameter
Estimate

Standard
Error Chi-Square Pr > ChiSq

0.37642
0.01718
-0.01497
-0.20906
0.17099
-0.24071
0.00186

0.15922
0.00575
0.00840
0.16439
0.13607
0.08860
0.00331

5.5892
8.9381
3.1769
1.6174
1.5792
7.3820
0.3150

0.0181
0.0028
0.0747
0.2035
0.2089
0.0066
0.5747

Hazard
Ratio
1.457
1.017
0.985
0.811
1.186
0.786
1.002

Three results stand out in Output 6.12:
 Those who acquired power by nonconstitutional means had
about a 46 percent higher risk of losing power (by means other
than natural death).
 Each additional year of age increased the risk of exit by about 1.7
percent.
 A 1 percent increase in GNP per capita yielded about a .24
percent decrease in the risk of exit.
In my judgment, of all the analyses done in this chapter, Output 6.11
and Output 6.12 give the most meaningful representation of the processes
governing the removal of leaders from power. To that, we may also wish to
add Output 6.7, which shows that only age is associated with the risk of a
death from natural causes.
Cumulative Incidence Functions

In equation (6.3) I defined the type-specific survivor function Sj(t),
but I warned that it did not apply to any well-defined, observable random
variable. Nevertheless, if you do Kaplan-Meier (or life table) estimation
with PROC LIFETEST, treating other events as censoring, what you get is
an estimate of the function Sj(t). This can be interpreted as the probability
that an individual makes it to time t without having an event of type j,
under the hypothetical presumption that other kinds of events cannot
occur. Similarly, Fj(t) = 1– Sj(t) can be interpreted as the probability that an
event of type j occurs to an individual by time t, in the absence of any
competing risks.
In some fields, especially those that are more oriented toward
applications, this is not regarded as a useful function to estimate, because
the competing risks cannot be removed. A further reason is that

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230

Competing Risks

∑ F (t ) ,
j

the sum of the probabilities of the different kinds of events occurring by
time t, may often exceed 1.
Instead, many people prefer to estimate cumulative incidence
functions, which are defined as
I j (t ) = Pr(Ti < t , J i = j ) .

This equation estimates the probability that an event of type j happens to
individual i before time t, in the presence of competing event types. As
described by Marubini and Valsecchi (1995), a consistent estimate of the
cumulative incidence function is given by

Iˆ j (t ) =

∑ Sˆ (t

k |t k ≤t

)

d jk
nk

where Sˆ (t ) is the Kaplan-Meier estimate of the overall survivor function,
djk is the number of events of type j that occurred at time tk, and nk is the
number at risk at time tk. The ratio of djk to nk is an estimate of the hazard
of event type j at time tk.
An attractive feature of this estimator is that

Sˆ (t ) + ∑ Iˆ j (t ) = 1 .
j

This says that at any time t, the probabilities for all the event types plus
the probability of no event add up to 1. Another attraction is that the
method does not depend on the assumption that each event type is noninformative for other event types.
Although cumulative incidence functions are not currently
implemented in any SAS procedure, there is a macro called CUMINCID
that is distributed with SAS and stored in its autocall location. That means
that the macro is available for use in any SAS program. Here’s how to use
the macro for the LEADERS data set to estimate the cumulative incidence
function for nonconstitutional exits (LOST=3):
%CUMINCID(DATA=leaders,
TIME=years,
STATUS=lost,
EVENT=3,
COMPETE=1 2,
CENSORED=0)

CHAPTER 6

Competing Risks

231

In this version of the data set, leaders whose YEARS value was 0
were recoded to have a value of .5. Output 6.13 displays the results. Like
the Kaplan-Meier method in PROC LIFETEST, the output produced by
CUMINCID contains one line for every observation, but most of these lines
contain no useful information. I’ve edited out all the lines that are not
informative. The cumulative incidence probabilities are substantially
lower than the cumulative failure probabilities reported by PROC
LIFETEST using the Kaplan-Meier method. For example, at 24 years, the
cumulative incidence estimate is .42 but the corresponding probability
reported by LIFETEST is .63.

Output 6.13

Estimates of the Cumulative Incidence Function
Cumulative Incidence Estimates
with Confidence Limits Based on the Log Transform

Obs

years

1
54
129
185
223
262
304
337
360
374
388
399
412
427
430
437
442
445
451
455

0.0000
0.5000
1.0000
2.0000
3.0000
4.0000
5.0000
6.0000
7.0000
8.0000
9.0000
10.0000
11.0000
13.0000
14.0000
15.0000
16.0000
17.0000
18.0000
19.0000

censor

CumInc

StdErr
CumInc

.
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

0.00000
0.11229
0.15979
0.20689
0.23473
0.25637
0.28356
0.29932
0.31055
0.32554
0.33208
0.34618
0.36890
0.38142
0.38568
0.39467
0.39972
0.40513
0.41100
0.41749

0.000000
0.014532
0.016919
0.018833
0.019810
0.020516
0.021330
0.021800
0.022151
0.022639
0.022874
0.023411
0.024258
0.024739
0.024896
0.025238
0.025468
0.025730
0.026042
0.026423

Lower95Pct
CumInc
.
0.08713
0.12984
0.17309
0.19895
0.21916
0.24469
0.25951
0.27004
0.28406
0.29014
0.30320
0.32429
0.33588
0.33985
0.34818
0.35280
0.35771
0.36300
0.36878

Upper95Pct
CumInc
.
0.14471
0.19664
0.24730
0.27696
0.29991
0.32860
0.34525
0.35715
0.37307
0.38008
0.39524
0.41964
0.43312
0.43770
0.44737
0.45289
0.45883
0.46534
0.47264

CHAPTER 6

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232

The CUMINCID macro also has a STRATA parameter that produces
cumulative incidence functions for multiple subgroups and plots the
functions on a single graph. However, it does not have the capability of
testing the null hypothesis that the cumulative incidence functions are
identical, although several such tests have been proposed (for example,
Gray, 1988). There are even regression methods based on cumulative
incidence functions rather than on hazard functions (Fine and Gray, 1999;
Scheike and Zhang, 2008). SAS macros that implement these methods can
readily be found on the Web.
Despite the popularity of these methods (and the insistence by some
that conventional methods are flatly incorrect), I think that cumulative
incidence functions have been oversold. As a descriptive device in applied
settings, the methods can certainly be useful. But if the aim is to
understand the causal mechanisms underlying event occurrence, I still
believe that the estimation of hazard functions (treating other events as
censoring) is the way to go (Pintilie, 2006).

CONCLUSION
As we have seen, competing risks analysis is easily accomplished
with conventional software by doing a separate analysis for each event
type, treating other events as censoring. The biggest drawback of
competing risks analysis is the requirement that times for different event
types be independent or at least that each event be noninformative for the
others. This requirement is exactly equivalent to the requirement for
random censoring discussed in Chapter 2. In either case, violations can
lead to biased coefficient estimates.
The seriousness of this problem depends greatly on the particular
application. For the LEADERS data set, I argued that death due to natural
causes is likely to be noninformative for the risk of either a constitutional
or a nonconstitutional exit. For the latter two types, however, the
presumption of noninformativeness may be unreasonable. In thinking
about this issue, it is helpful to ask the question, “Are there unmeasured
variables that may affect more than one event type?” If the answer is yes,
then there should be a presumption of dependence.
Unfortunately, there’s not a lot that can be done about the problem.
It’s possible to formulate models that incorporate dependence among event
types but, for any such model, there’s an independence model that does an
equally good job of fitting the data. Dependence models typically impose
parametric restrictions on the shape of the hazard functions, and the
results may heavily depend on those restrictions. Heckman and Honoré

CHAPTER 6

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233

(1989) showed that you can identify nonparametric dependence models,
so long as there is at least one continuous covariate with different
coefficients for different event types (along with some other mild
conditions), but the practical implications of their theorem have yet to be
explored. Just because you can do something doesn’t mean you can do it
well.
So for all practical purposes, we have little choice but to use a
method that rests on assumptions that may be implausible and cannot be
tested. In Chapter 8, “Heterogeneity, Repeated Events, and Other Topics,” I
describe a sensitivity analysis for informative censoring that may also be
useful for competing risks. Basically, this method amounts to redoing the
analysis under two worst-case scenarios and hoping that the qualitative
conclusions don’t change. The other thing to remember is that you can
reduce the problem of dependence by measuring and including those
covariates that are likely to affect more than one type of event. For
example, a diet that is high in saturated fat is thought to be a common risk
factor for both heart disease and cancer. If a measure of dietary fat intake is
included in the regression model, it can partially alleviate concerns about
possible dependence among these two event types.

CHAPTER 6

234

Competing Risks

CHAPTER 7
Analysis of Tied or Discrete Data with
PROC LOGISTIC
Introduction 235
The Logit Model for Discrete Time 236
The Complementary Log-Log Model for Continuous-Time Processes 240
Data with Time-Dependent Covariates 243
Issues and Extensions 246
Conclusion 255

INTRODUCTION
This chapter shows you how to use the LOGISTIC procedure to
analyze data in which many events occur at the same points in time. In
Chapter 5, “Estimating Cox Regression Models with PROC PHREG,” we
looked at several different methods for handling tied data with PROC
PHREG. There we saw that Breslow’s method—the standard formula for
partial likelihood estimation with tied data—is often a poor approximation
when there are many ties. This problem was remedied by two exact
methods, one that assumed that ties result from imprecise measurement
and another that assumed that events really occur at the same (discrete)
time. Unfortunately, both of these methods are computationally
demanding for large data sets with many ties. We also looked at tied data
in Chapter 4, “Estimating Parametric Regression Models with PROC
LIFEREG,” under the heading of interval censoring. While PROC LIFEREG
is adept at estimating parametric models with interval censoring, it cannot
incorporate time-dependent covariates.
The maximum likelihood methods described in this chapter do not
suffer from these limitations. They do not rely on approximations, the
computations are quite manageable even with large data sets, and they are
particularly good at handling large numbers of time-dependent covariates.
In addition, the methods make it easy to test hypotheses about the
dependence of the hazard on time.
The basic idea is simple. Each individual’s survival history is broken
down into a set of discrete time units that are treated as distinct
observations. After pooling these observations, the next step is to estimate
a binary regression model predicting whether an event did or did not

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236

occur in each time unit. Covariates are allowed to vary over time from one
time unit to another.
This general approach has two versions, depending on the form of
the binary regression model. By specifying a logit link, you get estimates of
the discrete-time proportional odds model proposed by Cox. This model is
identical to the model estimated when you specify the TIES=DISCRETE
option in PROC PHREG. Alternatively, by specifying a complementary loglog link, you get estimates of an underlying proportional hazards model in
continuous time. This is identical to the model that is estimated when you
specify the TIES=EXACT option in PROC PHREG.
The mechanics of this approach are similar to those of the piecewise
exponential model described in Chapter 4 and the counting process syntax
of Chapter 5. The main difference is that those methods assumed that you
know the exact time of the event within a given interval. By contrast, the
procedures in this chapter presume that you know only that an event
occurred within a given interval.

THE LOGIT MODEL

FOR

DISCRETE TIME

We begin with the logit version of the model because it is more
widely used and because logistic regression is already familiar to many
readers. In Chapter 5 (see The DISCRETE Method), we considered Cox’s
model for discrete-time data. In brief, we let Pit be the conditional
probability that individual i has an event at time t, given that an event has
not already occurred to that individual. The model says that Pit is related
to the covariates by a logistic regression equation:

 P 
log  it  = α t + β1 xit1 +  + β k xitk
1 − Pit 

(7.1)

where t =1, 2, 3,…. This model is most attractive when events can only
occur at regular, discrete points in time, but it has also been frequently
employed when ties arise from grouping continuous-time data into
intervals.
In Chapter 5, we saw how to estimate this model by the method of
partial likelihood, thereby discarding any information about the αts. Now
we are going to estimate the same model by maximum likelihood, so that
we get explicit estimates of the αts. The procedure is best explained by
way of an example. As in Chapter 5, we’ll estimate the model for 100

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Analysis of Tied or Discrete Data with PROC LOGISTIC

237

simulated job durations, measured from the year of entry into the job until
the year that the employee quit. Durations after the fifth year are censored.
We know only the year in which the employee quit, so the survival times
have values of 1, 2, 3, 4, or 5. These values are contained in a variable
called DUR, while the variable EVENT is coded 1 if the employee quit;
otherwise, it is coded 0. Covariates are ED (years of education), PRESTIGE
(a measure of the prestige of the occupation), and SALARY in the first year
of the job. None of these covariates are time-dependent.
The first task is to take the original data set (JOBDUR) with one
record per person and create a new data set (JOBYRS) with one record for
each year that each person was observed. Thus, someone who quit in the
third year gets three observations, while someone who still had not quit
after five years on the job gets five observations. The following DATA step
accomplishes this task:
DATA jobyrs;
SET jobdur;
DO year=1 TO dur;
IF year=dur AND event=1 THEN quit=1;
ELSE quit=0;
OUTPUT;
END;
RUN;

The DO loop creates 272 person-years that are written to the output
data set. The IF statement defines the dependent variable QUIT, which
equals 1 if the employee quit in that particular person-year; otherwise,
QUIT equals 0. Thus, if a person quit in the fifth year, QUIT is coded 0 for
the first four records and 1 in the last record. For people who don’t quit
during any of the 5 years, QUIT is coded 0 for all five records.
Output 7.1 shows the first 20 records produced by this DATA step.
Observation 1 is for a person who quit in the first year of the job.
Observations 2 through 5 correspond to a person who quit in the fourth
year. QUIT is coded 0 for the first three years and 1 for the fourth.
Observations 6 through 10 correspond to a person who still held the job at
the end of the fifth year.

CHAPTER 7

Analysis of Tied or Discrete Data with PROC LOGISTIC

238
Output 7.1 First 20 Cases of Person-Year Data Set for Job Durations

Obs
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

dur

event

1
4
4
4
4
5
5
5
5
5
2
2
3
3
3
1
1
1
2
2

1
1
1
1
1
0
0
0
0
0
1
1
1
1
1
1
1
1
1
1

7
14
14
14
14
16
16
16
16
16
12
12
9
9
9
11
13
10
12
12

prestige
3
62
62
62
62
70
70
70
70
70
43
43
18
18
18
31
26
1
28
28

salary
19
17
17
17
17
18
18
18
18
18
135
135
12
12
12
12
6
4
17
17

year
1
1
2
3
4
1
2
3
4
5
1
2
1
2
3
1
1
1
1
2

quit
1
0
0
0
1
0
0
0
0
0
0
1
0
0
1
1
1
1
0
1

Now we’re ready to estimate a logistic regression model for these
data. The following program accomplishes this task:
PROC LOGISTIC DATA=jobyrs;
CLASS year / PARAM=GLM;
MODEL quit(DESC)=ed prestige salary year;
RUN;

By specifying YEAR as a CLASS variable, we tell PROC LOGISTIC to treat
this variable as categorical rather than quantitative. The PARAM=GLM
option overrides the default effect coding and tells PROC LOGISTIC to
create a set of four indicator (dummy) variables, with the reference
category being YEAR=5 (the highest value). The DESC option specifies that
the model will predict the probability of a 1 for QUIT rather than a 0.
Output 7.2 shows the results. Comparing these estimates with the
partial likelihood estimates in Output 5.10, we see that the coefficients of
ED, PRESTIGE, and SALARY are similar, as are the chi-square statistics.
Again, this is not surprising because they are simply alternative ways of
estimating the same model.

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Output 7.2 Maximum Likelihood Estimates of Discrete-Time Logistic Model for Job Duration Data

Type 3 Analysis of Effects

Effect
ed
prestige
salary
year

Wald
Chi-Square

Pr > ChiSq

1
1
1
4

6.8392
46.5794
6.6791
23.2529

0.0089
<.0001
0.0098
0.0001

Analysis of Maximum Likelihood Estimates

Parameter

Estimate

Standard
Error

Wald
Chi-Square

Pr > ChiSq

Intercept
ed
prestige
salary
year
year
year
year
year

1
1
1
1
1
1
1
1
0

3.4443
0.2249
-0.1235
-0.0268
-2.6875
-1.4475
-0.0130
0.2355
0

1.1821
0.0860
0.0181
0.0104
0.8327
0.7671
0.7272
0.7779
.

8.4896
6.8392
46.5794
6.6791
10.4174
3.5605
0.0003
0.0916
.

0.0036
0.0089
<.0001
0.0098
0.0012
0.0592
0.9857
0.7621
.

1
2
3
4
5

Unlike partial likelihood, the maximum likelihood method also
gives us estimates for the effect of time on the odds of quitting, as reflected
in the αts in equation (7.1). The intercept in Output 7.2 is an estimate of
α5, the log-odds of quitting in year 5 for a person with values of 0 on all
covariates. For level j of the YEAR variable, the coefficient is an estimate of
αj – α5 (that is, the difference in the log-odds of quitting in year j and the
log-odds of quitting in year 5, controlling for the covariates). We see that
the log-odds is lowest in the first year of the job, rises steadily to year 3,
and then stays approximately constant for the next 2 years. Overall, the
effect of YEAR is highly significant with a Wald chi-square statistic of
23.25 with 4 d.f., as shown in the Type 3 table.
When the model in equation (7.1) is estimated by partial likelihood,
there can be no restrictions on the αts. With the maximum likelihood
method of this chapter, however, we can readily estimate restricted
versions of the model. In fact, because time (in this case, YEAR) is just
another variable in the regression model, we can specify the dependence
of the hazard on time as any function that SAS allows in the DATA
statement. For example, if we remove YEAR from the CLASS statement but

CHAPTER 7

Analysis of Tied or Discrete Data with PROC LOGISTIC

240

keep it in the MODEL statement, we constrain the effect of YEAR to be
linear on the log-odds of quitting. Alternatively, we can take the logarithm
of YEAR before putting it in the model, or we can fit a quadratic model
with YEAR and YEAR squared. To compare the fit of these models, we can
use the AIC (Akaike’s information criterion) or SC (Schwarz’s criterion)
statistics reported by PROC LOGISTIC. These are simply adjustments to –2
times the log-likelihood that penalize models for having more parameters.
Here are the results:

Unrestricted
Linear
Quadratic
Logarithmic

AIC

215.67
216.28
212.19
211.98

244.21
234.31
233.82
230.00

For both fit measures, smaller values are better. Clearly, the
logarithmic model is the best. The coefficients for ED, PRESTIGE, and
SALARY in the logarithmic model (not shown) hardly change at all from
the unrestricted model. The coefficient for the logarithm of YEAR is 2.00,
indicating that a 1 percent increase in time in the job is associated with a
2 percent increase in the odds of quitting.

THE COMPLEMENTARY LOG-LOG MODEL
PROCESSES

FOR

CONTINUOUS-TIME

As already noted, the logit model presumes that events can occur
only at discrete points in time. For most applications, however, ties occur
because event times are measured coarsely even though events can
actually occur at any point in time. Aside from the implausibility of the
logit model for such data, the logit model suffers from a lack of invariance
to the length of the time interval. In other words, switching from personmonths to person-years changes the model in a fundamental way, so that
coefficients are not directly comparable across intervals of different length.
To avoid these difficulties, we can first specify a model for
continuous-time data and from that derive a model for data grouped into
intervals. (In essence, that’s how the EXACT method was developed for
PROC PHREG.) Suppose that the intervals are of equal length beginning at
the origin. We’ll index them by t=1, 2, 3, …. As before, let Pit be the
probability that an event occurs to individual i in interval t, given that the
individual did not have events in any of the earlier intervals. If we now

CHAPTER 7

Analysis of Tied or Discrete Data with PROC LOGISTIC

241

assume that events are generated by Cox’s proportional hazards model, it
follows (Prentice and Gloeckler, 1978) that

log[− log(1 − Pit )] = α t + β1 xit1 +  + β k xitk .

(7.2)

This result is easily obtained using equation (5.14) in Chapter 5. The
transformation on the left side is called the complementary log-log
function, which is also what we call the model.
Like the logit function, the complementary log-log function takes a
quantity that varies between 0 and 1 and changes it to something that
varies between minus and plus infinity. Unlike the logit function, however,
the complementary log-log function is asymmetrical. For example, after
taking the logit transformation, a change in probability from .25 to .50 is
the same as from .50 to .75. On the complementary log-log scale, however,
the difference between probabilities of .25 and .50 is larger than the
difference between .50 and .75. This difference has an important practical
implication. In the logit model, switching the values of the binary
dependent variable merely changes the signs of the coefficients. In the
complementary log-log model, on the other hand, switching the values
produces completely different coefficient estimates, and it can even result
in nonconvergence of the maximum likelihood algorithm. It’s essential,
then, that the model be set up to predict the probability of an event rather
than a non-event.
Another important point about the model in equation (7.2) is that the
β coefficients are identical to the coefficients in the underlying
proportional hazards model. That doesn’t mean that the estimates that you
get from grouped data will be the same as those from the original
ungrouped data. What it does mean is that both are estimating the same
underlying parameters and are directly comparable to each other. It also
means that the complementary log-log coefficients have a relative hazard
interpretation, just like Cox model coefficients, and that the model (not the
estimates) is invariant to interval length.
Originally, people preferred the logit model because there was little
software for the complementary log-log model. Now there’s no excuse.
Many popular packages have complementary log-log options, and SAS
makes it available in the LOGISTIC, PROBIT, GENMOD, and GLIMMIX
procedures. In PROC LOGISTIC, PROC GENMOD, and PROC GLIMMIX,
you specify it with the LINK=CLOGLOG option in the MODEL statement.
In PROC PROBIT, you specify D=GOMPERTZ as an option in the MODEL
statement. (D=GOMPERTZ is a misnomer. The correct name of the
distribution function corresponding to the complementary log-log is
Gumbel, not Gompertz.)

CHAPTER 7

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242

To estimate the complementary log-log model for the job duration
data, we can use exactly the same SAS statements that we used for the
logit model, except for the addition of the LINK=CLOGLOG to the MODEL
statement. The coefficients in Output 7.3 are directly comparable to those
in Output 5.9 for the EXACT method in PROC PHREG. Indeed, they are
very similar—much closer to each other than either is to the two
approximate methods in Output 5.9.

Output 7.3 Maximum Likelihood Estimates of the Complementary Log-Log Model for Job Duration Data

Parameter

Estimate

Standard
Error

Wald
Chi-Square

Pr > ChiSq

Intercept
ed
prestige
salary
year
year
year
year
year

1
1
1
1
1
1
1
1
0

2.1952
0.1655
-0.0926
-0.0229
-2.0952
-1.1097
0.0489
0.2131
0

0.8868
0.0632
0.0126
0.00895
0.6600
0.6156
0.5840
0.6287
.

6.1270
6.8470
53.8990
6.5623
10.0777
3.2489
0.0070
0.1149
.

0.0133
0.0089
<.0001
0.0104
0.0015
0.0715
0.9333
0.7346
.

1
2
3
4
5

We can interpret the coefficients just as if this were a proportional
hazards model. In particular, exponentiating the coefficients gives us
hazard ratios. For example, a 1-year increase in education produces a
100(exp(.1655)−1)=18 percent increase in the hazard of quitting. A oneunit increase in prestige yields a 100(exp(–.0926)–1) = –9 percent change
in the hazard of quitting. For a thousand-dollar increase in salary, we get a
100(exp(–0.0229)–1) = –2 percent change in the hazard of quitting.
Again, we see a strong effect of YEAR, with approximately the same
pattern as in the logit model. As with that model, we can easily estimate
constrained effects of YEAR by including the appropriate transformed
variable as a covariate. Some of these constrained models have familiar
names. If the logarithm of YEAR is a covariate, then we are estimating a
Weibull model. If we just include YEAR itself (without the CLASS
statement), we have the Gompertz model described briefly in Chapter 2,
“Basic Concepts of Survival Analysis,” but not otherwise available in SAS.
Comparing the results in Output 7.3 with those for the logit model in
Output 7.2, we see that the coefficients are somewhat larger for the logit
model, but the chi-square statistics and p-values are similar. This is the

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Analysis of Tied or Discrete Data with PROC LOGISTIC

243

usual pattern. Only rarely do the two methods lead to different qualitative
conclusions.

DATA WITH TIME-DEPENDENT COVARIATES
Because the maximum likelihood method is particularly effective at
handling time-dependent covariates, let’s look at an example with three
covariates that change over time. The sample consists of 301 male
biochemists who received their doctorates in 1956 or 1963. At some point
in their careers, all of these biochemists had jobs as assistant professors at
graduate departments in the U.S. The event of interest is a promotion to
associate professor. The biochemists were followed for a maximum of 10
years after beginning their assistant professorships. For a complete
description of the data and its sources, see Long, Allison, and McGinnis
(1993). That article focuses on comparisons of men and women, but here I
look only at men in order to simplify the analysis. The original data set has
one record per person and includes the following variables:
DUR

the number of years from beginning of the job to promotion
or censoring.
EVENT
has a value of 1 if the person was promoted; otherwise,
EVENT has a value of 0.
UNDGRAD
a measure of the selectivity of undergraduate institution
(ranges from 1 to 7).
PHDMED
has a value of 1 if the person received his Ph.D. from a
medical school; otherwise, PHDMED has a value of 0.
PHDPREST
a measure of the prestige of the person’s Ph.D. institution
(ranges from 0.92 to 4.62).
ART1-ART10 the cumulative number of articles the person published in
each of the 10 years.
CIT1-CIT10
the number of citations in each of the 10 years to all
previous articles.
PREST1
a measure of the prestige of the person’s first employing
institution (ranges from 0.65 to 4.6).
PREST2
a measure of the prestige of the person’s second employing
institution (coded as missing for those who did not change
employers). No one had more than two employers during
the period of observation.
JOBTIME
the year of employer change, measured from the start of the
assistant professorship (coded as missing for those who did
not change employers).

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The covariates describing the biochemists’ graduate and
undergraduate institutions are constant over time, but article counts,
citation counts, and employer prestige all vary with time. The citation
counts, taken from Science Citation Index (Institute for Scientific
Information), are sometimes interpreted as a measure of the quality of a
scientist’s published work, but it may be more appropriate to regard them
as a measure of the impact on the work of other scientists.
The first and most complicated step is to convert the file of 301
persons into a file of person-years. The following DATA step accomplishes
that task:
DATA rankyrs;
INFILE 'c:rank.dat';
INPUT dur event undgrad phdmed phdprest art1-art10
cit1-cit10 prest1 prest2 jobtime;
ARRAY arts(*) art1-art10;
ARRAY cits(*) cit1-cit10;
IF jobtime=. THEN jobtime=11;
DO year=1 TO dur;
IF year=dur THEN promo=event;
ELSE promo=0;
IF year GE jobtime THEN prestige=prest2;
ELSE prestige=prest1;
articles=arts(year);
citation=cits(year);
OUTPUT;
END;
RUN;

This DATA step has the same basic structure as the one for the job
duration data. The DO loop creates and outputs a record for each personyear, for a total of 1,741 person-years. Within the DO loop, the dependent
variable (PROMO) is assigned a value of 1 if a promotion occurred in that
person-year; otherwise, PROMO has a value of 0.
What’s new is that the multiple values of each time-dependent
covariate must be read into a single variable for each of the person-years.
For the article and citation variables, which change every year, we create
arrays that enable us to refer to, say, articles in year 3 as ARTS(3). For the
two job prestige variables, we must test to see whether the current year (in
the DO loop) is greater than or equal to the year in which a change
occurred. If it is, we assign the later value to the variable PRESTIGE;
otherwise, PRESTIGE is assigned the earlier value. Note that for this to
work, we recode the JOBTIME variable so that missing values (for people
who didn’t have a second employer) are recoded as 11. That way the DO

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loop index, which has a maximum value of 10, never equals or exceeds
this value.
Now we can proceed to estimate regression models. For academic
promotions, which usually take effect at the beginning of an academic
year, it makes sense to view time as being truly discrete. A logit model,
then, seems entirely appropriate. Using PROC LOGISTIC, we specify the
model with a quadratic effect of YEAR:
PROC LOGISTIC DATA=rankyrs;
MODEL promo(DESC)=undgrad phdmed phdprest articles citation
prestige year year*year;
RUN;

Output 7.4 Estimates of Logit Model for Academic Promotions

Parameter

Estimate

Standard
Error

Wald
Chi-Square

Pr > ChiSq

Intercept
undgrad
phdmed
phdprest
articles
citation
prestige
year
year*year

1
1
1
1
1
1
1
1
1

-8.4845
0.1939
-0.2357
0.0271
0.0734
0.000126
-0.2535
2.0818
-0.1586

0.7756
0.0635
0.1718
0.0932
0.0181
0.00131
0.1138
0.2338
0.0203

119.6539
9.3236
1.8830
0.0843
16.3702
0.0091
4.9628
79.3132
61.0097

<.0001
0.0023
0.1700
0.7715
<.0001
0.9238
0.0259
<.0001
<.0001

Odds Ratio Estimates

Effect
undgrad
phdmed
phdprest
articles
citation
prestige

Point
Estimate
1.214
0.790
1.027
1.076
1.000
0.776

95% Wald
Confidence Limits
1.072
0.564
0.856
1.039
0.998
0.621

1.375
1.106
1.233
1.115
1.003
0.970

Output 7.4 shows the results. Not surprisingly, there is a strong effect
of number of years as an assistant professor. The odds of a promotion
increase rapidly with time, but at a decreasing rate; there is actually some
evidence of a reversal after 7 years. I also estimated a model with a set of

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246

10 dummy variables for years as an assistant professor, but a likelihood
ratio chi-square test showed no significant difference between that model
and the more restricted version shown here. Higher-order polynomials also
failed to produce a significant improvement. On the other hand, the chisquare test for comparing the model in Output 7.4 and a model that
excluded both YEAR and YEAR2 was 178.98 with 2 d.f.
Article counts had the next largest effect (as measured by the Wald
chi-square test): each additional published article is associated with an
increase of 7.6 percent in the odds of a promotion. But there is no evidence
of any effect of citations, suggesting that it’s the quantity of publications
that matters in promotions, not the importance or impact of the published
work.
Somewhat surprisingly, while there is no effect of the prestige of the
institution where a biochemist got his doctorate, there is a substantial
effect of the selectivity of his undergraduate institution. Each 1-point
increase on the 7-point selectivity scale is associated with a 21 percent
increase in the odds of a promotion, controlling for other covariates. There
is also a slightly negative effect of the prestige of the current employer,
suggesting that it may be harder to get promoted at a more prestigious
department.
Notice that once the person-year data set is created, the
time-dependent covariates are treated just like fixed covariates. Thus,
many models can be estimated with the saved data set without additional
data manipulations, making it especially convenient to estimate models
with large numbers of time-dependent covariates.

ISSUES AND EXTENSIONS
In this section, we look at a number of complications and concerns
that arise in the analysis of tied data using maximum likelihood methods.
Dependence Among the Observations?

A common reaction to the methods described in this chapter is that
there must be something wrong. In general, when multiple observations
are created for a single individual, it’s reasonable to suppose that those
observations are not independent, thereby violating a basic assumption
used to construct the likelihood function. The consequence of dependence
is usually biased standard error estimates and inflated test statistics. Even
worse, there are different numbers of observations for different
individuals, so some appear to get more weight than others.

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While concern about dependence is often legitimate, it is not
applicable here. In this case, the creation of multiple observations is not an
ad-hoc method; rather, it follows directly from factoring the likelihood
function for the data (Allison, 1982). The basic idea is this: In its original
form, the likelihood for data with no censoring can be written as a product
of probabilities over all n observations, as follows:
n

∏ Pr(T = t )
j =1

(7.3)

where Ti is the random variable and ti is the particular value observed for
individual i. Each of the probabilities in equation (7.3) can be factored in
the following way. If ti =5, we have

Pr(Ti = 5) = Pi 5 (1 − Pi 4 )(1 − Pi 3 )(1 − Pi 2 )(1 − Pi1 )

(7.4)

where, again, Pit is the conditional probability of an event at time t, given
that an event has not already occurred. This factorization follows directly
from the definition of conditional probability. Each of the five terms in
equation (7.4) behaves as if it came from a distinct, independent
observation.
For those who may still be unconvinced, a comparison of the
standard errors in Output 5.10 with those in Output 7.2 should be
reassuring. They are virtually identical, despite the fact that the partial
likelihood estimates in Output 5.10 are based on 100 persons, while the
maximum likelihood estimates in Output 7.2 are based on 272 personyears.
This lack of dependence holds only when no individual has more
than one event. When events are repeatable, as discussed in Chapter 8,
“Heterogeneity, Repeated Events, and Other Topics,” there is a real
problem of dependence. But the problem is neither more nor less serious
than it is for other methods of survival analysis.
Handling Large Numbers of Observations

Although the creation of multiple observations for each individual
does not violate any assumptions about independence, it may cause
practical problems when the number of individuals is large and the time
intervals are small. For example, if you have a sample of 1,000 persons
observed at monthly intervals over a 5-year period, you could end up with
a working data set of nearly 60,000 person-months. While this is not an
impossibly large number, it can certainly increase the time you spend
waiting for results, thereby inhibiting exploratory data analysis.
If you find yourself in this situation, there are several options you
may want to consider:

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First, you should ask yourself if you really need to work with
such small time intervals. If the time-dependent covariates are
only changing annually, you might as well switch from personmonths to person-years. True, there will be some loss of
precision in the estimates, but this is usually minimal. And by
switching to the piecewise exponential model described in
Chapter 4 (which has a similar data structure), you can even
avoid the loss of precision. On the other hand, if the timedependent covariates are changing at monthly intervals, you
should not aggregate to larger intervals.
 Second, if your covariates are all categorical or they at least have
a small number of levels, you can achieve great computational
economy by estimating the models from data that are grouped by
covariate values. You can accomplish this by using PROC
SUMMARY to create a grouped data set and then using the
grouped data syntax in PROC LOGISTIC. Alternatively, PROC
CATMOD will automatically group the data when you specify a
logit model, and you can save the grouped data set for further
analysis.
 Third, if you are estimating logit models, you may want to
sample on the dependent variable, at least for exploratory
analysis. Typically, the data sets created for the methods in this
chapter have a dichotomous dependent variable with an extreme
split—the number of nonevents will be many times larger than
the number of events. What you can do is take all the
observations with events and a random subsample of the
observations without events, so that the two groups are
approximately equal. Is this legitimate? Well, it is for the logit
model (but not for the complementary log-log model). It is well
known that disproportionate stratified sampling on the
dependent variable in a logit analysis does not bias coefficient
estimates (Prentice and Pike, 1979).


Unequal Intervals

To this point, we have assumed that the time intervals giving rise to
tied survival times are all of equal length. It’s not uncommon, however, for
some intervals to be longer than others, either by design or by accident.
For example, a study of deaths following surgery may have weekly followups soon after the surgery when the risk of death is highest and then
switch to monthly follow-ups later on. Some national panel studies, like
the Panel Survey of Income Dynamics, were conducted annually in most
years, but not in all. As a result, some intervals are 2 years long instead of

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1. Even if the intervals are equal by design, it often happens that some
individuals cannot be reached on some follow-ups.
Regardless of the reason, it should be obvious that, other things
being equal, the probability of an event will increase with the length of the
interval. And if time-dependent covariates are associated with interval
length, the result can be biased coefficient estimates. The solution to this
problem depends on the pattern of unequal intervals and the model being
estimated.
There is one case in which no special treatment is needed. If you are
estimating the models in equation (7.1) or equation (7.2), which place no
restrictions on the effect of time, and if the data are structured so that
every individual’s interval at time t is the same length as every other
individual’s interval at time t, then the separate parameters that are
estimated for every time interval automatically adjust for differences in
interval length. This situation is not as common as you might think,
however. Even when intervals at the same calendar time have the same
length, intervals at the same event time will have different lengths
whenever individuals have different origin points.
In all other cases, an ad-hoc solution will usually suffice: simply
include the length of the interval as a covariate in the model. If there are
only two distinct interval lengths, a single dummy variable will work. If
there are a small number of distinct lengths, construct a set of dummy
variables. If there are many different lengths, you will probably need to
treat length as a continuous variable but include a squared term in the
model to adjust for nonlinearity.
Empty Intervals

In some data sets, there are time intervals in which no individual
experiences an event. For example, in the LEADER data set that we
analyzed in Chapter 6, “Competing Risks,” none of the 472 leaders lost
power in the 22nd year of rule. Naturally, this is most likely to occur when
the number of time intervals is large and the number of individuals is
small. Whenever there are empty intervals, if you try to estimate a model
with an unrestricted effect of time, as in equation (7.1) or equation (7.2),
the maximum likelihood algorithm will not converge. This result is a
consequence of the following general principle. For any dichotomous
covariate, consider the 2 × 2 contingency table formed by that covariate
and the dichotomous dependent variable. If any of the four cells in that
table has a frequency of 0, the result is nonconvergence.
An easy solution is to fit a model with restrictions on the time effect.
The quadratic function that was estimated in Output 7.4, for example, will
not suffer from this problem. Alternatively, if you don’t want to lose the

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250

flexibility of the unrestricted model, you can constrain the coefficient for
any empty interval to be the same as that of an adjacent interval. The
simplest way to do this is to recode the variable containing the interval
values so that the adjacent intervals have the same value. For the LEADER
data, this requires a DATA step with the following statement:
IF year=22 THEN year=21;

Then specify YEAR as a CLASS variable in PROC LOGISTIC. Instead of
separate dummy (indicator) variables for years 21 and 22, this code
produces one dummy variable equal to 1 if an event time was equal to
either of those 2 years; otherwise, the variable is equal to 0.
Left Truncation

In Chapter 5, we discussed a problem known as left truncation, in
which individuals are not at risk of an event until some time after the
origin time. This commonly occurs when “survivors” are recruited into a
study at varying points in time. We saw how this problem can be easily
corrected with the partial likelihood method using PROC PHREG. The
solution is equally easy for the maximum likelihood methods discussed in
this chapter, although quite different in form. In creating the multiple
observations for each individual, you simply delete any time units in
which the individual is known not to be at risk of an event. For example, if
patients are recruited into a study at various times since diagnosis, no
observational units are created for time intervals that occurred before
recruitment. We can still include time since diagnosis as a covariate,
however.
This method also works for temporary withdrawals from the risk set.
If your goal is to predict migration of people from Mexico to the U.S., you
will probably want to remove anyone from the risk set when he or she was
in prison, in the military, and so on. Again, you simply exclude any time
units in which the individual is definitely not at risk.
Competing Risks

In Chapter 6, we saw that the likelihood function for data arising
from multiple event types can be factored into a separate likelihood
function for each event type, treating other event types as censoring.
Strictly speaking, this result only holds when time is continuous and
measured precisely, so that there are no ties. When time is discrete or
when continuous-time data are grouped into intervals, the likelihood
function does not factor. If you want full-information maximum likelihood
estimates, you must estimate a model for all events simultaneously. On the

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other hand, it’s also possible to do separate analyses for each event type
without biasing the parameter estimates and with only slight loss of
precision.
These points are most readily explained for the logit model for
competing risks. Let Pijt be the conditional probability that an event of type
j occurs to person i at time t, given that no event occurs before time t. The
natural extension of the logit model to multiple event types is the
multinomial logit model:

Pijt =

exp(β j xit )

1 + ∑ exp(β k xit )

j=1, 2, …

or, equivalently

 Pijt
log
 Pi 0t


 = β j x it


j=1, 2, …

where Pi0t is the probability that no event occurs at time t to individual i. As
in the case of a single event type, the likelihood for data arising from this
model can be manipulated so that each discrete-time point for each
individual appears as a separate observation. If there are, say, three event
types, these individual time units can have a dependent variable coded 1, 2,
or 3 if an event occurred and coded 0 if no event occurred. We can then use
PROC LOGISTIC to estimate the model simultaneously for all event types.
For the job duration data analyzed earlier in this chapter, suppose
that there are actually two event types, quitting (EVENT=1) and being fired
(EVENT=2), with EVENT=0 for censored cases. The following statements
produce a person-year data set with a dependent variable called
OUTCOME, which is coded 1 for quit, 2 for fired, and 0 for neither:
DATA jobyrs2;
SET jobdur;
DO year=1 TO dur;
IF year=dur THEN outcome=event;
ELSE outcome=0;
OUTPUT;
END;
RUN;

We can then use PROC LOGISTIC to estimate the multinomial logit model:
PROC LOGISTIC data=jobyrs2;
MODEL outcome(REF='0')=ed prestige salary year
/ LINK=GLOGIT;
RUN;

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252

The REF='0' option makes 0 the reference category for the dependent
variable. The default is to use the highest value as the reference category,
but in survival analysis we want each event type to be compared with no
event. The LINK=GLOGIT option requests an unordered multinomial logit
model, rather than the default cumulative logit model.
Output 7.5 displays the results. The Type 3 table gives statistics for
testing the null hypothesis that both coefficients for each covariate are 0, a
hypothesis that is clearly rejected for ED, PRESTIGE, and YEAR. The effect
of SALARY is less dramatic but still significant at the .03 level. The lower
portion of the table gives the coefficient estimates and their respective test
statistics. All the parameters labeled 1 pertain to the contrast between type
1 (quit) and no event, while all the parameters labeled 2 pertain to the
contrast between type 2 (fired) and no event. We see that education
increases the odds of quitting but reduces the odds of being fired, with
both coefficients highly significant. The prestige of the job, on the other
hand, reduces the risk of quitting while increasing the risk of being fired.
Again, both coefficients are highly significant. Finally, the odds of quitting
increase markedly with each year, but the odds of being fired hardly
change at all.

Output 7.5 PROC LOGISTIC Results for Competing Risks Analysis of Job Data

Type 3 Analysis of Effects

Effect
ed
prestige
salary
year

Wald
Chi-Square

Pr > ChiSq

2
2
2
2

21.0701
66.5608
6.7535
19.5623

<.0001
<.0001
0.0342
<.0001

(continued)

CHAPTER 7

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253
Output 7.5 (continued)

Analysis of Maximum Likelihood Estimates

Parameter

outcome

Intercept
Intercept
ed
ed
prestige
prestige
salary
salary
year
year

1
2
1
2
1
2
1
2
1
2

Estimate

Standard
Error

Wald
Chi-Square

Pr > ChiSq

1
1
1
1
1
1
1
1
1
1

0.3286
-1.5491
0.1921
-0.7941
-0.1128
0.1311
-0.0255
0.0110
0.7725
-0.00962

0.8196
1.6921
0.0836
0.2011
0.0169
0.0274
0.0103
0.0150
0.1748
0.2976

0.1608
0.8381
5.2809
15.5874
44.4661
22.8174
6.1743
0.5333
19.5367
0.0010

0.6885
0.3599
0.0216
<.0001
<.0001
<.0001
0.0130
0.4652
<.0001
0.9742

Now, we’ll redo the analysis with separate runs for each event type.
For this, we rely on a well-known result from multinomial logit analysis
(Begg and Gray, 1984). To estimate a model for event type 1, simply
eliminate from the sample all the person-years in which events of type 2
occurred. Then, do a binomial logit analysis for type 1 versus no event. To
estimate a model for event type 2, eliminate all the person-years in which
events of type 1 occurred. Then, do a binomial logit analysis for type 2
versus no event. Here’s the SAS code that accomplishes these tasks:
PROC LOGISTIC DATA=jobyrs2;
WHERE outcome NE 2;
MODEL outcome(DESC)=ed prestige salary year;
RUN;
PROC LOGISTIC DATA=jobyrs2;
WHERE outcome NE 1;
MODEL outcome(DESC)=ed prestige salary year;
RUN;

This procedure is justified as a form of conditional maximum
likelihood. The resulting estimates are consistent and asymptotically
normal, but there is some loss of precision, at least in principle. In
practice, both the coefficients and their estimated standard errors usually
differ only trivially from those produced by the simultaneous estimation
procedure. We can see this by comparing the results in Output 7.6 with
those in Output 7.5. The advantages of separating the estimation process
are that you can

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254

focus on only those event types in which you are interested
 specify quite different models for different event types, with
different covariates and different functional forms.


Output 7.6 PROC LOGISTIC Results for Separate Estimation of Quitting and Firing Models

Quittings
Parameter

Estimate

Standard
Error

Wald
Chi-Square

Pr > ChiSq

Intercept
ed
prestige
salary
year

1
1
1
1
1

0.3672
0.1895
-0.1125
-0.0255
0.7637

0.8254
0.0840
0.0169
0.0103
0.1742

0.1979
5.0911
44.3193
6.1688
19.2200

0.6564
0.0240
<.0001
0.0130
<.0001

Parameter

Estimate

Standard
Error

Wald
Chi-Square

Pr > ChiSq

Intercept
ed
prestige
salary
year

1
1
1
1
1

-1.5463
-0.7871
0.1300
0.0118
-0.0296

1.7054
0.1999
0.0274
0.0155
0.2913

0.8221
15.5022
22.5041
0.5845
0.0103

0.3646
<.0001
<.0001
0.4445
0.9190

Firings

Now, what about competing risks for the complementary log-log
model? Here, things are a little messier. It is possible to derive a
multinomial model based on a continuous-time proportional hazards
model, and this could be simultaneously estimated for all event types
using maximum likelihood. This is not a standard problem, however, and
no SAS procedure will do it without a major programming effort. Instead,
we can use the same strategy for getting separate estimates for each event
type that we just saw in the case of the logit model. That is, we delete all
individual time units in which events other than the one of interest
occurred. Then we estimate a dichotomous complementary log-log model
for the event of interest versus no event. In effect, we are deliberately
censoring the data at the beginning of any intervals in which other events
occur. This should not be problematic because, even if we have
continuous-time data, we have to assume that the different event types are
noninformative for one another. Once we eliminate all extraneous events
from the data, we reduce the problem to one for a single event type. Again,
there will be some slight loss of information in doing this.

CHAPTER 7

Analysis of Tied or Discrete Data with PROC LOGISTIC

CONCLUSION
The maximum likelihood methods discussed in this chapter are
attractive alternatives to partial likelihood when there are many ties and
many time-dependent covariates. Not only are they much more
computationally efficient, but they also give direct estimates of the effect of
time on the hazard of an event. The methods discussed here have
considerable intuitive appeal and are relatively straightforward to
implement. Once the expanded data set is constructed, the analyst can
proceed as in an ordinary logistic regression analysis with no need to treat
time-dependent covariates any differently than fixed covariates. This
approach also has advantages whenever there is ambiguity about the time
origin. Because time is treated just like any other covariate, there is great
flexibility in specifying and testing alternative functional forms, and
multiple time scales with different origins can be included in the model.

255

CHAPTER 7

256

Analysis of Tied or Discrete Data with PROC LOGISTIC

CHAPTER 8
Heterogeneity, Repeated Events, and Other Topics
Introduction 257
Unobserved Heterogeneity 257
Repeated Events 260
2

Generalized R

282

Sensitivity Analysis for Informative Censoring 283

INTRODUCTION
This chapter deals with several issues that arise for all the methods
that we have previously discussed. The first two issues—heterogeneity and
repeated events—are closely related. One problem with the models that we
have been considering is that biological and social entities usually differ in
ways that are not fully captured by the model. This unobserved
heterogeneity can produce misleading estimates of hazard functions and
attenuated estimates of covariate effects. When individuals can experience
more than one event, unobserved heterogeneity can also produce
dependence among the observations, leading to biased standard errors and
test statistics. We’ll survey a variety of methods for dealing with these
2
problems. Later in the chapter, we’ll see how to compute a generalized R
and how to gauge the possible consequences of informative censoring.

UNOBSERVED HETEROGENEITY
An implicit assumption of all the hazard models that we have
considered so far is that if two individuals have identical values on the
covariates, they also have identical hazard functions. If there are no
covariates in the model, then the entire sample is presumed to have a
single hazard function. Obviously, this is an unrealistic assumption.
Individuals and their environments differ in so many respects that no set
of measured covariates can possibly capture all the variation among them.
In an ordinary linear regression model, this residual or unobserved

CHAPTER 8

Heterogeneity, Repeated Events, and Other Topics

258

heterogeneity is explicitly represented by a random disturbance term, for
example,

y = βx + ε
where ε represents all unmeasured sources of variation in y. But in a Cox
regression model, for example, there is no disturbance term:

log h(t ) = α (t ) + βx .
The absence of a random disturbance term does not mean that the
model is deterministic. There is plenty of room for randomness in the
relationship between the unobserved hazard h(t) and the observed event
time. Nevertheless, the presence of unobserved heterogeneity can lead to
incorrect conclusions about the dependence of the hazard function on
time, and it can also produce biased estimates of the covariate coefficients.
The most serious problem is this: unobserved heterogeneity tends to
produce estimated hazard functions that decline with time, even when the
true hazard is not declining for any individual in the sample (Proschan,
1963; Heckman and Singer, 1985). This fact is most easily explained by an
example. Suppose we have a sample of 100 people, all of whom have
hazards that are constant over time. The sample is equally divided
between two kinds of people: those with a high hazard of death (h(t)=2.0)
and those with a low hazard of death (h(t)=0.5). Unfortunately, we don’t
know which people have which hazard, so we must estimate a hazard
function for the entire sample. Figure 8.1 shows what happens. The
empirical hazard function starts out, as you might expect, midway
between .5 and 2. But then it steadily declines until it approaches .5 as an
asymptote. What’s happening is that the high hazard people are dying
more rapidly at all points in time. As a result, as time goes by, the
remaining sample (the risk set) is increasingly made up of people with low
hazards. Because we can estimate the hazard function at time t with only
those who are still at risk at time t, the estimated hazard will be more and
more like the lower hazard.

CHAPTER 8

Heterogeneity, Repeated Events, and Other Topics

259
Figure 8.1 Empirical Hazard Function Produced by Mixing Two Constant Hazards

Of course, any real life situation will have more than two kinds of
people, but the basic principle is the same. Those with higher hazards will
tend to die (or experience whatever event is being studied) before those
with lower hazards, leaving a risk set that is increasingly made up of low
hazard people.
This problem has led to some potentially serious errors in
interpreting research results. In the management literature, for example,
there has been great interest in the hypothesis of the liability of newness
(Hannan and Freeman, 1984). This hypothesis says that when firms are
just starting out, they are prone to failure because they lack capital,
contacts, traditions, and so on. As firms get older and accumulate
resources, they become more resistant to failure. Empirical studies
invariably show that, in fact, younger firms do have higher rates of failure
than older firms. But this fact does not necessarily prove the hypothesis.
The results are equally consistent with the hypothesis that firms differ in
their initial vulnerability to failure. Weaker firms go out of business
quickly while stronger firms survive.
What can be done? As we’ll see in the next section, when events are
repeatable, it is quite feasible to separate the true hazard function from
unobserved heterogeneity. But when events are not repeatable, as with
both human and organizational death, the options are limited. If you find
an empirical hazard function that is increasing, then you can validly

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conclude that the true hazard is increasing for at least some fraction of the
sample over some interval of time. But decreasing hazard functions are
inherently ambiguous.
There have been numerous attempts to separate the hazard function
from unobserved heterogeneity by formulating models that incorporate
both. For example, we can insert a random disturbance term into a Weibull
hazard model:

log h(t ) = α log t + βx + ε .
But models like this are highly sensitive to the choice of a distribution for ε
or the form of the dependence on time. If such models are identified at all,
it is only because of the imposition of a particular functional form. That
situation is not conducive to drawing reliable conclusions.
What about estimates of the β coefficients? Are they affected by
unobserved heterogeneity? Given what we know about linear models, it’s a
foregone conclusion that coefficients may be severely biased if the
unobserved components are correlated with the measured covariates. The
more interesting question is what happens when the unobserved
disturbance is independent of the measured covariates. Early literature on
this question seemed to suggest that coefficient estimates could be grossly
biased in unexpected ways. But the work by Gail and colleagues (1984) is
more persuasive. They showed that unobserved heterogeneity tends to
attenuate the estimated coefficients toward 0. On the other hand, standard
errors and test statistics are not biased. Therefore, a test of the hypothesis
that a coefficient is 0 remains valid, even in the presence of unobserved
heterogeneity. It’s also important to realize that the attenuation of
coefficients is not a problem unique to hazard models, but it occurs with a
wide variety of nonlinear models, including logistic regression (Allison,
1987).

REPEATED EVENTS
All the models and analyses in the preceding chapters presume that
no individual experiences more than one event. That’s a reasonable
presumption if the event is death. But most events in the social sciences
are repeatable: births, marriages, job changes, promotions, arrests,
residence changes, and so on. There are also many repeatable events that
are of interest to biomedical scientists: tumor recurrences, seizures, urinary
infections, and hospitalizations, to name only a few. Because so much of
survival analysis has focused on deaths, good methods for handling
repeated events were relatively slow in coming. Nevertheless, several

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important developments have occurred in recent years, and some of those
methods are now available in SAS. Before examining the newer methods,
however, we need to consider what’s wrong with conventional methods.
Problems with Conventional Methods

There are basically two approaches to analyzing repeated events
with standard software. First, you can do a separate analysis for each
successive event. Suppose, for example, that you have reproductive
histories for a sample of women who are married or who have ever been
married, and you want to estimate a model for birth intervals. You start
with an analysis for the interval between marriage and the first birth. For
all those women who had a first birth, you then do a second analysis for
the interval between first birth and second birth. You continue in this
fashion until the number of women gets too small to reliably estimate a
model.
For a more detailed example, let’s look at some simulated data on job
changes. For 100 people, I generated repeated job durations, from the point
of entry into the labor market until 20 years had elapsed, at which time
observation was terminated. Thus, people whose first job lasted more than
20 years were censored at year 20. Everyone else had exactly one censored
job (the one still in progress in year 20) and at least one uncensored job. A
total of 395 jobs were produced.
Each person was assigned a certain number of years of schooling
(ED) that did not vary over jobs. Each job had a prestige score (ranging
from 1 to 100), as well as a salary (in thousands of dollars) that remained
constant during the job. The duration for each job was generated by a
Weibull model that included ED, PRESTIGE, and the logarithm of
SALARY as covariates. Coefficients were invariant across people and
across time.
Output 8.1 displays the first 20 cases out of the 395 jobs in the
working data set. ID is a variable that distinguishes different people. We
see that these 20 jobs were held by four people: 6 jobs for person 1, 1 job
for person 2, 3 for person 3, and 10 for person 4. EVENT is the censoring
indicator. The variable SEQ simply keeps track of where each job is in a
person’s sequence of jobs. LOGSAL is the natural logarithm of salary.

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Output 8.1 First 20 Cases in Repeated Job Duration Data Set

OBS

EVENT

SEQ

DURATION

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

1
1
1
1
1
1
2
3
3
3
4
4
4
4
4
4
4
4
4
4

1
1
1
1
1
0
0
1
1
0
1
1
1
1
1
1
1
1
1
1

7
7
7
7
7
7
14
16
16
16
12
12
12
12
12
12
12
12
12
12

1
2
3
4
5
6
1
1
2
3
1
2
3
4
5
6
7
8
9
10

2.3575
4.5454
1.0864
3.5893
4.6676
3.7538
20.0000
7.3753
11.2632
1.3615
1.1436
0.7880
1.0437
0.2346
0.7983
1.2168
0.9160
0.3174
0.6573
1.7036

PRESTIGE
27
31
8
22
30
24
69
69
60
56
27
35
30
15
39
41
24
26
45
41

LOGSAL
2.30259
1.38629
2.07944
2.83321
3.40120
3.13549
3.33220
3.58352
3.85015
3.82864
2.56495
2.19722
2.77259
2.56495
3.21888
3.52636
3.43399
3.36730
3.82864
3.91202

The easiest way to run separate models for each job in the sequence
is to sort the data by SEQ and then include a BY statement in the PROC
PHREG program:
PROC SORT DATA=jobmult;
BY seq;
PROC PHREG DATA=jobmult;
MODEL duration*event(0)=prestige logsal ed;
BY seq;
RUN;

In Output 8.2, we see results for the first five job durations. As is
typical of data like these, the number of observations declines
substantially for each successive interval until, by the sixth job, it’s really
too small to support a credible analysis. The effect of PRESTIGE is fairly
consistent over the five models, with more prestigious jobs having lower
hazards of termination. The effect of ED is not significant in the second
job, but it is clearly significant in the others. Its coefficient is much smaller
for the first two jobs than for the later jobs. The salary effect varies greatly
from job to job and hovers around the .05 level of significance, except for
job 4, when it is far from significant.

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Output 8.2 PROC PHREG Coefficients for Successive Jobs (Selected Output)

First Job (n=100)
Parameter DF
prestige
logsal
ed

1
1
1

Second Job (n=72)
Parameter DF
prestige
logsal
ed

Parameter
Estimate
-0.08714
-0.43249
0.17226

Parameter
Estimate

1
1
1

-0.07141
-0.77067
0.09044

Parameter DF

Parameter
Estimate

Third Job (n=52)

prestige
logsal
ed

1
1
1

Fourth Job (n=34)
Parameter DF
prestige
logsal
ed

-0.11049
-0.88703
0.43346

Parameter
Estimate

1
1
1

-0.10120
-0.19497
0.43956

Parameter DF

Parameter
Estimate

Fifth Job (n=34)

prestige
logsal
ed

1
1
1

-0.07592
-1.14765
0.34951

Standard
Error Chi-Square Pr > ChiSq
0.01212
0.21767
0.06597

<.0001
0.0469
0.0090

0.917
0.649
1.188

Standard
Error Chi-Square Pr > ChiSq

Hazard
Ratio

0.01555
0.28446
0.06747

51.6530
3.9478
6.8171

Hazard
Ratio

<.0001
0.0067
0.1801

0.931
0.463
1.095

Standard
Error Chi-Square Pr > ChiSq

Hazard
Ratio

0.02365
0.41098
0.14648

21.0889
7.3399
1.7968

<.0001
0.0309
0.0031

0.895
0.412
1.543

Standard
Error Chi-Square Pr > ChiSq

Hazard
Ratio

0.02613
0.44619
0.12108

21.8185
4.6583
8.7569

0.0001
0.6621
0.0003

0.904
0.823
1.552

Standard
Error Chi-Square Pr > ChiSq

Hazard
Ratio

0.02744
0.58605
0.11327

14.9978
0.1909
13.1784

7.6559
3.8349
9.5216

0.0057
0.0502
0.0020

0.927
0.317
1.418

While this approach has some things going for it, it’s inefficient in a
couple of respects. It’s tedious to do multiple analyses, and the more
numbers you have to interpret, the more room there is for ambiguity and
confusion. While it is tempting to interpret the job-to-job variations as real
differences, I know for a fact that the underlying process is constant over

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time—after all, I generated the data! If the process is, in fact, invariant from
one interval to the next, it’s also statistically inefficient to produce several
redundant estimates. That may not be much of a concern if you’re working
with census data and have 100,000 cases. But if you have only 100 cases,
you want to use your limited information to the best advantage. An
additional problem is that later jobs are a biased sample: the only people
who had a fifth job were those who already had four jobs in the 20-year
period. Of necessity, those four jobs were shorter than average, so it’s likely
that their fifth job will also be short.
There is a second general approach to repeated events that avoids
these problems by treating each interval as a distinct observation, pooling
all the intervals together, and estimating a single model. Output 8.3 shows
the results of doing that for the 395 job durations, also including SEQ as
one of the predictors. We now see that all four variables have highly
significant coefficients. This model is essentially equivalent to the gaptime model of Prentice and colleagues (1981), except that their model
would have stratified on SEQ instead of including it as a covariate.

Output 8.3

PROC PHREG Results for Pooled Job Durations

Parameter DF
prestige
logsal
ed
seq

1
1
1
1

Parameter
Estimate
-0.07865
-0.59694
0.18663
0.26911

Standard
Error Chi-Square Pr > ChiSq
0.00633
0.11443
0.02887
0.02946

154.3027
27.2127
41.7959
83.4457

<.0001
<.0001
<.0001
<.0001

Hazard
Ratio
0.924
0.550
1.205
1.309

Unfortunately, this method introduces a new problem—dependence
among the multiple observations. It’s well-known that whenever two or
more observations come from the same unit (person, litter, organization),
they tend to be more alike than two randomly chosen observations. In the
birth interval example, we would expect that women who have short first
intervals will also tend to have short second intervals, and so on. Pooling
these observations without taking the dependence into account can lead to
standard error estimates that are biased downward and test statistics that
are biased upward. In essence, the estimation procedure is fooled into
thinking it has more information than it actually does. Because
observations are correlated, some of the apparent information in the
sample is redundant.

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Dependence among observations can be thought of as arising from
unobserved heterogeneity. Second intervals tend to be like first intervals
because there are unmeasured, stable factors affecting both intervals. If we
could measure all of these factors and include them as covariates, the
dependence problem would disappear. But, of course, that’s not going to
happen. As we saw in the preceding section, unobserved heterogeneity
leads to artifactually declining hazard functions and coefficients that are
attenuated toward 0. These two problems are still present with repeated
events, but with the added problem of biased standard errors and test
statistics. That’s important to remember because some methods correct
only the standard errors, while others can also correct biases in the
coefficients and hazard functions.
Before we look at ways of correcting for dependence, let’s first look
at a simple way to detect it. Even though dependence is always likely to be
present to some degree, it may be so small that it has trivial effects on the
estimates. Hence, we need some way to judge just how substantial the
problem is. Here’s a simple ad-hoc way to do just that: Estimate a model
for the second interval with the length of the first interval as a covariate.
You should also include the covariates you would otherwise put in the
model because the important question is whether there is residual
dependence after the effects of any covariates have been removed. Here is
the code:
PROC SORT DATA=jobmult;
BY id seq;
DATA joblag;
SET jobmult;
durlag=LAG1(duration);
PROC PHREG DATA=joblag;
WHERE seq = 2;
MODEL duration*event(0)=prestige logsal ed durlag;
RUN;

Output 8.4 shows the results for the 72 second jobs in the job
duration data set. Two things are noteworthy here. First, the duration of
the first job has a highly significant negative coefficient, indicating that
long first jobs are associated with low hazards, implying long second jobs.
Clearly, there is dependence here that needs to be corrected in some way. If
the coefficient had not been significant, we could ignore the dependence
without much fear of error. Second, the coefficients for the other covariates
have changed substantially from the values in Output 8.2 for second jobs.
While this is a further indication that dependence may be influencing the
results in some way, these coefficients are not necessarily any better than

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those that we have already estimated, and they may even be worse. In
other words, this technique is only good for diagnosing the problem, not
for correcting it.

Output 8.4 Model for Second Interval with First Interval as a Covariate

Variable

Parameter
Estimate

prestige
logsal
ed
dur1

1
1
1
1

-0.060441
-0.283335
0.036648
-0.269221

Standard
Error

Chi-Square

0.01576
0.32508
0.06953
0.07639

14.71353
0.75966
0.27785
12.42143

Pr >
Chi-Square
0.0001
0.3834
0.5981
0.0004

Hazard
Ratio
0.941
0.753
1.037
0.764

Given the dependence in the job durations, we can’t accept the
pooled estimates in Output 8.3 at face value. At a minimum, the standard
errors are likely to be too low and the chi-square statistics too high. The
coefficients may also be attenuated toward 0, and there may be other
biases if the unobserved heterogeneity is correlated with the measured
covariates. (Note that these coefficient biases will also occur in the
unpooled estimates in Output 8.2.) What can we do about it? One
approach is to ignore the possible biases and concentrate on getting better
estimates of the standard errors and test statistics. The other approach is to
formulate a model that incorporates unobserved heterogeneity and
estimate that model by maximum likelihood or conditional likelihood.
While this second approach can correct biases in the coefficients, it
depends more on a correct specification of the model.
Robust Standard Errors

PROC PHREG has an option called COVSANDWICH that makes it
easy to correct for dependence when there are repeated observations. This
option invokes a method variously known as the robust variance estimator
or the modified sandwich estimator, developed for Cox regression by Lin
and Wei (1989) and described in some detail in Therneau and Grambsch
(2000). (The name sandwich comes from the matrix formula for the robust
variance-covariance estimator, which has the general form ABA where A is
the inverse of the observed information matrix.) The technique is
sometimes described as a population-averaged method. One attraction of
this method is that there is no need to make assumptions about the

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nature or structure of the dependence. On the other hand, there is no
correction for biases in the coefficients that arise from unobserved
heterogeneity.
Here’s a PROC PHREG program for the job duration data that
includes this option:
PROC PHREG DATA=jobmult COVSANDWICH(AGGREGATE);
MODEL duration*event(0)=prestige logsal ed seq;
ID id;
RUN;

The option COVSANDWICH can be abbreviated to COVS. To correct for
dependence, it’s necessary to include both the AGGREGATE option and an
ID statement that gives the name of the variable containing the ID number
that has the same value for all observations in the same cluster (a person,
in this example).
In Output 8.5, we see that the coefficients (and hazard ratios) are
identical to what we got in Output 8.3 without the COVSANDWICH
option. What has changed are the standard errors, chi-squares, and pvalues. We also see a new column, “StdErr Ratio,” that gives the ratio of
the new, robust standard errors to the original standard errors. Three of the
four standard errors are larger with the robust correction, which is what
you typically find when you correct for dependence among repeated
observations. However, the standard error for PRESTIGE is a little smaller.
It’s also common to find the greatest increase in the standard errors for
those variables, like ED, that do not change over time.
Another change in the output is that there are now two versions of
the score chi-square and the Wald chi-square for testing the null
hypothesis that all the coefficients are 0. The model-based versions are
what you get when you don’t use the COVSANDWICH option. The
sandwich chi-squares correct for dependence and will usually be smaller
than the model-based chi-squares. In this example, the score statistic gets a
much larger correction than the Wald statistic, although it’s not obvious
why this should be so.

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Output 8.5 PROC PHREG Results with Robust Standard Errors

Testing Global Null Hypothesis: BETA=0
Test
Likelihood Ratio
Score (Model-Based)
Score (Sandwich)
Wald (Model-Based)
Wald (Sandwich)

Chi-Square

Pr > ChiSq

433.3143
386.7275
51.6611
349.3395
334.6795

4
4
4
4
4

<.0001
<.0001
<.0001
<.0001
<.0001

Analysis of Maximum Likelihood Estimates

Parameter DF
prestige
logsal
ed
seq

1
1
1
1

Parameter
Estimate
-0.07865
-0.59694
0.18663
0.26911

Standard StdErr
Error Ratio Chi-Square Pr > ChiSq
0.00615
0.14249
0.04604
0.03057

0.972
1.245
1.595
1.038

163.4016
17.5509
16.4301
77.5025

<.0001
<.0001
<.0001
<.0001

Hazard
Ratio
0.924
0.550
1.205
1.309

Although we have corrected the standard errors for dependence, we
have done nothing to improve the coefficients. Not only are they subject to
attenuation bias (which we shall correct in the next session), they are also
less than fully efficient. That means that the true standard errors are larger
than they need to be. This inefficiency can be corrected using methods
similar to those of Wei and colleagues (1989). The basic idea is to get
coefficient estimates for each successive event (just as we did in Output
8.1) and then calculate a weighted average of those coefficients using the
robust covariance matrix to calculate optimal weights.
The trick is to do this all in one run of PROC PHREG. (A SAS macro
called WLW, which automates the following steps, can be downloaded at
www.pauldallison.com.) The first step is to create interactions between
each covariate and a set of dummy variables that indicate whether it’s the
first event, the second event, and so on. This can most easily be done in a

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DATA step by creating arrays to hold the new variables and then defining
the interactions in a DO loop:
DATA job;
SET my.jobmult;
ARRAY p (*) p1-p10;
ARRAY s (*) s1-s10;
ARRAY e (*) e1-e10;
DO i=1 TO 10;
p(i)=prestige*(seq=i);
s(i)=logsal*(seq=i);
e(i)=ed*(seq=i);
END;
RUN;

We then fit a model with PROC PHREG using all 30 of these
covariates (plus SEQ itself), while adjusting the standard errors with the
COVSANDWICH option. Three TEST statements are then used to calculate
a weighted average for each set of coefficients:
PROC PHREG DATA=job COVS(AGGREGATE);
MODEL duration*event(0)=p1-p10 s1-s10 e1-e10 seq;
ID id;
Prestige: TEST p1,p2,p3,p4,p5,p6,p7,p8,p9,p10 / AVERAGE;
Salary: TEST s1,s2,s3,s4,s5,s6,s7,s8,s9,s10 / AVERAGE;
Education: TEST e1,e2,e3,e4,e5,e6,e7,e8,e9,e10 / AVERAGE;
RUN;

There’s little point in looking at the 30 coefficients in the fitted
model. What’s important are the weighted averages reported in Output
8.6. These estimates are similar in magnitude to those in Output 8.5
(although the weighted coefficient for salary is 43 percent larger than the
unweighted one). The standard errors are slightly lower, which is what
you hope for with a method that is theoretically more efficient.

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Output 8.6

PROC PHREG Results with Robust Standard Errors
Average Effect for Test Prestige

Estimate

Standard
Error

z-Score

Pr > |z|

-0.0730

0.0057

-12.7635

<.0001

Average Effect for Test Salary

Estimate

Standard
Error

z-Score

Pr > |z|

-0.8601

0.1117

-7.6995

<.0001

Average Effect for Test Education

Estimate

Standard
Error

z-Score

Pr > |z|

0.2534

0.0332

7.6352

<.0001

Random-Effects Models

A second approach to the problem of dependence not only corrects
the standard errors and test statistics, but it also corrects for some or all of
the bias in the coefficients caused by unobserved heterogeneity. The basic
idea is to formulate a model that explicitly introduces a disturbance term
representing unobserved heterogeneity. Models of this sort are sometimes
described as subject-specific.
Letting hij(t) be the hazard for the jth event for individual i at time t,
we can write

log hij (t ) = α (t ) + βx ij + ε i

(8.1)

where εi represents unobserved heterogeneity. Notice that ε is subscripted
by i but not by j, indicating that the unobserved component is constant
from one job to the next. At this point, there are two ways to proceed. In
the next section, we will consider a fixed-effects approach. Here we shall
assume that εi is a random variable with a specified distribution,
independent of xij. This leads to random-effects or frailty models that can
be estimated by maximum likelihood or partial likelihood (Klein, 1992;

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McGilchrist, 1993). When events are repeated, such models are well
identified and are not highly sensitive to the choice of a distribution for ε.
Unfortunately, SAS has no procedure that is explicitly designed to
estimate models like this. However, with a bit of programming, it is
possible to estimate parametric random-effects models using the NLMIXED
procedure. PROC NLMIXED does maximum likelihood estimation of nonlinear mixed models (that is, models that incorporate both random effects
and fixed effects). For example, we can estimate a Weibull random-effects
model, which sets α(t) = µ + α log t in equation (8.1). To estimate this
model in PROC NLMIXED, you must specify the log-likelihood for a single
individual. Let ti be the event or censoring time for individual i, and let δi =
1 if individual i is uncensored and 0 if i is censored. Define λi = exp(βxi +
εi), where εi has a normal distribution with a mean of 0 and a variance σ2. It
can be shown (using formulas in Chapter 4) that the contribution to the
log-likelihood (conditional on ε) for a single individual is
log Li = −λi ti

(α +1)

+ δ i (log(α + 1) + α log ti + log λi ) .

Here is the corresponding PROC NLMIXED program:
PROC NLMIXED DATA=jobmult;
lambda=exp(b0+bed*ed+bpres*prestige+bsal*logsal+bseq*seq+e);
ll=-lambda*duration**(alpha+1)+ event*(LOG(alpha+1)+
alpha*LOG(duration)+LOG(lambda));
MODEL duration~GENERAL(ll);
RANDOM e~NORMAL(0,s2) SUBJECT=id;
PARMS b0=1 bed=0 bpres=0 bsal=0 btime=0 s2=1 alpha=0;
RUN;

Because PROC NLMIXED can estimate such a wide range of non-linear
models, it is necessary to specify the model in considerably more detail
than in most other procedures. The statement beginning with LAMBDA
defines λi = exp(βxi + εi). In PROC NLMIXED, every parameter must be
assigned a name. It’s advisable to give the coefficients names that contain
the corresponding variable names so that the output can be more easily
interpreted. The E at the end of the statement corresponds to ε, the random
effect. The statement beginning with LL defines the log-likelihood. Note
that EVENT is the censoring indicator, equivalent to δi.
The MODEL statement sets DURATION as the response variable,
with a distribution corresponding to the log-likelihood LL. The RANDOM
statement declares E to be a normally distributed random variable with a
mean of 0 and a variance S2 (which is just the name of the parameter to be
estimated). The SUBJECT option says that E is a different random variable

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for each value of the ID variable. The PARMS statement sets the starting
values for the parameters. If you omit the PARMS statement, the default
starting values are set to 1 for all parameters, which often doesn’t work. As
shown here, I usually get results by setting all coefficients to 0, setting the
intercept to 1 and the variance parameter to 1. But if you run into
problems, you may do better by using coefficient estimates from PROC
PHREG as starting values.
The key results are shown in Output 8.7. The coefficient for
PRESTIGE (BPRES) is about the same as in Output 8.5, with similar
standard error. LOGSAL has a coefficient (BSAL) that is substantially
larger in magnitude than in Output 8.5 and with a slightly smaller
standard error. The big change is in the effect of education (ED). The
coefficient in Output 8.5 was positive and highly significant, but here the
coefficient (BED) is negative and almost significant at the .05 level. Last,
the coefficient for SEQ, which was large and highly significant in Output
8.5, is now much smaller and not quite statistically significant. All of these
discrepancies between Outputs 8.5 and 8.7 are due to the presence of the
random error term ε. If you remove E and the RANDOM statement from the
PROC NLMIXED program, you get estimates for a conventional Weibull
model that are very similar to those in Output 8.5.
Besides the coefficient estimates, we also get an estimate of S2, the
variance of ε, which is large and highly significant. This tells us that there
is definitely unobserved heterogeneity across persons or, equivalently, that
there is dependence among the repeated observations. The last line is an
estimate of α, the coefficient of log t. This can be interpreted by saying that
each 1 percent increase in time is associated with a 2.2 percent increase in
the hazard of job termination.

Output 8.7 PROC NLMIXED Results for Weibull Model with Random Effects

Parameter

Estimate

Standard
Error

b0
bpres
bsal
bed
bseq
s2
alpha

2.5528
-0.08032
-0.9810
-0.2265
0.06754
6.7986
2.1943

1.3258
0.008202
0.1523
0.1178
0.03637
1.5026
0.1627

t Value

Pr > |t|

99
99
99
99
99
99
99

1.93
-9.79
-6.44
-1.92
1.86
4.52
13.48

0.0570
<.0001
<.0001
0.0575
0.0663
<.0001
<.0001

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Fixed-Effects Models

Instead of treating εi in equation (8.1) as a set of random variables
with a specified probability distribution, we can treat it as representing a
set of fixed constants (that is, a fixed-effects model). The attraction of this
approach is that, rather than assuming that εi is independent of xi, we
allow εi to be freely correlated with xi. That implies that the estimates
actually control for all stable, unobserved characteristics of the individual.
How can we estimate such a model? One possibility is to put dummy
variables in the model for all individuals (except one). This method works
well for linear regression models but not for Cox regression, which is
subject to something called incidental parameters bias (Allison, 2002).
When the average number of intervals per person is less than three,
regression coefficients are inflated by approximately 30 to 90 percent,
depending on the level of censoring (a higher proportion of censored cases
produces greater inflation).
Fortunately, there is a simple alternative method that does the job
very well. First, we modify equation (8.1) to read
log hij (t ) = α i (t ) + βx ij .

In this equation, the fixed effect εi has been absorbed into the unspecified
function of time, which is now allowed to vary from one individual to
another. Thus, each individual has her own hazard function, which is
considerably less restrictive than allowing each individual to have her
own constant.
This model can be estimated by partial likelihood using the method
of stratification, discussed in Chapter 5. Stratification allows different
subgroups to have different baseline hazard functions, while constraining
the coefficients to be the same across subgroups. We can implement this
method in PROC PHREG by using the STRATA statement, with an
identification variable that distinguishes individuals. This is called the
fixed-effects partial likelihood (FEPL) method. For the job duration data
set, the SAS statements are
PROC PHREG DATA=jobmult NOSUMMARY;
MODEL duration*event(0)=prestige logsal seq;
STRATA id;
RUN;

The NOSUMMARY option suppresses the information that is usually
reported for each stratum. Otherwise, you get a line of output for every
individual in the sample.

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Notice that the variable ED is not included as a covariate. One
drawback of the FEPL method is that it can only estimate coefficients for
those covariates that vary across (or within) the successive spells for each
individual. And because education does not vary over time for this
sample, its effect cannot be estimated. On the other hand, the fixed-effect
method implicitly controls not only for education but for all constant
covariates, regardless of whether they are measurable and regardless of
whether they are correlated with the measured covariates. In other words,
this method controls for things like race, ethnicity, sex, religion, region of
origin, personality—anything that’s stable over time.
As Output 8.8 shows, a fixed-effects analysis for the job duration
data yields results that are similar to those for the random-effects model in
Output 8.6, except that the effect of SEQ is even smaller.

Output 8.8 Results of Fixed-Effects Partial Likelihood for Repeated Job Durations

Parameter

Parameter
Estimate

Standard
Error

Chi-Square

Pr > ChiSq

Hazard
Ratio

prestige
logsal
seq

1
1
1

-0.05594
-0.86566
0.02073

0.00979
0.17432
0.03941

32.6427
24.6616
0.2769

<.0001
<.0001
0.5988

0.946
0.421
1.021

Fixed-effects partial likelihood was proposed by Chamberlain (1985),
who expressed reservations about its use when the number of intervals
varied across individuals and when the censoring time (the time between
the start of an interval and the termination of observation) depended on
the lengths of preceding intervals. Both of these conditions exist for the job
duration data set and for most other applications to repeated events.
Whatever the theoretical merit of these concerns, however, my own (1996)
Monte Carlo simulations have convinced me that there is little or no
problem in practice.
There is one exception to that conclusion: the FEPL method does not
do a good job of estimating the effect of the number of previous events,
such as the SEQ variable in this example. Specifically, FEPL tends to yield
negative estimates for the number of prior events, even when there is no
real effect. On the other hand, conventional partial likelihood produces
estimates for the number of prior events that are much more strongly
biased but in the opposite direction (as we found in Output 8.3).
As already noted, a major advantage of the FEPL method is that the
unobserved disturbance term is allowed to be correlated with the
measured covariates. However, when the disturbance term is not

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correlated with any of the covariates, the random-effects approach
produces more efficient estimates (that is, with smaller standard errors).
This differential efficiency is especially great when the average number of
events per individual is less than two. The reason is that the FEPL method
excludes two types of individuals from the partial likelihood function:
 those with no events (that is, those with only a single censored
spell)
 those with one uncensored spell and one censored spell, if the
censored spell is shorter than the uncensored spell.
One circumstance in which the unobservable characteristics of
individuals are not correlated with their measured characteristics is a
randomized experiment. But with nonexperimental data, it’s much safer to
assume that they are correlated.
In sum, the FEPL method is the preferred technique for repeated
events whenever
 the data do not come from a randomized experiment
 interest is centered on covariates that vary across intervals for
each individual
 most individuals have at least two events
 there is a reasonable presumption that the process generating
events is invariant over time.
If the data are produced by a randomized experiment or if the main
interest is in covariates that are constant for each individual, a randomeffects method is preferable.
Specifying a Common Origin for All Events

Another issue that arises in the analysis of repeated events is
whether the hazard varies as a function of time since the last event or as a
function of time since the process began. All the analyses so far have
made the former assumption, that the clock is reset to 0 whenever an event
occurs. But in many applications, it is reasonable to argue that the hazard
depends on the time since the individual first became at risk, regardless of
how many intervening events have occurred. For example, the hazard for a
job change may depend on the time in the labor force rather than the time
in the current job.
PROC PHREG makes it easy to specify models in which the hazard
depends on a single origin for all of an individual’s events. As before, we
create a separate record for each interval for each individual. But instead
of a single variable containing the length of the interval, the record must
contain a starting time (measured from the common origin) and a stopping
time (also measured from the origin). Output 8.9 shows what the records

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look like for the job duration data set (omitting the covariates). Here is the
program that produced this data set and output:
DATA strtstop;
SET jobmult;
RETAIN stop;
IF seq=1 THEN stop=0;
start=stop;
stop=duration+stop;
PROC PRINT;
VAR id event seq start stop;
RUN;

Output 8.9

First 20 Cases for Job Duration Data with Start and Stop Times
OBS

EVENT

SEQ

START

STOP

1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20

1
1
1
1
1
1
2
3
3
3
4
4
4
4
4
4
4
4
4
4

1
1
1
1
1
0
0
1
1
0
1
1
1
1
1
1
1
1
1
1

1
2
3
4
5
6
1
1
2
3
1
2
3
4
5
6
7
8
9
10

0.0000
2.3575
6.9029
7.9893
11.5786
16.2462
0.0000
0.0000
7.3753
18.6385
0.0000
1.1436
1.9315
2.9752
3.2098
4.0081
5.2249
6.1409
6.4583
7.1156

2.3575
6.9029
7.9893
11.5786
16.2462
20.0000
20.0000
7.3753
18.6385
20.0000
1.1436
1.9315
2.9752
3.2098
4.0081
5.2249
6.1409
6.4583
7.1156
8.8192

The model is then specified by the following program:
PROC PHREG DATA=strtstop COVSANDWICH(AGGREGATE);
MODEL (start,stop)*event(0)=prestige logsal ed seq;
ID id;
RUN;

Notice that I have also used the COVSANDWICH option to adjust for
possible dependence among the multiple events for each person. The
results in Output 8.10 are quite similar to those in Output 8.5, which reset
the origin for the hazard function at each job termination, although the

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coefficient magnitudes are somewhat smaller. These are essentially
estimates of the counting process model proposed by Andersen and Gill
(1982), although they did not correct the standard errors for dependence.
If we stratified by SEQ rather than included it as a covariate, we would get
estimates of the total time model described by Prentice and colleagues
(1981).

Output 8.10 PROC PHREG Estimates with a Common Origin for Pooled Job Durations

Parameter DF
prestige
logsal
ed
seq

1
1
1
1

Parameter
Estimate
-0.05223
-0.37332
0.12815
0.17951

Standard StdErr
Error Ratio Chi-Square Pr > ChiSq
0.00382
0.08465
0.03439
0.02104

0.671
0.710
1.170
0.709

187.0645
19.4493
13.8827
72.7975

<.0001
<.0001
0.0002
<.0001

Hazard
Ratio
0.949
0.688
1.137
1.197

An alternative approach is to use the stop times for each
observational record but not the start times. If this is done, however, it is
essential that one also stratify on the sequence number, as in the marginal
model of Wei and colleagues (1989). Otherwise, the same individual
could appear more than once in the same risk set. For example, we can
use
PROC PHREG DATA=strtstop COVSANDWICH(AGGREGATE);
MODEL stop*event(0)=prestige logsal ed;
ID id;
STRATA seq;
RUN;

This approach produces the results in Output 8.11, which are quite similar
to those in Output 8.10.

Output 8.11 PROC PHREG with Common Origin and No Adjustment of Start Times

Parameter DF
prestige
logsal
ed

1
1
1

Parameter
Estimate
-0.06167
-0.52963
0.13875

Standard StdErr
Error Ratio Chi-Square Pr > ChiSq
0.00628
0.13594
0.04859

0.987
1.123
1.567

96.5662
15.1788
8.1543

<.0001
<.0001
0.0043

Hazard
Ratio
0.940
0.589
1.149

One disadvantage of specifying a single time origin for all events is
that you cannot then do fixed-effects partial likelihood. That is because

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each individual is treated as a separate stratum in FEPL. For each of those
strata, a common time origin implies that there is only one observation in
the risk set at any given point in time, so the partial likelihood becomes
degenerate. Furthermore, although it’s theoretically possible to fit a
random-effects model with a common time origin, I am not aware of any
software that will do this.
Repeated Events for Discrete-Time Maximum Likelihood

In Chapter 7, we saw how to use PROC LOGISTIC to estimate models
for discrete-time survival data using the method of maximum likelihood.
These methods can easily be extended to handle repeated events. In fact,
there are even more methods available for discrete-time maximum
likelihood, and some of those methods are simpler to implement than the
ones that we have already discussed in this chapter.
Before discussing the analytic methods, we must first consider the
construction of the data set. In Chapter 7, for non-repeated events, the data
set consisted of one record for each unit of time that each individual was
observed, up to and including the time unit in which an event occurred.
But once an event occurred, no more observational records were created.
When events are repeatable, however, you don’t stop with the first event
occurrence. An observational record is created for each time unit that the
individual is observed, regardless of how many events occur before or
after.
Once the data set is created, you can simply do a binary regression
using PROC LOGISTIC, either with the default logit link (for truly discrete
time) or the complementary log-log link (for grouped continuous-time
data). Unlike in Chapter 7, however, methods are needed to correct for
dependence among the repeated observations. These methods are
discussed in detail in Chapter 3 of my book Fixed Effects Regression
Methods for Longitudinal Data Using SAS (SAS Institute, 2005).
Here is an example of how to do it for the job duration data. In the
following DATA step, I first convert the DURATION variable into discrete
years by using the CEIL function (which rounds up to the next higher
integer). Then, using a DO loop, I produce a set of multiple records for
each person, for a total of 2,072 person years:
DATA discrete;
SET jobmult;
durdis=CEIL(duration);
DO time=1 TO durdis;
term=0;
IF time=durdis THEN term=event;

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279

OUTPUT;
END;
RUN;

To implement the method of robust standard errors, you can use
PROC SURVEY LOGISTIC with the CLUSTER statement:
PROC SURVEYLOGISTIC DATA=discrete;
MODEL term(DESC)=prestige logsal ed seq time time*time;
CLUSTER id;
RUN;

Notice that, unlike the PROC PHREG models, it’s important to include the
time since the last event as a covariate. (Actually, I’ve allowed for a quadratic
function by including both TIME and TIME squared.) In PROC PHREG,
dependence on time is automatically allowed via the α(t) function. As
expected, the results in Output 8.11 are very similar to those in Output 8.5. If
you prefer to estimate a proportional hazards model rather than a logit
model, just put the / LINK=CLOGLOG option in the MODEL statement.

Output 8.11 PROC SURVEYLOGISTIC with Robust Standard Errors

Parameter

Estimate

Standard
Error

Intercept
prestige
logsal
ed
seq
time
time*time

1
1
1
1
1
1
1

-0.4169
-0.0894
-0.6253
0.2114
0.3317
0.4012
-0.0162

0.6629
0.00840
0.1807
0.0602
0.0385
0.0857
0.00492

Wald
Chi-Square

Pr > ChiSq

0.3954
113.2848
11.9773
12.3411
74.2487
21.9219
10.8633

0.5295
<.0001
0.0005
0.0004
<.0001
<.0001
0.0010

Random-effects models (for either link function) can be estimated
with PROC NLMIXED, although the syntax is a little simpler than what we
used for the Weibull model earlier in this chapter. Even easier is PROC
GLIMMIX, which has a much simpler syntax than PROC NLMIXED. Here
is the code:
PROC GLIMMIX DATA=discrete METHOD=QUAD;
MODEL term=prestige logsal ed seq time
/DIST=BIN SOLUTION LINK=LOGIT;
RANDOM INTERCEPT / SUBJECT=id;
RUN;

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By default, PROC GLIMMIX does pseudo-likelihood estimation, which is
very fast but may not be sufficiently accurate for binary outcomes.
However, you can force it to do true maximum likelihood estimation by
specifying METHOD=QUAD in the MODEL statement. This option uses
Gaussian quadrature to calculate the likelihood function, the same method
used by PROC NLMIXED.
Results in Output 8.12 are very similar to those produced by PROC
NLMIXED in Output 8.7. The covariance parameter of 9.2179 is the
estimated variance of εi.

Output 8.12 PROC GLIMMIX Estimates for the Random-Effects Model

Covariance Parameter Estimates

Cov Parm

Subject

Intercept

Estimate

Standard
Error

9.2179

2.6330

Solutions for Fixed Effects

Effect

Estimate

Standard
Error

Intercept
prestige
logsal
ed
seq
time
time*time

3.0191
-0.09723
-0.9589
-0.2857
0.08710
1.2281
-0.04313

1.6496
0.01266
0.2514
0.1532
0.05818
0.1275
0.005883

t Value

Pr > |t|

98
1967
1967
1967
1967
1967
1967

1.83
-7.68
-3.81
-1.86
1.50
9.63
-7.33

0.0703
<.0001
0.0001
0.0624
0.1345
<.0001
<.0001

To estimate a fixed-effects model, simply use PROC LOGISTIC with
the STRATA statement, as follows:
PROC LOGISTIC DATA=discrete;
MODEL term(DESC)=prestige logsal ed seq time time*time;
STRATA id;
RUN;

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However, unlike the robust standard error method, the fixed-effects
method is only available for the logit link, not the complementary log-log
link. Results in Output 8.13 are very similar to those in Output 8.8. Note
that the coefficient for education cannot be estimated in a fixed-effects
model because that variable does not vary over time.

Output 8.13 Fixed-Effects Estimates with PROC LOGISTIC

Analysis of Maximum Likelihood Estimates

Parameter

Estimate

Standard
Error

Wald
Chi-Square

Pr > ChiSq

prestige
logsal
ed
seq
time
time*time

1
1
0
1
1
1

-0.0762
-1.0499
-4.05E-6
-0.00772
1.3716
-0.0346

0.0131
0.2648
.
0.0571
0.1400
0.00765

33.5857
15.7198
.
0.0183
96.0533
20.4496

<.0001
<.0001
.
0.8924
<.0001
<.0001

Besides these methods, one can also adjust for dependence using the
method of generalized estimating equations (GEE), which is available in
PROC GENMOD. Invoked with the REPEATED statement, this method
produces robust standard error estimates using the same method that is
used by PROC PHREG. However, GENMOD also produces GEE estimates
of the coefficients, which are more statistically efficient than conventional
logistic regression coefficients. Although GEE estimates have less sampling
variability than conventional estimates, they do not adjust for bias due to
unobserved heterogeneity.
Specifying the hazard as a function of time since a common origin is
easy with any of these models for discrete-time data. That’s because time is
treated like any other variable on the right-hand side of the equation. So
you simply create another variable, which is the time since origin, and
include that in the model. You can also include the time since the last
event in the same model. And you can specify nonlinear functions of
either of these variables.

CHAPTER 8

282

Heterogeneity, Repeated Events, and Other Topics

GENERALIZED R2
People who do a lot of linear regression tend to become attached to
R as a measure of how good their models are. When they switch to PROC
LIFEREG or PROC PHREG, they may experience withdrawal symptoms
because no similar statistic is reported. In this section, I show how a
2
generalized R can be easily calculated from statistics that are reported.
2
Before doing that, I want to caution readers that R is not all it’s cracked up
to be, regardless of whether it’s calculated for a linear model or a
2
proportional hazards model. In particular, R does not tell you anything
2
about how appropriate the model is for the data. You can obtain an R of
only .05 for a model whose assumptions are perfectly satisfied by the data
2
and whose coefficients are precisely unbiased. Similarly, an R of .95 does
not protect you against severe violations of assumptions and grossly biased
2
coefficients. All R tells you is how well you can predict the dependent
variable with the set of covariates. And even for prediction, some
authorities argue that the standard error of the estimate is a more
meaningful and useful measure.
2
Still, other things being equal, a high R is definitely better than a
2
low R , and I happen to be one of those who miss it if it’s not there. With
that in mind, let’s see how we can calculate one for survival models.
2
Unfortunately, there’s no consensus on the best way to calculate an R for
2
nonlinear models. In my opinion, many of the R s that are reported by
some widely used packages are virtually worthless. The statistic that I
describe here was proposed by Cox and Snell (1989) and was also one of
three statistics endorsed by Magee (1990). It’s the same statistic that is
available as an option for PROC LOGISTIC.
2
Let G be the likelihood ratio chi-square statistic for testing the null
2
hypothesis that all covariates have coefficients of 0. This G is reported
directly by PROC PHREG and PROC LOGISTIC. For PROC LIFEREG, you
must calculate it yourself by fitting models both with and without the
covariates and then taking twice the positive difference in the log2
likelihoods. With that statistic in hand, you can calculate the R as
2

 − G2 

R 2 = 1 − exp
 n 

(8.2)

where n is the sample size. If you are using a method that breaks the
individual’s time to event into multiple records (for example, the
piecewise exponential model or the maximum likelihood logistic model),
the n should be the number of individuals, not the number of
observational records.

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The justification for this formula is simple. For an ordinary linear
regression model with a normal error term, it’s possible to calculate a
likelihood ratio chi-square statistic for the null hypothesis that all
coefficients are 0 (although what’s usually reported is an F statistic). That
2
chi-square statistic is related to the standard R by the formula in equation
(8.2). By analogy, we use the same formula for nonlinear models.
You can use this statistic for any regression model estimated by
maximum likelihood or partial likelihood. Unlike the linear model,
however, it cannot be interpreted as a proportion of variation in the
dependent variable that is explained by the covariates. It’s just a number
between 0 and 1 that is larger when the covariates are more strongly
associated with the dependent variable. Nevertheless, it seems to behave
2
in similar ways to the usual R . In samples with no censored data, I have
compared OLS linear regression models for the logarithm of time with
various accelerated failure time models estimated by maximum likelihood
and with Cox models estimated by partial likelihood. In all cases that I
2
have examined, the generalized R s from the likelihood-based procedures
2
are similar in magnitude to the R from the ordinary regression model.
Here are some examples. In Chapter 4, “Estimating Parametric
Regression Models with PROC LIFEREG,” we used PROC LIFEREG to
estimate a Weibull model for the 432 cases in the recidivism data set, and
we calculated a likelihood ratio chi-square statistic of 33.48 for the test that
2
all coefficients are 0. Applying the formula above, we get an R of .0746.
For the same data set, we used partial likelihood in Chapter 5 to estimate a
proportional hazards model. As shown in Output 5.1, the likelihood ratio
2
chi-square statistic is 33.13, so the R is virtually identical. For the 65 heart
transplant cases, the PROC PHREG model in Output 5.3 had a likelihood
2
ratio chi-square statistic of 16.6, yielding an R of .23.

SENSITIVITY ANALYSIS

FOR

INFORMATIVE CENSORING

In Chapters 2 and 6, we discussed the relatively intractable problem
of informative censoring. Let’s quickly review the problem. Suppose that
just before some particular time t, there are 50 individuals who are still at
risk of an event. Of those 50 individuals, 5 are censored at time t. Suppose
further that 20 of the 50 at risk have covariate values that are identical to
those of the 5 who are censored. We say that censoring is informative if the
5 who are censored are a biased subsample of the 20 individuals with the
same covariate values. That is, they have hazards that are systematically
higher or lower than those who were not censored. Informative censoring
can lead to parameter estimates that are seriously biased.

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When censoring is random (that is, not under the control of the
investigator), it’s usually not difficult to imagine scenarios that would lead
to informative censoring. Suppose, for example, that you’re studying how
long it takes rookie policemen to be promoted, and those who quit are
treated as censored. It doesn’t take much insight to suspect that those who
quit before promotion have, on average, poorer prospects for promotion
than those who stay. Unfortunately, there’s no way to test this hypothesis.
You can compare the performance records and personal characteristics of
those who quit and those who stayed, but these are all things that would
probably be included as covariates in your model. Remember that what
we’re concerned about is residual informativeness, after the effects of
covariates have been taken into account. And even if we could
discriminate between informative and noninformative censoring, there are
no standard methods for handling informative censoring. The best that can
be done by way of correction is to include as covariates any factors that are
believed to affect both event times and censoring times.
There is, however, a kind of sensitivity analysis that can give you
some idea of the possible impact that informative censoring might have on
your results. The basic idea is to redo the analysis under two extreme
assumptions about censored cases. One assumption is that censored
observations experience events immediately after they are censored. This
corresponds to the hypothesis that censored cases are those that tend to be
at high risk of an event. The opposite assumption is that censored cases
have longer times to events than anyone else in the sample. Obviously, this
corresponds to the hypothesis that censored cases are those that tend to be
at low risk of an event.
This is the general strategy. Some care needs to be taken in
implementation, however. Let’s consider the LEADERS data set that we
analyzed extensively in Chapter 6. There were four outcomes, as coded in
the variable LOST:
Code
0
1
2
3

Frequency
115
165
27
165

Reason
leader still in power at the end of the study
leader left office by constitutional means
leader died of natural causes
leader left office by nonconstitutional
means

We saw that types 1 and 3 were similar in many respects. Output
6.12 displayed estimates for a Cox model that combined types 1 and 3 but
that treated types 0 and 2 as censoring. Now we want to see how sensitive
those estimates in Output 6.12 are to possible informative censoring.

CHAPTER 8

Heterogeneity, Repeated Events, and Other Topics

285

The censoring that occurred because the study ended is, in fact,
random censoring because the leaders began their spells in power at
varying points in time. It’s possible that those who entered power later had
higher or lower risks of exit than those who entered earlier. Nonetheless,
we can control for this kind of random censoring by including the START
time as a covariate, which we did in Output 6.12. We can’t do anything
like that for deaths, however, so our sensitivity analysis will focus on the
27 cases that have a value of 2 for the LOST variable.
The first reanalysis assumes that if the 27 people had not died in a
particular year, they would have immediately been removed from power
by constitutional or nonconstitutional means. We can accomplish this
reanalysis by removing the value 2 from the list of censored values in the
MODEL statement, thereby treating it as if it were a 1 or a 3:
PROC PHREG DATA=leaders;
MODEL years*lost(0)=manner age start military conflict
loginc literacy / TIES=EXACT;
STRATA region;
RUN;

The second reanalysis assumes that if those 27 leaders had not died
of natural causes, they would have remained in power at least as long as
anyone else in the sample. We still treat them as censored, but we change
their censoring time to the largest event time in the sample, which, in this
case, is 24 years. Because we’re doing a partial likelihood analysis, we
could change it to any number greater than or equal to 24, and the result
would be the same. If we were using PROC LIFEREG, which uses the exact
times of all observations, we might want to change the censoring time to
the longest observed time, either censored or uncensored. The longest
censored time in this sample is 27 years.
Here’s the SAS code for changing the censoring times to 24 and then
running the model:
DATA leaders2;
SET leaders;
IF lost=2 THEN years=24;
PROC PHREG DATA=leaders2;
MODEL years*lost(0,2)=manner age start military conflict
loginc literacy / TIES=EXACT;
STRATA region;
RUN;

Compare the results for these two models, shown in Output 8.14,
with those in Output 6.12. For each covariate, the two new estimates

CHAPTER 8

Heterogeneity, Repeated Events, and Other Topics

286

bracket the original estimate. The biggest difference is in the effect of age,
which gets larger in the top panel but much smaller in the bottom panel, to
the point where it is no longer significant at the .05 level. This should not
be surprising because AGE was the only variable of any importance in the
model for deaths due to natural causes (Output 6.7). For the most part,
however, the results in Output 8.14 are reassuring. We can be reasonably
confident that treating natural deaths as noninformative censoring has no
appreciable affect on the conclusions.

Output 8.14

Sensitivity Analysis of LEADERS Data Set for Informative Censoring

Censored Cases Treated as Uncensored Cases
Parameter
manner
age
start
military
conflict
loginc
literacy

Parameter
Estimate

Standard
Error

Chi-Square

Pr > ChiSq

Hazard
Ratio

1
1
1
1
1
1
1

0.38528
0.02307
-0.01774
-0.22637
0.12992
-0.18263
0.0006600

0.15490
0.00547
0.00816
0.16041
0.13073
0.08235
0.00320

6.1868
17.8215
4.7251
1.9913
0.9876
4.9181
0.0426

0.0129
<.0001
0.0297
0.1582
0.3203
0.0266
0.8366

1.470
1.023
0.982
0.797
1.139
0.833
1.001

Censoring Times Recoded as 24
Parameter
manner
age
start
military
conflict
loginc
literacy

Parameter
Estimate

Standard
Error

Chi-Square

Pr > ChiSq

Hazard
Ratio

1
1
1
1
1
1
1

0.31912
0.01001
-0.01316
-0.14295
0.20979
-0.27022
0.00378

0.15669
0.00568
0.00826
0.16305
0.13708
0.08939
0.00334

4.1479
3.1128
2.5394
0.7686
2.3421
9.1382
1.2819

0.0417
0.0777
0.1110
0.3806
0.1259
0.0025
0.2575

1.376
1.010
0.987
0.867
1.233
0.763
1.004

CHAPTER 8

Heterogeneity, Repeated Events, and Other Topics

287

In interpreting output like this, you should remember that these are
worst-case scenarios, and it’s unlikely that either of the extremes is an
accurate depiction of reality. The usual estimates are still your best guess
of what’s really going on. It’s also worth noting that for many applications,
one of these extremes may be much more plausible than the other.
Naturally, you’ll want to focus your attention on the extreme that is most
sensible.

CHAPTER 8

288

Heterogeneity, Repeated Events, and Other Topics

CHAPTER 9
A Guide for the Perplexed
How to Choose a Method 289
Conclusion 292

HOW TO CHOOSE A METHOD
In this book, we’ve examined many different approaches to the
analysis of survival data: Kaplan-Meier estimation, log-rank tests,
accelerated failure time models, piecewise exponential models, Cox
regression models, logit models, and complementary log-log models. Each
of these methods is worth using in some situations. Along the way I have
tried to point out their relative advantages and disadvantages, but those
discussions are scattered throughout the book. It’s not easy to keep all the
points in mind when designing a study or planning an analysis.
Many readers of early versions of this book urged me to provide a
concluding roadmap for choosing a method of survival analysis. Although
I give this kind of advice all the time, I do so here with some hesitation.
While statisticians may have great consensus about the characteristics of
various statistical methods, the choice among competing methods is often
very personal, especially when dealing with methods as similar in spirit
and results as those presented here. Five equally knowledgeable
consultants could easily give you five different recommendations. I’m
going to present some rules of thumb that I rely on myself in giving advice,
but please don’t take them as authoritative pronouncements. Use them
only to the degree that you find their rationales persuasive.
Make Cox Regression Your Default Method

Given the relative length of Chapter 5, “Estimating Cox Regression
Models with PROC PHREG,” it will come as no surprise that I have a
strong preference for Cox regression using PROC PHREG. This particular
method
 is more robust than the accelerated failure time methods
 has excellent capabilities for time-dependent covariates
 handles both continuous-time and discrete-time data
 allows for late entry and temporary exit from the risk set

CHAPTER 9

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290

has a facility for nonparametric adjustment of nuisance variables
(stratification).
PROC PHREG can also do log-rank tests (using a single dichotomous
covariate). PROC PHREG will even do Kaplan-Meier estimation (by fitting
a model with no covariates and using the BASELINE statement to produce
a table of survival probabilities).
Beyond these intrinsic advantages, Cox regression has the
considerable attraction of being widely used, accepted, and understood.
What’s the point of choosing a marginally superior method if your
audience is confused or skeptical and you have to waste valuable time and
space with explanation and justification? For better or worse (mostly for
better), Cox regression has become the standard, and it makes little sense
to choose a different method unless you have a good reason for doing so.
Having said that, I’ll now suggest some possible good reasons. They are
presented in the form of questions that you should ask yourself when
deciding on a method.


Is the Sample Large with Heavily Tied Event Times?

As we saw in Chapter 5, PROC PHREG can deal with situations in
which there are many events occurring at the same recorded times using
either the DISCRETE method or the EXACT method. Unfortunately, those
options can take a great deal of computer time that increases rapidly with
sample size. If you have a sample of 10,000 observations with only 10
distinct event times, you can expect to wait a long time before you see any
output. Why wait when the alternatives are so attractive? The
complementary log-log and logit methods of Chapter 7 estimate exactly the
same models as the EXACT and DISCRETE methods with statistical
efficiency that is at least as good as that provided by PROC PHREG.
Do You Want to Study the Shape of the Hazard Function?

In some studies, one of the major aims is to investigate hypotheses
about the dependence of the hazard on time. Cox regression is far from
ideal in such situations because it treats the dependence on time as a
nuisance function that cancels out of the estimating equations. You can
still produce graphs of the baseline survival and hazard functions, but
those graphs don’t provide direct tests of hypotheses.
With PROC LIFEREG, on the other hand, you can produce formal
hypothesis tests that answer the following sorts of questions:
 Is the hazard constant over time?
 If not constant, is the hazard increasing, decreasing, or nonmonotonic?

CHAPTER 9

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291

If increasing (or decreasing), is the rate of change going up or
down?
All of these questions are addressed within the context of smooth
parametric functions. If that seems too restrictive, you can get much more
flexibility with the piecewise exponential, or with maximum likelihood
estimation of the logit or complementary log-log models. With these
models, the time scale is chopped into intervals, and the least restrictive
models have a set of indicator variables to represent those intervals.
Restrictions can easily be imposed to represent functions of almost any
desired shape. A further advantage of these multiple-record methods is
that two or more time axes can be readily introduced into a single model.
A model for promotions, for example, could include the time since the last
promotion, the time since initial employment by the firm, the time in the
labor force, and the time since completion of education.
So if you want to study the dependence of the hazard on time, there
are good alternatives to Cox regression. But whatever method you use, I
urge you to be cautious in interpreting the results. As we saw in Chapter 8,
“Heterogeneity, Repeated Events, and Other Topics,” the hazard function is
strongly confounded with uncontrolled heterogeneity, which makes
hazards look like they are declining with time even when they are constant
or increasing. As a result, any declines in the hazard function may be
purely artifactual, a possibility that you can never completely rule out.


Do You Want to Generate Predicted Event Times or
Survival Probabilities?

As we saw in Chapter 5, you can use output from the BASELINE
statement in PROC PHREG to get predicted median survival times or
survival probabilities for any specified set of covariates. Because the
BASELINE statement produces a complete set of survivor function
estimates, however, getting predicted values or 5-year survival
probabilities for a large number of observations can be rather cumbersome.
Furthermore, predicted median survival times may be unavailable for
many or all of the observations if a substantial fraction of the sample is
censored. With PROC LIFEREG, on the other hand, you can easily generate
predicted median survival times (or any other percentile) for all
observations using the OUTPUT statement. Using my PREDICT macro, you
can also produce estimated survival probabilities for a specified survival
time.

CHAPTER 9

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292

Do You Have Left-Censored Data?

The only SAS procedure that allows for left censoring is PROC
LIFEREG. In principle, it’s possible to adapt Cox regression to handle left
censoring, but I know of no commercial program that does this.

CONCLUSION
While it’s good to consider these questions carefully, you shouldn’t
become obsessed with making the right choice. For most applications, a
reasonable case can be made for two or more of these methods.
Furthermore, the methods are so similar in their underlying philosophy
that they usually give similar results. When they differ, it’s typically in
situations where the evidence is not strong for any conclusion. If you have
the time, it’s always worthwhile trying out two or more methods on the
same data. If they lead you to the same conclusions, then your confidence
is increased. If they are discrepant, your confidence is appropriately
reduced. If they are widely discrepant, you should carefully investigate the
reasons for the discrepancy. Perhaps you made an error in setting up one
method or the other. Maybe there’s a serious peculiarity in the data. Search
for outliers and examine residuals and influence diagnostics.
Discrepancies are often a powerful indication that there is something
important to be learned.

APPENDIX 1
Macro Programs
Introduction 293
The LIFEHAZ Macro 293
The PREDICT Macro 296

INTRODUCTION
In the main text, I introduced two macros for carrying out certain
auxiliary analyses with PROC LIFEREG. They are listed here with
instructions and background information. They are also available on the
Web at www.pauldallison.com.
Both of the macro programs use keywords for the required and
optional parameters. Default values (if any) are given after the equal sign in
the parameter list. Thus, you only need to specify parameters that differ
from the default value, and you can specify these parameters in any order
in the macro call.

THE LIFEHAZ MACRO
The LIFEHAZ macro produces parametric plots of hazard functions
based on models fitted by PROC LIFEREG. In the LIFEREG procedure, you
must specify OUTEST=name1 in the PROC LIFEREG statement. You must
also use the OUTPUT statement with OUT=name2 and XBETA=name3. By
default, the hazard is plotted for the mean value of the XBETA option (the
linear predictor). If you want a plot for a specific observation, you must
specify the observation number (OBSNO) when you invoke the macro. The
macro requires SAS/GRAPH software.
To use the macro, you must first read it into your SAS session. This
can be done by simply copying the text of the macro into the editor
window and then running it. Alternatively, you can use the %INCLUDE
statement to specify the name and location of the file containing the macro
(which must have a .sas extension). Once the macro has been read in to
SAS, you can use it repeatedly during your SAS session. For an example
using the LIFEHAZ macro, see the section Generating Predictions and

APPENDIX 1

Macro Programs

294

Hazard Functions in Chapter 4, “Estimating Parametric Regression Models
with PROC LIFEREG.”
Parameters
OUTEST=

OUT=_LAST_

XBETA=

OBSNO=

specifies the name of the data set produced by the
OUTEST= option in PROC LIFEREG.
specifies the name of the data set produced by the OUT=
option in PROC LIFEREG.
specifies the name used with the XBETA= option in the
OUTPUT statement.
specifies the sequence number of the observation for
which you want plots of the hazard function. (The default
method is to use the mean of the linear predictor to
generate the hazards.)

Program
%macro lifehaz(outest=,out=,obsno=0,xbeta=lp);
/********************************************************************
Version 2.0 (9-14-01)
This version of LIFEHAZ works for SAS Release 6.12 through
Release 9.2.
********************************************************************/
data;
set &outest;
call symput('time',_NAME_);
run;
proc means data=&out noprint;
var &time &xbeta;
output out=_c_ min(&time)=min max(&time)=max mean(&xbeta)=mean;
run;
data;
set &outest;
call symput('model',_dist_);
s=_scale_;
d=_shape1_;
_y_=&obsno;
set _c_ (keep=min max mean);
if _y_=0 then m=mean;
else do;
set &out (keep=&xbeta) point=_y_;
m=&xbeta;
end;
inc=(max-min)/300;
g=1/s;

APPENDIX 1

Macro Programs

295

alph=exp(-m*g);
_dist_=upcase(_dist_);
if _dist_='LOGNORMAL' or _dist_='LNORMAL' then do;
do t=min to max by inc;
z=(log(t)-m)/s;
f=exp(-z*z/2)/(t*s*sqrt(2*3.14159));
Surv=1-probnorm(z);
h=f/Surv;
output;
end;
end;
else if _dist_='GAMMA' then do;
k=1/(d*d);
do t=min to max by inc;
u=(t*exp(-m))**(1/s);
f=abs(d)*(k*u**d)**k*exp(-k*u**d)/(s*gamma(k)*t);
Surv=1-probgam(k*u**d,k);
if d lt 0 then Surv=1-Surv;
h=f/Surv;
output;
end;
end;
else if _dist_='WEIBULL' or _dist_='EXPONENTIAL' or _dist_='EXPONENT' then do;
do t=min to max by inc;
h=g*alph*t**(g-1);
output;
end;
end;
else if _dist_='LLOGISTIC' or _dist_='LLOGISTC' then do;
do t=min to max by inc;
h=g*alph*t**(g-1)/(1+alph*t**g);
output;
end;
end;
else put 'ERROR:DISTRIBUTION NOT FITTED BY LIFEREG';
run;
proc gplot;
plot h*t / haxis=axis2 vaxis=axis1 vzero;
symbol1 i=join v=none c=black;
axis1 label=(f=titalic angle=90 'Hazard');
axis2 label=(f=titalic justify=c 'time' f=titalic justify=c "&model");
run; quit;
%mend lifehaz;

APPENDIX 1

Macro Programs

296

THE PREDICT MACRO
The PREDICT macro produces predicted survival probabilities for all
individuals in a sample for specified survival times, based on models
fitted by PROC LIFEREG. You must specify OUTEST=name1 in the PROC
LIFEREG statement. You must also use the OUTPUT statement with
OUT=name2 and XBETA=name3. The probabilities are written to the
output window and also to a temporary SAS data set named _PRED_.
To use the macro, you must first read it into your SAS session. This
can be done by simply copying the text of the macro into the editor
window and then running it. Alternatively, you can use the %INCLUDE
statement to specify the name and location of the file containing the macro
(which must have a .sas extension). Once the macro has been read in to
SAS, you can use it repeatedly during your SAS session. For an example
using the PREDICT macro, see the section Generating Predictions and
Hazard Functions in Chapter 4.
Parameters
OUTEST=

OUT=_LAST_

XBETA=

TIME=

specifies the name of the data set produced by the
OUTEST= option in PROC LIFEREG.
specifies the name of the data set produced by the OUT=
option in PROC LIFEREG.
specifies the name used with the XBETA= option in the
OUTPUT statement.
specifies the survival time to be evaluated. This time must
be in the same metric as the event times.

Program
%macro predict (outest=, out=_last_,xbeta=,time=);
/********************************************************************
Example: To get 5-year survival probabilities for every individual
in the sample (assuming that actual survival times are measured in
years);
%predict(outest=a, out=b, xbeta=lp, time=5).
*********************************************************************/

APPENDIX 1

Macro Programs

297

data _pred_;
_p_=1;
set &outest point=_p_;
set &out;
lp=&xbeta;
t=&time;
gamma=1/_scale_;
alpha=exp(-lp*gamma);
prob=0;
_dist_=upcase(_dist_);
if _dist_='WEIBULL' or _dist_='EXPONENTIAL' or _dist_='EXPONENT' then prob=exp(alpha*t**gamma);
if _dist_='LOGNORMAL' or _dist_='LNORMAL' then prob=1-probnorm((log(t)-lp)/_scale_);
if _dist_='LLOGISTIC' or _dist_='LLOGISTC' then prob=1/(1+alpha*t**gamma);
if _dist_='GAMMA' then do;
d=_shape1_;
k=1/(d*d);
u=(t*exp(-lp))**gamma;
prob=1-probgam(k*u**d,k);
if d lt 0 then prob=1-prob;
end;
drop lp gamma alpha _dist_ _scale_ intercept
_shape1_ _model_ _name_ _type_ _status_ _prob_ _lnlike_ d k u;
run;
proc print data=_pred_;
run;
%mend predict;

APPENDIX 1

298

Macro Programs

APPENDIX 2
Data Sets
Introduction 299
The MYEL Data Set: Myelomatosis Patients 299
The RECID Data Set: Arrest Times for Released Prisoners 300
The STAN Data Set: Stanford Heart Transplant Patients 301
The BREAST Data Set: Survival Data for Breast Cancer Patients 302
The JOBDUR Data Set: Durations of Jobs 302
The ALCO Data Set: Survival of Cirrhosis Patients 302
The LEADERS Data Set: Time in Power for Leaders of Countries 303
The RANK Data Set: Promotions in Rank for Biochemists 304
The JOBMULT Data Set: Repeated Job Changes 305

INTRODUCTION
This appendix provides information about the data sets used as
examples in this book. They are listed in the order in which they are
discussed in the main text. All data are written in free format, with the
variables appearing in the order given in each of the following sections.
These data sets are available on the Web at
support.sas.com/publishing/authors/allison.html.

THE MYEL DATA SET: MYELOMATOSIS PATIENTS
The MYEL data set contains survival times for 25 patients diagnosed
with myelomatosis (Peto et al., 1977). The patients were randomly
assigned to two drug treatments. The entire data set is listed in Output 2.1.
These data are used extensively in Chapter 3 to illustrate the LIFETEST
procedure and briefly in Chapter 5. The variables are as follows:
DUR

STATUS
TREAT

is the time in days from the point of randomization to
either death or censoring (which could occur either by loss
to follow up or termination of the observation).
has a value of 1 if dead and a value of 0 if censored.
specifies a value of 1 or 2 to correspond to the two
treatments.

APPENDIX 2

Data Sets

300

RENAL

has a value of 1 if renal functioning was normal at the time
of randomization; it has a value of 0 for impaired
functioning.

THE RECID DATA SET: ARREST TIMES FOR RELEASED PRISONERS
The RECID data set contains information about 432 inmates who
were released from Maryland state prisons in the early 1970s. This data set
is used in Chapters 3, 4, and 5. The first 20 cases are listed in Output 3.10.
The data were kindly provided to me by Dr. Kenneth Lenihan who was
one of the principal investigators. The aim of this research was to
determine the efficacy of financial aid to released inmates as a means of
reducing recidivism. Results from the study are described in Rossi, Berk,
and Lenihan (1980). (This book also reports results from a much larger
follow-up study done in Texas and Georgia.) Half the inmates were
randomly assigned to receive financial aid (approximately the same
amount as unemployment compensation). They were followed for 1 year
after their release and were interviewed monthly during that period. Data
on arrests were taken from police and court records. The data set used here
contains the following variables:
WEEK
ARREST
FIN

AGE
RACE
WEXP
MAR
PARO
PRIO

is the week of first arrest; WEEK has a value of 52 if not
arrested.
has a value of 1 if arrested; otherwise, ARREST has a value
of 0.
has a value of 1 if the inmate received financial aid after
release; otherwise, FIN has a value of 0. FIN is randomly
assigned, with equal numbers in each category.
is the age in years at the time of release.
has a value of 1 if the inmate is black; otherwise, RACE has
a value of 0.
has a value of 1 if the inmate had full-time work experience
before incarceration; otherwise, WEXP has a value of 0.
has a value of 1 if the inmate was married at the time of
release; otherwise, MAR has a value of 0.
has a value of 1 if released on parole; otherwise, PARO has
a value of 0.
is the number of convictions before the current
incarceration.

APPENDIX 2

Data Sets

301

EDUC

is the highest level of completed schooling, coded as
2 = 6th grade or less
3 = 7th to 9th grade
4 = 10th to 11th grade
5 = 12th grade
6 = some college
EMP1−EMP52 represents the employment status in each of the first 52
weeks after release. These variables have values of 1 if the
inmate was employed full time; otherwise, EMP1–EMP52
have values of 0. Data are missing for weeks after the first
arrest.

THE STAN DATA SET: STANFORD HEART TRANSPLANT PATIENTS
Various versions of data from the Stanford Heart Transplant study
have been reported in a number of publications. The data set used in
Chapter 5, “Estimating Cox Regression Models with PROC PHREG,”
appeared in Crowley and Hu (1977). The sample consisted of 103 cardiac
patients who were enrolled in the transplantation program between 1967
and 1974. After enrollment, patients waited varying lengths of time until a
suitable donor heart was found. Patients were followed until death or until
the termination date of April 1, 1974. Of the 69 transplant recipients, only
24 were still alive at termination. At the time of transplantation, all but
four of the patients were tissue typed to determine the degree of similarity
with the donor. The data set contains the following variables:
DOB
DOA
DOT
DLS
DEAD
SURG
M1
M2
M3

is the date of birth.
is the date of acceptance into the program.
is the date of transplant.
is the date last seen (death date or censoring date).
has a value of 1 if the patient is dead at DLS; otherwise,
DEAD has a value of 0.
has a value of 1 if the patient had open-heart surgery before
DOA; otherwise, SURG has a value of 0.
is the number of donor alleles with no match in the
recipient (1 through 4).
has a value of 1 if the donor and recipient mismatch on the
HLA-A2 antigen; otherwise, M2 has a value of 0.
is the mismatch score.

All date variables are in the mm/dd/yy format.

APPENDIX 2

Data Sets

302

THE BREAST DATA SET: SURVIVAL DATA FOR BREAST CANCER
PATIENTS
The BREAST data set (displayed in Output 5.4) contains survival
data for 45 breast cancer patients. The data are taken from Collett (1994)
except that survival time for the 8th patient is changed from 26 to 25 to
eliminate ties. The variables are as follows:
SURV
DEAD
X

is the survival time (or censoring time) in months,
beginning with the month of surgery.
1 = dead; 0=alive.
has a value of 1 if the tumor had a positive marker for
metastasis; otherwise, X has a value of 0.

THE JOBDUR DATA SET: DURATIONS OF JOBS
The JOBDUR data set consists of 100 simulated observations of jobs
and job holders. Results of analyzing these data were reported in Output
5.9 and 5.10. The variables are as follows:
DUR
EVENT
ED
PRESTIGE
SALARY

is the length of the job in years (integer values only).
1 = quit; 2 = fired; 0 = censored.
is the number of years of completed schooling.
is a measure of the prestige of the job.
is the salary in thousands of dollars.

For the analyses in Chapter 5, firings were treated as censoring.

THE ALCO DATA SET: SURVIVAL OF CIRRHOSIS PATIENTS
The ALCO data set, displayed in Output 5.15, consists of simulated
data for 29 alcoholic cirrhosis patients. The variables are as follows:
SURV
DEAD
TIME2−TIME10

is the number of months from diagnosis until death or
termination of the study.
has a value of 1 if the patient died; otherwise, DEAD has
a value of 0.
represent the time of clinic visits in months since
diagnosis.

APPENDIX 2

Data Sets

303

PT1−PT10

is the blood coagulation measure at diagnosis and at
subsequent clinic visits.

THE LEADERS DATA SET: TIME IN POWER FOR LEADERS OF
COUNTRIES
The LEADERS data set, analyzed in Chapter 6, “Competing Risks,” is
described in detail by Henry Bienen and Nicolas van de Walle (1991), who
generously provided me with the data. Each record in the data set
corresponds to a spell in power for a primary leader of a country. I used a
subset of the original data, restricting the spells to countries outside of
Europe, North America, and Australia; spells that began in 1960 or later;
and only the first leadership spell for those leaders with multiple spells.
This left a total of 472 spells.
Each record includes the following variables:
YEARS
LOST
MANNER
START
MILITARY
AGE
CONFLICT
LOGINC
GROWTH
POP
LAND
LITERACY
REGION

is the number of years in power, integer valued. Leaders in
power less than 1 year have a value of 0.
0 = still in power in 1987; 1 = exit by constitutional means;
2 = death by natural causes; and 3 = nonconstitutional exit.
specifies how the leader reached power: 0 = constitutional
means; 1 = nonconstitutional means.
is the leader’s year of entry into power.
is the background of the leader: 1 = military; 0 = civilian.
is the age of the leader, in years, at the time of entry into
power.
is the level of ethnic conflict: 1 = medium or high; 0 = low.
is the natural logarithm of GNP per capita (dollar
equivalent) in 1973.
is the average annual rate of per capita GNP growth
between 1965 and 1983.
is the population, in millions (year not indicated).
is the land area, in thousands of square kilometers.
is the literacy rate (year not indicated).
0 = Middle East; 1 = Africa; 2 =Asia; 3 = Latin America.

APPENDIX 2

Data Sets

304

THE RANK DATA SET: PROMOTIONS IN RANK FOR BIOCHEMISTS
The RANK data set, analyzed in Chapter 7, “Analysis of Tied or
Discrete Data with PROC LOGISTIC,” contains records for 301 biochemists
who at some point in their careers were assistant professors at graduate
departments in the U.S. The event of interest is a promotion to associate
professor. The data set includes the following variables:
DUR

is the number of years from the beginning of the job to
promotion or censoring (ranges from 1 to 10, in integers).
EVENT
has a value of 1 if the person was promoted; otherwise,
EVENT has a value of 0.
UNDGRAD
specifies the selectivity of the undergraduate institution
(ranges from 1 to 7).
PHDMED
has a value of 1 if the person received a Ph.D. from a
medical school; otherwise, PHDMED has a value of 0.
PHDPREST
is a measure of the prestige of the Ph.D. institution (ranges
from 0.92 to 4.62).
ART1−ART10 specify the cumulative number of articles published in
each of the 10 years.
CIT1−CIT10 specify the number of citations in each of the 10 years to all
previous articles.
PREST1
is a measure of the prestige of the first employing
institution (ranges from 0.65 to 4.6).
PREST2
is the measure of the prestige of the second employing
institution (same as PREST1 for those who did not change
employers). No one had more than two employers during
the period of observation.
JOBTIME
is the year of employer change, measured from the start of
the assistant professorship (coded as missing for those who
did not change employers).

APPENDIX 2

Data Sets

305

THE JOBMULT DATA SET: REPEATED JOB CHANGES
The JOBMULT data set contains simulated information for 395 jobs
observed over a period of 20 years for 100 people. The first 20 cases are
displayed in Output 8.1, with analytic results in Output 8.2 through 8.9.
ID
EVENT
ED
SEQ

DURATION
PRESTIGE
LOGSAL

is a unique identification number for each person.
has a value of 1 if the job was terminated; EVENT has a
value of 0 if censored.
is the number of years of schooling completed.
is the number of the job in each person’s sequence.
is the length of the job until termination or censoring.
is a measure of the prestige of the job, ranging from 1 to
100.
is the natural logarithm of the annual salary at the
beginning of the job.

APPENDIX 2

306

Data Sets

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Index
A
academic promotions study 243–246, 304
accelerated failure time model
See AFT (accelerated failure time) model
actuarial method
See life-table method
ADJUST= option, LIFETEST procedure 46,
48
AFT (accelerated failure time) model
alternative distributions 77–89
competing risks analysis and 220–227
exponential model 78–80
functionality 72–77
gamma model 85–86, 98–100
hypothesis tests and 74–75
LIFEREG procedure and 72–73, 220
log-logistic model 83–85
OLS and 73
parametric regression models 72–77
PHREG procedure and 220
Weibull distribution model 80–82
age criteria 24
AGGREGATE suboption (COVSANDWICH
option, PHREG) 267
AICC (AIC corrected) 74, 118
AIDS patients study 162–163
Akaike's information criterion (AIC)
defined 74
example 118
logit model for discrete time 240
partial likelihood estimation and 131
albumin measurement study 166–169
ALCO data set 302–303
alcoholic cirrhosis study 170–172, 302–303
ALPHA= option, LIFETEST procedure 37
ASSESS statement, PHREG procedure
173–178
RESAMPLE option 174
ATRISK option, LIFETEST procedure 35

B
baseline hazard function 127
BASELINE statement, PHREG procedure
186–187, 189–192, 290–291
LOWER= option 187, 191
OUT= option 187
SURVIVAL= option 187, 192
UPPER= option 187, 191

BAYES statement
LIFEREG procedure 120
PHREG procedure 197–199
Bayesian analysis
Cox regression method and 197–200
LIFEREG procedure 117–124, 131, 197
parametric regression models and
117–124, 131
PHREG procedure and 197
Bayesian information criterion (BIC)
defined 74
example 118
logit model for discrete time 240
partial likelihood estimation and 131
BEST= option, PHREG procedure 201
Bonferroni method 48, 97
breast cancer study 137–141, 302
BREAST data set 302
Breslow approximation
comparison of methods 152–153
defined 131, 142
heart transplant study and 155
limitations of 235
partial likelihood method 142
recidivism study and 145–147
results for job duration data 148–149
BY statement, PHREG procedure 263

C
calendar time 24
cancer study 137–141, 302
categorical data analysis 6
categorical variables 87–89
CATMOD procedure 248
cause-specific hazards 204–206
c.d.f. (cumulative distribution function)
15–16
CEIL function 114, 167, 278
censoring
functionality 4–5, 9–14
in ML estimation 73, 91
informative 14–15, 283–287
interval 10, 71, 106–108, 235
left 9–10, 71, 103–106, 292
LIFEREG procedure and 6, 103–108
maximum likelihood and 73
parametric regression models and
103–108
PHREG procedure and 6
piecewise exponential model and 113

INDEX
314

censoring (continued)
random 12–13
right 9–15, 71
Type I 11–13
Type II 12–13
chi-square statistics
computing 40–42
hypothesis tests and 95–97
partial likelihood estimation and
131–132
Pearson's chi-square test 208
testing for differences in survivor
functions 47
testing for effects of covariates 61–63
cirrhosis study 170–172, 302–303
CL option, LIFETEST procedure 35
CL=PL option, HAZARDRATIO statement
(PHREG) 196
CLASS statement
LIFEREG procedure 87–89, 94–96, 192
LOGISTIC procedure 238
PHREG procedure 192–193
CLUSTER statement, SURVEYLOGISTIC
procedure 279
competing risks analysis
accelerated failure time models
220–227
alternative approaches to multiple event
types 227–232
background information 203–204
covariate effects via Cox models
213–220
estimates and tests without covariates
208–213
exponential distribution model and
221–223
generalized gamma model and 222
job duration data 252
LIFEREG procedure and 203
LIFETEST procedure and 203
log-normal model 222
LOGISTIC procedure 251–254
logit model 226, 228
maximum likelihood methods and
250–254
PHREG procedure and 203
time in power example 207–213
type-specific hazards 204–206
complementary log-log function 240–243
conditional density, and hazard function
17
confidence limits 36, 68

CONFTYPE= option, LIFETEST procedure
38
constant hazard function 78
continuous-time methods
complementary log-log function
240–243
LIFEREG procedure and 6
PHREG procedure and 6
proportional hazards model and 236
CONTRAST statement, PHREG procedure
192–194
CONVERGE= option, MODEL statement
(LIFEREG) 94
convergence criterion 140
CORR procedure 176
counting process method
albumin measurement study 167–168
alcoholic cirrhosis study 171–172
functionality 154
heart transplant study 156, 158, 184
PHREG procedure and 154, 156, 158
recidivism study 165
covariates
See also time-dependent covariates
representing time origins 160–161
testing for effects 186–192
time-constant 6, 72
COVB option, MODEL statement (LIFEREG)
93, 115
COVSANDWICH option, PHREG procedure
266–269, 276
AGGREGATE suboption 267
Cox regression models
See also PHREG procedure
as default method 289–290
background information 125–126
Bayesian analysis 197–200
covariate effects and 213–220
customized hazard ratios 195–197
estimating survivor functions 186–192
functionality 6, 29
Gompertz distribution and 127
job duration data 147–151
late entry into the risk set 183–186
left truncation and 183–186
partial likelihood method and 128–141
piecewise exponential model and 112
proportional hazards model and
126–128, 172–179
robust sandwich estimator 266–270
tied data 142–153
time-dependent covariates 153–172
with nonproportional hazards 172–183

INDEX
315

CUMINCID macro 230–232
cumulated variables 165
cumulative distribution function (c.d.f.)
15–16
cumulative hazard function 64
cumulative incidence functions 229–232
customized hazard ratios 195–197

D
D= option, MODEL statement (PROBIT) 241
data structure, basic requirements 26–28
death, study of
See survival analysis
DIC (deviance information criterion) 119
DISCRETE method
See TIES=DISCRETE option, MODEL
statement (PHREG)
discrete-time hazard 150, 278
discrete-time methods
choosing 290
functionality 235–236
GENMOD procedure and 6
LOGISTIC procedure and 6
logit model for 236–240
PHREG procedure and 6
repeated events and 278–281
DISTRIBUTION=EXPONENTIAL option,
MODEL statement (LIFEREG) 78
DISTRIBUTION=GAMMA option, MODEL
statement (LIFEREG) 86
DISTRIBUTION=LLOGISTIC option,
MODEL statement (LIFEREG) 85
DISTRIBUTION=WEIBULL option, MODEL
statement (LIFEREG) 80
double exponential distribution 78

E
Efron's approximation
comparing methods for handling ties
148–149, 152–153
heart transplant study and 155
recidivism study and 146–147
ELSE DO statement 157
empirical score process 173
empty intervals 249–250
equal precision method 36
estimated hazard function 53–55
event types
alternative approaches 227–232
conditional processes 227–229
covariate effects and 213–220

cumulative incidence functions
229–232
estimates and tests without covariates
208–213
time in power for leaders study
207–208
type-specific hazards 204–206
events
defined 2
repeated 260–281
EXACT method
See TIES=EXACT option, MODEL
statement (PHREG)
exponential distribution model
accelerated failure time model 78–80
competing risks analysis 221–223
double exponential distribution 78
functionality 20, 78–80
generalized gamma model and 85,
98–99
hazard functions and 20
heterogeneity and 79
LIFEREG procedure and 221
piecewise 112–116, 199
probability plot for 101

F
F statistic 283
FEPL (fixed-effects partial likelihood)
273–275, 278
FIRTH option, MODEL statement (PHREG)
141
fixed-effects models 273–275, 281
fixed-effects partial likelihood (FEPL)
273–275, 278
Fleming-Harrington test 43–44
frailty models 271
FREQ statement, LIFETEST procedure
56–57

G
gamma model
See generalized gamma model
GEE (generalized estimating equation)
methods 6, 281
generalized gamma model
accelerated failure time model 85–86,
98–100
competing risks analysis and 222
exponential distribution model and 85,
98–99
functionality 85–86

INDEX
316

generalized gamma model (continued)
goodness-of-fit and 98–100
hazard functions and 85–86
log-normal model 98–99
nesting and 98–99
2
generalized R 282–283
GENMOD procedure
complementary log-log options 241
discrete-time methods and 6
functionality 6
MODEL statement 241
REPEATED statement 281
Geweke statistic 120
Gibbs sampler 118
GLIMMIX procedure 241, 278, 279–280
Gompertz distribution
complementary log-log model and 242
Cox regression models and 127
functionality 20–22
hazard functions and 20–22
goodness-of-fit tests 98–100
gradient, defined 93
Greenwood's formula 33
GROUP option, STRATA statement
(LIFETEST) 63
grouped data 55–59, 248
Gumbel distribution 78, 241
H
Hall-Wellner method 36
hazard functions
alternative distributions and 78
baseline 127
cause-specific hazards 204–206
conditional density and 17
constant 78
cumulative 64
defined 16–18
discrete-time 150
estimated 53–55
exponential model and 20
gamma model and 85–86
generating predictions and 108–112
Gompertz distribution 20–22
interpretations of 18–20
kernel smoothing method 66
log-logistic model and 83–84
log-normal model and 82–83, 98
origin of time and 22–26
parametric regression models and
108–112
smoothed 67–69, 210–211
studying shape of 290–291

type-specific hazards 204–206
uncontrolled heterogeneity and 70
unobserved heterogeneity and 259
Weibull distribution model 21–22
hazard ratios
customized 195–197
interpreting 132, 135
HAZARDRATIO statement, PHREG
procedure 195–197
CL=PL option 196
heart transplant study
See Stanford Heart Transplantation
Program
Hessian matrix 93–94
heterogeneity
exponential model and 79
hazard functions and 70
uncontrolled 70
unobserved 79, 82, 257–260
hypothesis tests
AFT model and 74–75
chi-square statistics and 95–97
liability of newness 259
LIFEREG procedure and 71, 95–98
likelihood-ratio tests 95–96
linear 192–194
parametric regression models and
95–98

I
ID statement, PHREG procedure 267
IF-DO statement 157
IF statement 155, 171
IML procedure 96
incidental parameters bias 273
independence hypothesis 42
informative censoring 14–15, 283–287
intensity functions
See hazard functions
intercept estimates 132
interval censoring
defined 10
LIFEREG procedure and 71, 106–108,
235
intervals
empty 249–250
unequaled 248–249
INTERVALS= option, LIFETEST procedure
49
ITPRINT option, MODEL statement
(LIFEREG) 93

INDEX
317

J
job duration data
common time origins and 276
competing risks analysis 252
Cox regression models and 147–151
fixed-effects models 274
JOBDUR data set 302
JOBMULT data set 305
logit model for discrete time 238–239
maximum likelihood estimates 242
PHREG procedure 264
repeated events and 262, 264
JOBDUR data set 302
JOBMULT data set 305

K
Kaplan-Meier method
functionality 29–38
LIFEREG procedure and 31, 101
LIFETEST procedure and 29, 31, 187,
229–231
PHREG procedure and 31, 290
kernel smoothing method 66, 210
kidney function study 179–180
Kolmogorov-Smirnov type test 174

L
late entry into the risk set 183–186
LEADERS data set 303
leadership, time in power
covariate effects via Cox models
213–220
estimates and tests without covariates
208–213
LEADERS data set 303
study overview 207–208
left censoring
choosing methods 292
defined 9–10
LIFEREG procedure and 71, 103–106,
292
origin of time and 10
left truncation 183–186, 250
liability of newness hypothesis 259
life-table method
functionality 29, 49–55
grouped data 55–59
heart transplant study and 56–59
LIFETEST procedure and 5, 29
LIFEHAZ macro 111, 293–295
LIFEREG procedure

See also MODEL statement, LIFEREG
procedure
See also parametric regression models
additional distributions 77–89
AFT model and 72–73, 220
BAYES statement 120
Bayesian analysis and 117–124, 131,
197
censoring and 6, 103–108
choosing methods 290–291
CLASS statement 87–89, 94–96, 192
competing risks analysis and 203
continuous-time methods and 6
convergence problems 94
data structure requirements 26
exponential models and 221
functionality 5–6
2
generalized R 282
generating predictions 108–112
hypothesis testing 71, 95–98
informative censoring and 285
interval censoring and 71, 106–108, 235
Kaplan-Meier method and 31, 101
left censoring and 71, 103–106, 292
maximum likelihood estimation 89–95
Newton-Raphson algorithm 140
OLS and 93
ORDER option 88
OUTEST= option 110–111
OUTPUT statement 108–111, 291
partial likelihood method and 129–133
PHREG procedure comparison 71–72
piecewise exponential model 112–116
PROBPLOT statement 100–101
QUANTILE keyword 109
STD keyword 109
testing for effects of covariates 60
time-dependent covariates and 6, 235
TIMELIST option 49
Weibull model and 283
LIFETEST procedure
See also PLOTS= option, LIFETEST
procedure
See also survival curves
ADJUST= option 46, 48
ALPHA= option 37
ATRISK option 35
CL option 35
competing risks analysis and 203
CONFTYPE= option 38
data structure requirements 26
FREQ statement 56–57
functionality 5–6, 29

INDEX
318

LIFETEST procedure (continued)
INTERVALS= option 49
Kaplan-Meier method and 29, 31, 187,
229–231
kernel smoothing option 210
life-table method 5, 29
METHOD= option 31
NOCENSOR option 35
OUTSURV= option 37–38
STRATA statement 39, 43–45, 47,
62–63
TEST option 39
TEST statement 62–63
testing differences in survival curves
141
time in power study 209
TIME statement 39, 74
likelihood-ratio tests
goodness-of-fit and 98–100
hypothesis testing and 95–96
p-values for 40
partial likelihood estimation and 131
linear hypothesis tests 192–194
LINK= option
MODEL statement (LOGISTIC) 212,
241, 252
MODEL statement (SURVEYLOGISTIC)
279
log-hazard format 78, 132
log-log function, complementary 240–243
log-log survival (LLS) plots
myelomatosis study 64–66
time in leadership study 209–210
log-logistic model 83–85
log-normal model
competing risks analysis 222
functionality 73, 82–83
generalized gamma model and 98–99
hazard functions and 82–83, 98
LIFEREG procedure and 5
probability plot for 102
log-rank statistic
partial likelihood estimation and 141
testing for differences in survivor
functions 40–43, 46–47
testing for effects of covariates 59, 61
log-survival (LS) plot 64–65, 132
LOGISTIC procedure
CLASS statement 238
competing risks analysis and 251–254
complementary log-log functions 241
discrete-time hazards 150, 278
discrete-time methods and 6

functionality 6
2
generalized R 282
grouped data syntax 248
logit model for discrete time 238–240
MODEL statement 212, 239, 241, 252
multinomial logit model and 212
repeated events and 278–281
STRATA statement 280
testing proportionality 212
logit model
competing risks analysis 226, 228
complementary log-log model 240–243
for academic promotions 243–246
for discrete time 236–240
multinomial 212, 251–252
LOWER= option, BASELINE statement
(PHREG) 187, 191

M
Markov Chain Monte Carlo (MCMC)
methods 117–118, 120
martingale residuals 173
maximum likelihood estimation
See ML (maximum likelihood)
estimation
maximum partial likelihood method 126
MAXITER= option, MODEL statement
(LIFEREG) 94
MCMC methods 117–118, 120
MDY function 28
METHOD= option
LIFETEST procedure 31
MODEL statement (GLIMMIX) 278
ML (maximum likelihood) estimation
censoring and 73, 91
competing risks analysis 250–254
complications and concerns 246–254
functionality 89–90
job duration data 242
LIFEREG procedure and 89–95
mathematics 90–93
parametric regression models and
89–95
piecewise exponential model and 199
practical details 93–95
random-effects models and 271
repeated events and 278–281
time-dependent covariates and 243–246
MMDDYY9. format 134
MODEL statement, GENMOD procedure
241
MODEL statement, GLIMMIX procedure
241, 278

INDEX
319

MODEL statement, LIFEREG procedure
categorical variables 89
censoring and 104–105
CONVERGE= option 94
COVB option 93, 115
DISTRIBUTION=EXPONENTIAL option
78
DISTRIBUTION=GAMMA option 86
DISTRIBUTION=LLOGISTIC option 85
DISTRIBUTION=WEIBULL option 80
hypothesis tests and 96
ITPRINT option 93
MAXITER= option 94
MODEL statement, LOGISTIC procedure
LINK= option 212, 241, 252
logit model for discrete time 239
REF= option 252
MODEL statement, NLMIXED procedure
271
MODEL statement, PHREG procedure
BAYES statement and 197
customized hazard ratios 196
estimating survivor function 188, 192
FIRTH option 141
informative censoring and 285
programming statements method and
155
TIES=DISCRETE option 143, 149–153,
236
TIES=EFRON option 146–147, 217
TIES=EXACT option 142, 143–149,
152–153, 228, 236
MODEL statement, PROBIT procedure 241
MODEL statement, SURVEYLOGISTIC
procedure 279
modified Peto-Peto test 43–44
modified sandwich estimator 266–270
Monte Carlo simulations 274
multinomial logit model 212, 251–252
MYEL data set 299–300
myelomatosis data
Cox regression model and 181
Kaplan-Meier method 31–38
log-survival plots 64–65
MYEL data set 299–300
testing for differences in survivor
functions 39–44

N
nesting 98–99
newness, liability of 259
Newton, Isaac 92
Newton-Raphson algorithm 92–93, 140

NLMIXED procedure 271–272, 279–280
NOCENSOR option, LIFETEST procedure
35
noninformativeness 206
nonproportional hazards
functionality 172–173
interactions with time 177–179
stratification and 179–183
testing proportionality assessments
173–177
NOSUMMARY option, PHREG procedure
273

O
observations
dependence among 246–247
large numbers of 247–248
odds ratios, and hazard ratios 132
ODS (output delivery system) 34–35
ODS GRAPHICS 118, 188
OLS (ordinary least squares)
AFT model and 73
comparing with logarithm of time 283
LIFEREG procedure and 93
ORDER option, LIFEREG procedure 88
ordinary least squares
See OLS
origin of time
See time origins
OUT= option, BASELINE statement
(PHREG) 187
OUTEST= option, LIFEREG procedure
110–111
output delivery system (ODS) 34–35
OUTPUT statement
LIFEREG procedure 108–111, 291
PHREG procedure 194
programming statements method and
157
OUTSURV= option, LIFETEST procedure
37–38
OVERLAY suboption (PLOTS= option,
LIFETEST) 190

P
p-values 40, 89
PARAM= option, CLASS statement
(LOGISTIC) 238
parametric regression models
See also LIFEREG procedure
accelerated failure time model 72–77
alternative distributions 77–86

INDEX
320

parametric regression models (continued)
Bayesian estimation and timing
117–124, 131
categorical variables 87–89
censoring and 103–108
CLASS statement and 87–89
evaluating model fit 100–103
functionality 71–72
generating predictions 108–112
goodness-of-fit tests and 98–100
hazard functions and 108–112
hypothesis tests 95–98
maximum lilkelihood estimation and
89–95
piecewise exponential model 112–116
PARMS statement, NLMIXED procedure
272
partial likelihood method
Bayesian information criterion (BIC)
131
Breslow approximation 142
chi-square statistics and 131–132
Cox regression models and 126–128
examples 129–136
functionality 150
heart transplant study and 133–136
intercept estimates and 132
LIFEREG procedure and 129–133
likelihood-ratio tests 131
log-rank statistic 141
mathematical and computational details
137–141
maximum 126
PHREG procedure 129–136
random-effects models and 271
score tests and 131–132
semiparametric regression analysis and
71
time-dependent covariates and 158–160
p.d.f. (probability density function) 16, 91
Pearson's chi-square test 208
Peto-Peto test 43–44
PHREG procedure
See also Cox regression models
See also MODEL statement, PHREG
procedure
accelerated failure time models 220
ASSESS statement 173–178
BASELINE statement 186–187,
189–192, 290–291
BAYES statement 197–199
Bayesian analysis and 197
BEST= option 201

breast cancer study 141
Breslow approximation 142
BY statement 263
censoring and 6
choosing methods 289–290
CLASS statement 192–193
common time origins and 275–277, 279
comparison of methods 152–153
competing risks analysis and 203
continuous-time methods and 6
CONTRAST statement 192–194
convergence criterion 140
counting process method 154, 156, 158
covariate effects via Cox models
214–217
COVSANDWICH option 266–269, 276
cumulated variables 165
data structure requirements 26
DISCRETE method 143, 149–153
dicrete-time methods and 6
EXACT method 143–149, 152–153
functionality 5–6, 125–126
2
generalized R 282
HAZARDRATIO statement 195–197
ID statement 267
job duration data 264
Kaplan-Meier method and 31, 290
LIFEREG procedure comparison 71–72
log-hazard form 78
NOSUMMARY option 273
OUTPUT statement 194
partial likelihood method and 129–136
piecewise exponential model and 116
PLOTS= option 186, 188
programming statements method
154–155, 157
SELECTION= option 60, 201
STRATA statement 180–182, 218, 273
TEST statement 192–194, 269
testing for effects of covariates 60, 63
time-dependent covariates and
154–155, 159–168, 170–171
piecewise exponential model 112–116, 199
PIECEWISE option, BAYES statement
(PHREG) 199
PLOTS= option, LIFETEST procedure
equal precision method 36
evaluating model fit 100
Hall-Wellner method 36
hazard functions 66–68
log survival plots 64
OVERLAY suboption 190

INDEX
321

plots of survival and hazard estimates
53
plots with confidence limits 36, 68
time in leadership study 209
PLOTS= option, PHREG procedure 186,
188
pointwise limits 36
population-averaged method 266–270
posterior distribution 117–124
power of county leaders
See leadership, time in power
PREDICT macro 110–111, 291, 296–297
predictions, generating 108–112
prior distribution 117
probability density function (p.d.f.) 16, 91
probability distributions
Bayesian analysis and 117
cumulative distribution function 15–16
hazard function 16–22
LIFEREG procedure and 72
probability density function 16
probability plots 101–103
PROBIT procedure 241
PROBPLOT statement, LIFEREG procedure
100–101
product-limit estimator
See Kaplan-Meier method
programming statements method
albumin measurement study 167
alcoholic cirrhosis study 170–171
heart transplant study 154–155, 157
proportional hazards model
continuous-time method and 236
Cox regression models and 126–128,
172–179
defined 22, 42, 126–128
estimating 279
exponential model and 20
Gompertz distribution 20–22, 127
PHREG procedure 5
time-invariant covariates and 140
type-specific hazards and 205
Weibull distribution model 21–22, 127
proportionality, test of 173–177, 212
prospective analysis 3

Q
qualitative changes 2
QUANTILE keyword, LIFEREG procedure
109

R
2

R 282–283
random censoring 12–13
random-effects models 6, 270–272, 280
RANDOM statement, NLMIXED procedure
271–272
RANK data set 304
Raphson, Joseph 92
rate, defined 18
RECID data set 300–301
recidivism study
AFT model and 73–77
Bayesian analysis and 118–124, 197
Breslow approximation and 145–147
censoring and 104–108
counting process method and 165
covariate tests for 61
customized hazard ratios 195
Efron's approximation and 146–147
exponential model and 79
gamma model and 86, 99
generating predictions 108–112
life-table method 47–55
likelihood-ratio tests 96, 100
log-logistic model and 85
log-normal model and 82–83
partial likelihood estimation and 130
piecewise exponential model and
113–115
probability plots for 101–103
RECID data set 300–301
smoothed hazard functions 67
survivor function estimates 187, 189,
191
testing for differences in survivor
functions 45–48
testing for effects of covariates 60–61
tied data 143
ties 142
time-dependent covariates and
163–164, 178–179
Weibull distribution model 81, 283
REF= option, MODEL statement (LOGISTIC)
252
REG procedure 73
regression
See also Cox regression models
See also parametric regression models
semiparametric regression analysis 71
renal function study 179–180

INDEX
322

repeated events
background information 260–261
discrete-time maximum likelihood
278–281
fixed-effects models 273–275
job duration data and 262, 264
LOGISTIC procedure 278–281
ML estimation 278–281
problems with conventional methods
261–266
random-effects models 270–272
robust standard errors 266–270
specifying common origins for 275–278
REPEATED statement, GENMOD procedure
281
RESAMPLE option, ASSESS statement
(PHREG) 174
residuals, martingale 173
retrospective data 3
right censoring
defined 9–10
patterns of 11–15
PHREG procedure and 71
robust variance estimator 266–270

S
sandwich estimator, modified 266–270
Schoenfeld residuals 175–178
Schwarz's criterion
See Bayesian information criterion (BIC)
score, defined 93
score tests
defined 95
partial likelihood estimation and
131–132
SELECTION= option, PHREG procedure 60,
201
semiparametric regression analysis 71
sensitivity analysis for informative censoring
283–287
singly Type I censored 11–12
slash (/) 74
smoothed hazard functions 66–69, 210–211
spell, defined 207
STAN data set 301
Stanford Heart Transplantation Program
counting process method and 156, 158,
184
Efron’s approximation and 155
left truncation and 184–186
life-table method 56–59
partial likelihood estimation 133–136
STAN data set 301

time-dependent covariates 154–161,
169
STD keyword, LIFEREG procedure 109
STRATA statement
LIFETEST procedure 39, 43–45, 47,
62–63
LOGISTIC procedure 280
PHREG procedure 180–182, 218, 273
stratification technique 179–183
study of death
See survival analysis
SUBJECT option, RANDOM statement
(NLMIXED) 272
subject-specific models 270
SUMMARY procedure 248
SURVEYLOGISTIC procedure 279
survival analysis
approaches to 5–6
breast cancer study 137–141
censoring 4–5, 9–14
choice of origin in measuring time
23–26
data structure requirements 26–28
defined 1
DISCRETE method and 149
interpreting hazard function 18–20
representing survival distributions
15–18
simple hazard models 20–22
suitable uses 1
survival data in 2–3
time-dependent covariates 4–5
usage considerations 4–5
survival curves
See also LIFETEST procedure
functionality 5–6
Kaplan-Meier method 30–38
life-table method 49–55
life tables from grouped data 55–59
log survival plots 64–66
smoothed hazard function 67–69
testing for differences in 38–48, 141
testing for effects of covariates 59–63
survival data
analysis considerations 2–3
censoring considerations 4–5
time-dependent covariates 4–5
timing of events 23–26
survival distributions
See probability distributions
SURVIVAL= option, BASELINE statement
(PHREG) 187, 192
survivor functions

INDEX
323

AFT model and 72
defined 5, 15
estimating 186–192
log-logistic model 84
PHREG procedure and 71
testing for differences in 38–48
type-specific hazards 205–206
Weibull distribution model 80

T
Tarone-Ware test 43–44
TEST option, LIFETEST procedure 39
TEST statement
LIFETEST procedure 62–63
PHREG procedure 192–194, 269
testing
See also hypothesis tests
for differences in survival curves 38–48,
141
for effects of covariates 59–63
goodness-of-fit tests 98–100
linear hypotheses 192–194
proportionality 173–177, 212
THIN= option, BAYES statement (LIFEREG)
120
tied data
Breslow approximation 131, 142,
145–149
Cox regression models and 142–153
defined 142–143
DISCRETE method 143, 149–153
EXACT method 143–149, 152–153
interval censoring and 235
TIES=DISCRETE option, MODEL statement
(PHREG)
background information 142
comparing methods for handling ties
152–153
discrete-time method and 236
functionality 149–151
TIES=EFRON option, MODEL statement
(PHREG) 146–147, 217
TIES=EXACT option, MODEL statement
(PHREG)
background information 142
comparing methods for handling ties
152–153
functionality 144–149
multiple event types and 228
proportional hazards model and 236
time-constant covariates 6, 72
time-dependent covariates
ad-hoc estimates 166–169

changing at irregular intervals 169–172
Cox regression models and 153–172
data with 243–246
functionality 4–5, 153–154
heart transplant study and 135–136,
154–161
interactions with time 177–179
LIFEREG procedure and 6, 235
LOGISTIC procedure and 6
measuring at regular intervals 161–166
ML estimation 243–246
partial likelihood method and 158–160
PHREG procedure and 6, 72
time in power for leaders study
See leadership, time in power
time origins
age criteria and 24
calendar time and 24
covariates representing 160–161
hazard functions and 22–26
job duration data 276
left censoring and 10
specifying common 275–278, 279
timing of events and 23–26
TIME statement, LIFETEST procedure 39,
74
TIMELIST option, LIFEREG procedure 49
transplant study
See Stanford Heart Transplantation
Program
truncation, left 183–186, 250
Tukey's method 46
Type I censoring 11–13
Type II censoring 12–13
TYPE option, STRATA statement (PHREG)
218
type-specific hazards 204–206

U
uncontrolled heterogeneity 70
unequaled intervals 248–249
univariate analysis 5
unobserved heterogeneity 79, 82, 257–260
UPPER= option, BASELINE statement
(PHREG) 187, 191

V
variables
categorical 87–89
cumulated 165
vector notation 7

INDEX
324

W
Wald tests
competing risks analysis 218–219
defined 95–96
partial likelihood estimation and 132
piecewise exponential model and
115–116
Weibull distribution model
accelerated failure time model 80–82
Bayesian analysis and 118
competing risks analysis and 222–224
complementary log-log model and 242
estimating 283
functionality 21–22, 80–82
generalized gamma model and 85,
98–99
goodness-of-fit and 98–99
hazard functions and 21–22
left censoring and 106
LIFEREG procedure and 283
likelihood-ratio tests and 96
log survival plots and 65
partial likelihood estimation and
132–133
probability plot 101–103
proportional hazards model and 127
random-effects models and 272
WHERE statement 164, 225
Wilcoxon statistic 40–44, 59
WLW macro 268

X
XBETA= option, OUTPUT statement
(LIFEREG) 110

Z
z-statistic 120

Symbols
/ (slash) 74

Allison, Paul D. Survival Analysis Using SAS®: A Practical Guide, Second Edition. Copyright © 2010, SAS Institute Inc.,
Cary, North Carolina, USA. ALL RIGHTS RESERVED. For additional SAS resources, visit support.sas.com/publishing.

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