Confidence Intervals Instructions

instructions-confidence-intervals

instructions-confidence-intervals

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Confidence intervals

Rona Axelrod and Danny Kaplan

StatPREP Instructor Guide

Activities

•What is a confidence interval?

•Sampling bias and confidence intervals

•Comparing two groups

•Comparing two confidence intervals

•Two-sample t test

Learning objectives connected to confidence intervals

1. Master the relevant vocabulary ... population, sample, sample statistic,

point estimate, margin of error, confidence interval, confidence level.

2. Recognize the estimation process that leads to a confidence interval:

1. a sample is selected randomly from a population

2. a statistic is calculated from that sample

3. the margin of error and, correspondingly, the confidence interval are

calculated from that sample.

3. Distinguish between the “summary interval” (which is a summary of the

raw values of the variable) and the “confidence interval” (which describes

uncertainty in a sample statistic or model parameter).

4. Understand that each time that estimation process is carried out, a new

confidence interval will be created.

5. Recognize and appropriately make use of the confidence interval as a

presentation of uncertainty due to sampling.

6. Appropriately compare confidence intervals not by whether their endpoints

are approximately the same but by whether or not they overlap.

i. Interpret a lack of overlap as an indication that the populations underly-

ing the two samples are different.

7. Be able to translate the width of a confidence interval from a preliminary

study of size n into a reasonable guess for the margin of error for a study of

size N.

8. Recognize common sources of sampling bias – e.g. self-selection, pre-

screening, survival – and that the possible existence of sampling bias is not

incorporated into the confidence interval.

Additional resources

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•Instructor orientation

•Role in statistical practice

•Classroom discussion

•Assessment

•Tips for an active classroom

•Student pre-requisites

•Pitfalls

Orientation

A fundamental aspect of statistical reasoning is recognizing that estimates

computed from a sample are, to some extent, random. That is, another sample

will likely produce a different estimate.

Remarkably, we can estimate the amount of uncertainty due to the random-

ness of sampling from just a single sample. We quantify this uncertainty with

amargin of error. As a rule, the margin of error is inversely proportional to

the square root of the sample size: √n. That proportionality with n plays out

against a base value that depends on the particular statistic being calculated,

e.g. the mean, the median, the standard deviation, the 75th percentile, and so

on.

There are two conventional notations for the margin of error:

•In text the margin of error is presented after the point estimate with a

±serving as punctuation. Example: the mean height of a group of 100

adults would be written as 55.6 ±0.3 inches, where 55.6 inches is the point

estimate and 0.3 inches is the margin of error. The whole assembly – that

is, 55.6 ±0.3 inches – is called a confidence interval.

•In graphics, the margin of error is the half-length of the I-shaped error bar

drawn centered on the point estimate. The ends of the error-bar glyph

demark the ends of the confidence interval.

The idea of a margin of error is closely tied to the use of the normal proba-

bility distribution to model the sampling variability of the sample statistic. But

many important sample statistics do not have a sampling distribution that is

meaningfully approximated as normal. An example is the risk ratio: the ratio

of two proportions that is often used in medical or epidemiological commu-

nication but which is much more widely applicable. With such statistics, the

confidence interval is not centered on the point estimate.

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Role in statistical practice

Much of the use of margins of error and confidence intervals in statistical com-

munication is based on a misconception. Consider this definition of confidence

interval from the widely used Elementary Statistics text by Triola:

A confidence interval is a range of values used to estimate the true value of a

population parameter. – p. 300, 13th edition

This quote correctly captures how confidence intervals are generally used:

as a prediction interval for the “true value of a population parameter.” That is,

many people interpret a confidence interval as marking the probability distri-

bution of the “true value” given the sample. This is wrong, simply because the

“true value” isn’t random. But the interpretation isn’t horrible and is arguably

much better than ignoring the uncertainty in the sample statistic.

Perhaps it would be better simply to regard confidence intervals as a marker

of an abstract quantity, precision. Then we can focus on how that abstract

quantity can be used in practice, for instance,

1. Determine when another estimate of the same statistical quantity (using a

different sample) is inconsistent with the estimate we have made with our

sample. Such inconsistency is indicated when the confidence intervals of

the two estimates do not overlap. Or, if we are looking for consistency with

a point estimate, the inconsistency is marked when our confidence interval

doesn’t include the point estimate.

2. Anticipate how the precision would change if more data were collected or if

an estimate were made using a new sample entirely.

Conceptual pitfalls

Many students (and professionals) will acquire a completely incorrect interpre-

tation of confidence intervals and treat them as if they were summary intervals.

For instance, consider an estimate of mean commute time and its confidence

interval. Many people will mistakenly think that “most” commute times fall

within the confidence interval. In reality the fraction that do can be vanishingly

small for large n.

We suspect that one reason for this misconception is that conventional

intro stats book do not have a name for the interval that contains “most” of the

raw values. This is why we introduce the summary interval in these StatPREP

101 lessons.

Similarly, we need to deal with the usual failure of introductory statistics to

include reference to Bayesian ideas. Usually this is justified by pointing out the

dominance of frequentist inference ideas in published research. That may be

so, but there is a strong connection between informal Bayesian ideas and the

ways people think about knowledge and uncertainty.

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Keep in mind that the way most people (mis-)interpret statistical inferential

objects such as confidence intervals and p-values is fundamentally Bayesian.

That is, they seek to know the probability of an event, given the data. That

event might be “the ‘true value’ falls within the interval,” or “the null hypothesis

is false.”

It can be tempting to try to correct the common misconceptions about con-

fidence intervals and p-values. For instance, some textbooks include a process

view of confidence intervals by imagining a large collection of samples, each

of which produces its own confidence interval. Then, when the “true value”

of the population parameter is revealed to us, we will find that 95% of those

intervals include the “true value.” Unfortunately, the “correct” interpretation of

frequentist objects of inference does not inform practice any differently than

the “incorrect” Bayesian interpretation. Perhaps correctives to such miscon-

ceptions should be saved for courses about the development of statistical

methods rather than their application.

We often forget that confidence intervals are an expression of uncertainty

and seek to make them precise beyond any reasonable use. This is perhaps

why many educators favor 1.96 over 2 or spend so much time on special tech-

niques for small data (such as t-intervals). The search for extra precision is

also what prevents the introduction of simple techniques for comparing es-

timates – do the confidence intervals overlap? – in favor of more complex

techniques such as the t-test.

The confidence interval can be calculated from data and is a reliable mea-

sure of the amount of uncertainty created by a random sample of size n. There

are other sources of uncertainty – biases – which can be important but which

are not captured by a confidence interval. In particular, the canonical example

used for teaching confidence intervals, survey or poll results, are in today’s

world so heavily influenced by self-selection bias to the extent that the con-

fidence interval itself does not provide a valid indication of the reliability of

results.

Student prerequisites

1. Familiarity with how the process of random sampling introduces uncer-

tainty: the sample statistic will likely be different from one sampling trial to

another.

2. Acquaintance with the idea of an interval representing the vast majority

(taken usually as 95%) of the possibilities.

Creating an active classroom

Some specific discussion topics/themes for confidence intervals:

•Where is the mean located relative to the lower and upper bounds of the

confidence interval?

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•How do the following factors affect the distance between the lower and

upper bounds of the confidence interval: sample size, spread of the data,

and confidence level?

•Do you notice any overlap of the confidence intervals? Generate new sam-

ples and notice whether the same behaviors occur. What does the overlap

or lack of overlap tell us?

Assessment items

•Given a population parameter and a single confidence interval for a sample

of size n ...

1. Draw 20 plausible confidence intervals for new samples of size n

2. Draw 20 plausible confidence intervals for new samples of size 4n

•Given a population parameter and a confidence interval on the mean for size

n, estimate and draw a 95% summary interval.

•Given a small set of data, construct a bootstrap trial

Pushing the envelope/advancing the field

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