Paramap Manual

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Package ‘paramap’
March 17, 2019
Type Package
Title Factor Analysis Functions for Assessing Dimensionality
Version 1.9
Date 2019-03-16
Author Brian P. O'Connor
Maintainer Brian P. O'Connor
Description Factor analysis-related functions and datasets for assessing dimensionality.
Imports stats, utils, psych, polycor
Suggests lattice
LazyLoad yes
LazyData yes
License GPL (>= 2)
NeedsCompilation no

R topics documented:
paramap-package .
CONGRUENCE .
data_Harman . . .
data_NEOPIR . . .
data_RSE . . . . .
data_WISC . . . .
EXTENSION_FA .
FACTORABILITY
IMAGE_FA . . . .
LOCALDEP . . .
MAP . . . . . . . .
MAXLIKE_FA . .
NEVALSGT1 . . .
PARALLEL . . . .
PA_FA . . . . . . .
PCA . . . . . . . .
POLYCHORIC_R
PROCRUSTES . .
PROMAX . . . . .
RAWPAR . . . . .

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CONGRUENCE
ROOTFIT .
SALIENT .
SESCREE .
VARIMAX

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Index

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paramap

Description
This package provides factor analysis-related functions for assessing dimensionality. Users can request that the analyses be conducted using polychoric correlations, which are preferable to Pearson
correlations for item-level data, and a polychoric correlation matrix is returned for possible further
analyses. There are also functions for conducting principal components analysis, principal axis factor analysis, maximum likelihood factor analysis, image factor analysis, and extension factor analysis, all of which can take raw data or correlation matrices as input and with options for conducting
the analyses using Pearson correlations, Kendall correlations, Spearman correlations, or polychoric
correlations. Varimax rotation, promax rotation, and Procrustes rotations can be performed.

CONGRUENCE

factor solution congruence

Description
This function aligns two factor loading matrices and computes the factor solution congruence and
the root mean square residual.
Usage
CONGRUENCE(target, loadings, display)
Arguments
target

The target loading matrix.

loadings

The loading matrix that will be aligned with the target.

display

Display the results? The options are TRUE or FALSE.

Details
The function first searches for the alignment of the factors from the two loading matrices that has
the highest factor solution congruence. It then aligns the factors in "loadings" with the factors
in "target" without changing the loadings. The alignment is based solely on the positions and
directions of the factors. The function then produces the Tucker-Wrigley-Neuhaus factor solution
congruence coefficient as an index of the degree of similarity between between the aligned loading
matrices (see Guadagnoli & Velicer, 1991; and ten Berge, 1986, for reviews).

data_Harman

Value
A list with the following elements:
rcBefore

The factor solution congruence before factor alignment

rcAfter

The factor solution congruence after factor alignment

rcFactors

The congruence for each factor

rmsr

The root mean square residual

residmat

The residual matrix

loadingsNew

The aligned loading matrix

Author(s)
Brian P. O’Connor
References
Guadagnoli, E., & Velicer, W. (1991). A comparison of pattern matching indices. Multivariate
Behavior Research, 26, 323-343.
ten Berge, J. M. F. (1986). Some relationships between descriptive comparisons of components
from different studies. Multivariate Behavioral Research, 21, 29-40.
Examples
# RSE data
loadings <- PCA(data_RSE[1:150,],
corkind='pearson', nfactors = 3,
rotate='varimax', display=FALSE)
target
<- PCA(data_RSE[151:300,], corkind='pearson', nfactors = 3,
rotate='varimax', display=FALSE)
CONGRUENCE(target$loadingsROT, loadings$loadingsROT, display=TRUE)
## Not run:
# NEO-PI-R data
loadings <- PCA(data_NEOPIR[1:500,], corkind='pearson', nfactors = 3,
rotate='varimax', display=FALSE)
target <- PCA(data_NEOPIR[501:1000,], corkind='pearson', nfactors = 3,
rotate='varimax', display=FALSE)
CONGRUENCE(target$loadingsROT, loadings$loadingsROT, display=TRUE)
## End(Not run)

data_Harman

Correlation matrix from Harman (1967, p. 80).

Description
The correlation matrix for eight physical variables for 305 cases from Harman (1967, p. 80).
Usage
data(data_Harman)

data_NEOPIR

References
Harman, H. H. (1967). Modern factor analysis (2nd. ed.). Chicago: University of Chicago Press.
Examples
## Not run:
# MAP test on the Harman correlation matrix
MAP(data_Harman, display=TRUE)
# parallel analysis of the Harman correlation matrix
RAWPAR(data_Harman, extract='PCA', Ndatasets=100, percentile=95,
Ncases=305, display=TRUE)
## End(Not run)

data_NEOPIR

Description
A data frame with scores for 1000 cases on 30 variables that have the same intercorrelations as
those for the Big 5 facets on pp. 100-101 of the NEO-PI-R manual (Costa & McCrae, 1992).
Usage
data(data_NEOPIR)
References
Costa, P. T., & McCrae, R. R. (1992). Revised NEO personality inventory (NEO-PIR) and NEO
five-factor inventory (NEO-FFI): Professional manual. Odessa, FL: Psychological Assessment Resources..
Examples
head(data_NEOPIR)
## Not run:
# MAP test on the data_NEOPIR data
MAP(data_NEOPIR, corkind='pearson', display=TRUE)
# parallel analysis of the data_NEOPIR data
RAWPAR(data_NEOPIR, extract='PCA', Ndatasets=100, percentile=95,
corkind='pearson', display=TRUE)
## End(Not run)

data_RSE

Item-level dataset for the Rosenberg Self-Esteem scale

Description
A data frame with 300 observations on the 10 items from the Rosenberg Self-Esteem scale.
Usage
data(data_RSE)
Examples
head(data_RSE)
## Not run:
# MAP test on the Rosenberg Self-Esteem Scale (RSE) data
MAP(data_RSE, corkind='pearson', display=TRUE)
# parallel analysis of the Rosenberg Self-Esteem Scale (RSE) data
RAWPAR(data_RSE, extract='PCA', Ndatasets=100, percentile=95,
corkind='pearson', display=TRUE)
## End(Not run)

data_WISC

Description
A data frame with scores for 175 cases on 10 WISC-R subscales, used by Tabacknick & Fidell
(2013, p. 737) in their section on confirmatory factor analysis.
Usage
data(data_WISC)
References
Tabachnik, B. G., & Fidell, L. S. (2014). Using multivariate statistics. New York, NY: Pearson.
Examples
head(data_WISC)
## Not run:
# MAP test on the data_WISC data
MAP(data_WISC, corkind='pearson', display=TRUE)
# parallel analysis of the data_WISC data
RAWPAR(data_WISC, extract='PCA', Ndatasets=100, percentile=95,

EXTENSION_FA
corkind='pearson', display=TRUE)
## End(Not run)

EXTENSION_FA

extension factor analysis

Description
Extension factor analysis provides correlations between nonfactored items and the factors that exist
in a set of core items. The extension item correlations are then used to decide which factor, if any,
a prospective item belongs to.
Usage
EXTENSION_FA(data, Ncore, Next, higherorder, roottest,
corkind, corkindRAND, extract, rotate, Nfacts,
NfactsHO, Ndatasets, percentile, salvalue, numsals,
iterpaf, iterml, tolerml, ppower)
Arguments
data

An all-numeric dataframe where the rows are cases & the columns are the variables.

Ncore

An integer indicating the number of core variables. The function will run the
factor analysis on the data that appear in column #1 to column #Ncore of the
data matrix.

An integer indicting the number of extension variables, if any. The function
will run extension factor analyses on the remaining columns in data, i.e., using
column #Ncore+1 to the last column in data. Enter zero if there are no extension
variables.

corkind

The kind of correlation matrix to be used. The options are ’pearson’, ’kendall’,
’spearman’, and ’polychoric’.

corkindRAND

The kind of correlation matrix to be used for the random data when roottest =
’parallel’. The options are ’pearson’, ’kendall’, ’spearman’, and ’polychoric’.
These options are included for research purposes. In most applications, it is
probably best to use Pearson correlations, which is the default.

higherorder

Should a higher-order factor analysis be conducted? The options are TRUE or
FALSE.

roottest

The method for determing the number of factors. The options are: ’Nsalient’ for
number of salient loadings (see salvalue & numsals below); ’parallel’ for parallel analysis (see Ndatasets & percentile below); ’MAP’ for Velicer’s minimum
average partial test; ’SEscree’ for the standard error scree test; ’nevals>1’ for
the number of eigenvalues > 1; and ’user’ for a user-specified number of factors
(see Nfacts & NfactsHO below).

Nfacts

An integer indicating the user-determined number of factors (required only if
roottest = ’user’).

NfactsHO

An integer indicating the user-determined number of higher order factors (required only if roottest = ’user’ and higherorder = TRUE).

EXTENSION_FA

extract

The factor extraction method. The options are: ’PAF’ for principal axis / common factor analysis; ’PCA’ for principal components analysis; ’ML’ for maximum likelihood.

rotate

The factor rotation method. The options are: ’promax’, ’varimax’, and ’none’.

Ndatasets

An integer indicating the # of random data sets for parallel analyses (required
only if roottest = ’parallel’).

percentile

An integer indicating the percentile from the distribution of parallel analysis
random eigenvalues to be used in determining the # of factors (required only if
roottest = ’parallel’). Suggested value: 95

salvalue

The minimum value for a loading to be considered salient (required only if
roottest = ’Nsalient’). Suggested value: .40

numsals

The number of salient loadings required for the existence of a factor i.e., the
number of loadings > or = to salvalue (see above) for the function to identify a
factor. Required only if roottest = ’Nsalient’. Gorsuch (1995a, p. 545) suggests:
3

iterpaf

The maximum # of iterations for a principal axis / common factor analysis (required only if extract = ’PAF’). Suggested value: 100

iterml

The maximum # of iterations for a maximum likelihood analysis (required only
if extract = ’ML’). Suggested value: 100

tolerml

The tolerance value for a maximum likelihood analysis (required only if extract
= ’ML’). Suggested value: .001

ppower

The power value to be used in a promax rotation (required only if rotate = ’promax’). Suggested value: 3

Details
Traditional scale development statistics can produce results that are baffling or misunderstood by
many users, which can lead to inappropriate substantive interpretations and item selection decisions. High internal consistencies do not indicate unidimensionality; item-total correlations are
inflated because each item is correlated with its own error as well as the common variance among
items; and the default number-of-eigenvalues-greater-than-one rule, followed by principal components analysis and varimax rotation, produces inflated loadings and the possible appearance of
numerous uncorrelated factors for items that measure the same construct (Gorsuch, 1997a, 1997b).
Concerned investigators may then neglect the higher order general factor in their data as they use
misleading statistical output to trim items and fashion unidimensional scales.
These problems can be circumvented in exploratory factor analysis by using more appropriate factor analytic procedures and by using extension analysis as the basis for adding items to scales.
Extension analysis provides correlations between nonfactored items and the factors that exist in a
set of core items. The extension item correlations are then used to decide which factor, if any, a
prospective item belongs to. The decisions are unbiased because factors are defined without being
influenced by the extension items. One can also examine correlations between extension items and
any higher order factor(s) in the core items. The end result is a comprehensive, undisturbed, and
informative picture of the correlational structure that exists in a set of core items and of the potential
contribution and location of additional items to the structure.
Extension analysis is rarely used, at least partly because of limited software availability. Furthermore, when it is used, both traditional extension analysis and its variants (e.g., correlations between
estimated factor scores and extension items) are prone to the same problems as the procedures
mentioned above (Gorsuch, 1997a, 1997b). However, Gorusch (1997b) described how diagonal
component analysis can be used to bypass the problems and uncover the noninflated and unbiased
extension variable correlations – all without computing factor scores.

EXTENSION_FA

Value
A list with the following elements:
fits1

eigenvalues & fit coefficients for the first set of core variables

rff

factor intercorrelations

corelding

core variable loadings on the factors

extcorrel

extension variable correlations with the factors

fits2

eigenvalues & fit coefficients for the higher order factor analysis

rfflding

factor intercorrelations from the first factor analysis and the loadings on the
higher order factor(s)

ldingsef

variable loadings on the lower order factors and their correlations with the higher
order factor(s)

extsef

extension variable correlations with the lower order factor(s) and their correlations with the higher order factor(s)

Author(s)
Brian P. O’Connor
References
Gorsuch, R. L. (1997a). Exploratory factor analysis: Its role in item analysis. Journal of Personality
Assessment, 68, 532-560.
Gorsuch, R. L. (1997b). New procedure for extension analysis in exploratory factor analysis. Educational and Psychological Measurement, 57, 725-740.
Dwyer, P. S. (1937) The determination of the factor loadings of a given test from the known factor
loadings of other tests. Psychometrika, 3, 173-178.
Horn, J. L. (1973) On extension analysis and its relation to correlations between variables and
factor scores. Multivariate Behavioral Research, 8(4), 477-489.
O’Connor, B. P. (2001). EXTENSION: SAS, SPSS, and MATLAB programs for extension analysis.
Applied Psychological Measurement, 25, p. 88.
Examples
## Not run:
EXTENSION_FA(data_RSE, Ncore=7, Next=3, higherorder=TRUE, roottest='MAP',
corkind='pearson', extract='PCA', rotate='promax', Nfacts=4,
NfactsHO=1, Ndatasets=100, percentile=95, salvalue=.40, numsals=3,
iterpaf=200, iterml=30, tolerml=.001, ppower=4)
EXTENSION_FA(data_NEOPIR, Ncore=12, Next=6, higherorder=TRUE, roottest='MAP',
corkind='pearson', extract='PCA', rotate='promax', Nfacts=4,
NfactsHO=1, Ndatasets=100, percentile=95, salvalue=.40, numsals=3,
iterpaf=200, iterml=30, tolerml=.001, ppower=4)
## End(Not run)

FACTORABILITY

The factorability of a correlation matrix

Description
Three methods for assessing the factorability of a correlation matrix
Usage
FACTORABILITY(data, corkind='pearson', Ncases=NULL, display=TRUE)
Arguments
data

An all-numeric dataframe where the rows are cases & the columns are the variables, or a correlation matrix with ones on the diagonal. The function internally
determines whether the data are a correlation matrix.

corkind

The kind of correlation matrix to be used if data is not a correlation matrix. The
options are ’pearson’, ’kendall’, ’spearman’, and ’polychoric’. Required only if
the entered data is not a correlation matrix.

Ncases

The number of cases for a correlation matrix. Required only if the entered data
is a correlation matrix.

display

Display the results? The options are TRUE (the default) or FALSE.

Details
This function provides results from three methods of assessing whether a dataset or correlation
matrix is suitable for factor analysis:
1 – whether the determinant of the correlation matrix is > 0.00001;
2 – Bartlett’s test of whether a correlation matrix is significantly different an identity matrix; and
3 – the Kaiser-Meyer-Olkin measure of sampling adequacy
Value
A list with the following elements:
chisq

The chi-squared value for Bartlett’s test

The degrees of freedom for Bartlett’s test

pvalue

The significance level for Bartlett’s test

Rimage

The image correlation matrix

KMO

The overall KMO value

KMOvars

The KMO values for the variables

Author(s)
Brian P. O’Connor

IMAGE_FA

References
Bartlett, M. S. (1951). The effect of standardization on a chi square approximation in factor analysis, Biometrika, 38, 337-344.
Cerny, C. A., & Kaiser, H. F. (1977). A study of a measure of sampling adequacy for factoranalytic correlation matrices. Multivariate Behavioral Research, 12(1), 43-47.
Dziuban, C. D., & Shirkey, E. C. (1974). When is a correlation matrix appropriate for factor analysis? Psychological Bulletin, 81, 358-361.
Kaiser, H. F., & Rice, J. (1974). Little Jiffy, Mark IV. Educational and Psychological Measurement, 34, 111-117.
Examples
FACTORABILITY(data_RSE, corkind='pearson')

IMAGE_FA

image factor analysis

Description
image factor analysis
Usage
IMAGE_FA(data, corkind, nfactors, rotate, ppower, display)
Arguments
data

An all-numeric dataframe where the rows are cases & the columns are the variables, or a correlation matrix with ones on the diagonal.The function internally
determines whether the data are a correlation matrix.

corkind

nfactors

The number of factors to extract.

rotate

The factor rotation method. The options are: ’promax’, ’varimax’, and ’none’.

ppower

The power value to be used in a promax rotation (required only if rotate = ’promax’). Suggested value: 3

display

Display the results? The options are TRUE or FALSE.

Value
A list with the following elements:
eigenvalues

The eigenvalues

loadingsNOROT

The unrotated factor loadings

loadingsROT

The rotated factor loadings (for varimax rotation)

LOCALDEP

structure

The structure matrix (for promax rotation)

pattern

The pattern matrix (for promax rotation)

correls

The correlations between the factors (for promax rotation)

Author(s)
Brian P. O’Connor
Examples
IMAGE_FA(data_NEOPIR, corkind='pearson', nfactors=5, rotate='varimax', ppower=3, display=TRUE)

LOCALDEP

Provides the residual correlations after partialling the first component
out of a correlation matrix.

Description
Item response theory models are based on the assumption that the items display local independence.
The latent trait is presumed to be responsible for the associations between the items. Once the latent
trait is partialled out, the residual correlations between pairs of items should be negligible. Local
dependence exists when there is additional systematic covariance among the items. It can occur
when pairs of items have highly similar content or between sequentially presented items in a test.
Local dependence distorts IRT parameter estimates, it can artificially increase scale information, and
it distorts the latent trait, which becomes too heavily defined by the locally dependent items. The
LOCALDEP function partials out the first component (not the IRT latent trait) from a correlation
matrix. Examining the residual correlations is a preliminary, exploratory method of determining
whether local dependence exists. The function also displays the number of residual correlations
that are >= a range of values.
Usage
LOCALDEP(data, corkind, display)
Arguments
data

corkind

display

Display the results? The options are TRUE or FALSE.

Value
A list with the following elements:
correlations

The corrrelation matrix

residcor

The residualized corrrelation matrix

MAP

Author(s)
Brian P. O’Connor
Examples
# Residual correlations for the Rosenberg Self-Esteem Scale (RSE)
LOCALDEP(data_RSE, corkind = 'pearson', display=TRUE)

MAP

Velicer’s minimum average partial (MAP) test for number of factors

Description
Velicer’s minimum average partial (MAP) test for determining the number of factors focuses on
the common variance in a correlation matrix. It involves a complete principal components analysis followed by the examination of a series of matrices of partial correlations. Specifically, on the
first step, the first principal component is partialled out of the correlations between the variables of
interest, and the average squared coefficient in the off-diagonals of the resulting partial correlation
matrix is computed. On the second step, the first two principal components are partialled out of
the original correlation matrix and the average squared partial correlation is again computed. These
computations are conducted for k (the number of variables) minus one steps. The average squared
partial correlations from these steps are then lined up, and the number of components is determined
by the step number in the analyses that resulted in the lowest average squared partial correlation.
The average squared coefficient in the original correlation matrix is also computed, and if this coefficient happens to be lower than the lowest average squared partial correlation, then no components
should be extracted from the correlation matrix. Statistically, components are retained as long as
the variance in the correlation matrix represents systematic variance. Components are no longer
retained when there is proportionately more unsystematic variance than systematic variance (see
O’Connor, 2000, p. 397).
The MAP test is often more appropriate for factor analyses than it is for principal components
analyses. In Velicer’s words, "Component analysis has a variety of purposes. It can be used to find
a parsimonious description of the total variance of the variables involved; in this case, the [MAP
test] is not applicable. Principal component analysis is frequently used to express the variance
shared among variables in a set; that is, it is used as kind of a factor analysis" (1976, p. 321). "...
if component analysis is employed as an alternative to factor analysis or as a first-stage solution for
factor analysis, the stopping rule proposed here would seem the most appropriate." (1976, p. 326)’
Usage
MAP(data, corkind, display)
Arguments
data

corkind

display

Display the results? The options are TRUE or FALSE.

MAXLIKE_FA

Value
A list with the following elements:
eigenvalues

eigenvalues

avgsqrs

Velicer’s average squared correlations

nfMAP

number of factors according to the original (1976) MAP test

nfMAP4

number of factors according to the revised (2000) MAP test

Author(s)
Brian P. O’Connor
References
Velicer, W. F. (1976). Determining the number of components from the matrix of partial correlations. Psychometrika, 41, 321-327.
Velicer, W. F., Eaton, C. A., and Fava, J. L. (2000). Construct explication through factor or component analysis: A review and evaluation of alternative procedures for determining the number of
factors or components. In R. D. Goffin & E. Helmes, eds., Problems and solutions in human assessment (p.p. 41-71). Boston: Kluwer.
O’Connor, B. P. (2000). SPSS and SAS programs for determining the number of components
using parallel analysis and Velicer’s MAP test. Behavior Research Methods, Instrumentation, and
Computers, 32, 396-402.
Examples
# MAP test on the Harman correlation matrix
MAP(data_Harman, corkind='pearson', display=TRUE)
## Not run:
# MAP test on the Rosenberg Self-Esteem Scale (RSE) using Pearson correlations
MAP(data_RSE, corkind='pearson', display=TRUE)
# MAP test on the Rosenberg Self-Esteem Scale (RSE) using polychoric correlations
MAP(data_RSE, corkind='polychoric', display=TRUE)
# MAP test on the NEO-PI-R data
MAP(data_NEOPIR, display=TRUE)
## End(Not run)

MAXLIKE_FA

maximum likelihood factor analysis

Description
maximum likelihood factor analysis

MAXLIKE_FA

Usage
MAXLIKE_FA(data, corkind, nfactors, tolerml, iterml, rotate, ppower, display)

Arguments
data

An all-numeric dataframe where the rows are cases & the columns are the variables, or a correlation matrix with ones on the diagonal.The function internally
determines whether the data are a correlation matrix.

corkind

nfactors

The number of factors to extract.

tolerml

The tolerance level.

iterml

The maximum number of iterations.

rotate

The factor rotation method. The options are: ’promax’, ’varimax’, and ’none’.

ppower

The power value to be used in a promax rotation (required only if rotate = ’promax’). Suggested value: 3

display

Display the results? The options are TRUE or FALSE.

Value
A list with the following elements:
eigenvalues

The eigenvalues

loadingsNOROT

The unrotated factor loadings

loadingsROT

The rotated factor loadings (for varimax rotation)

structure

The structure matrix (for promax rotation)

pattern

The pattern matrix (for promax rotation)

correls

The correlations between the factors (for promax rotation)

Author(s)
Brian P. O’Connor

Examples
MAXLIKE_FA(data_RSE, corkind='pearson', nfactors = 2,
tolerml = .001, iterml = 50, rotate='promax', ppower=3, display=TRUE)

NEVALSGT1

The number of eigenvalues greater than 1 in a correlation matrix.

Description
This function returns the count of the number of eigenvalues greater than 1 in a correlation matrix.
This value is often referred to as the "Kaiser", "Kaiser-Guttman", or "Guttman-Kaiser" rule for
determining the number of components or factors in a correlation matrix.
The rationale is that a component with an eigenvalue of 1 accounts for as much variance as a
single variable. Extracting components with eigenvalues of 1 or less than 1 would defeat the usual
purpose of component and factor analyses. Furthermore, the reliability of a component will always
be nonnegative when its eigenvalue is greater than 1. This rule is the default retention criteria in
SPSS and SAS.
There are a number of problems with this rule of thumb. Monte Carlo investigations have found that
its accuracy rate is not acceptably high (Zwick & Velicer, 1986)). The rule was originally intended
to be an upper bound for the number of components to be retained, but it is most often used as the
criterion to determine the exact number of components or factors. Guttman’s original proof applies
only to the population correlation matrix and the sampling error that occurs in specific samples
results in the rule often overestimating the number of components. The rule is also considered
overly mechanical, e.g., a component with an eigenvalue of 1.01 achieves factor status whereas a
component with an eigenvalue of .999 does not.
This function is included in this package for curiosity and research purposes.
Usage
NEVALSGT1(data, corkind, display)
Arguments
data

corkind

display

Display the eigenvalues and the number that are greater than one? The options
are TRUE or FALSE.

Value
The number of eigenvalues greater than 1.
Author(s)
Brian P. O’Connor

PARALLEL

References
Kaiser, H. F. (1960). The application of electronic computer to factor analysis. Educational and
Psychological Measurement, 20, 141-151.
Guttman, L. (1954). Some necessary conditions for common factor analysis. Psychometrika, 19,
149-161.
Fabrigar, L. R., Wegener, D. T., MacCallum, R. C., & Strahan, E. J. (1999). Evaluating the use
of exploratory factor analysis in psychological research. Psychological Methods, 4, 272-299.
Hayton, J. C., Allen, D. G., Scarpello, V. (2004). Factor retention decisions in exploratory factor analysis: A tutorial on parallel analysis. Organizational Research Methods, 7, 191-205.
Zwick, W. R., & Velicer, W. F. (1986). Comparison of five rules for determining the number of
components to retain. Psychological Bulletin, 99, 432-442.
Examples
## Not run:
# test on the Harman correlation matrix
NEVALSGT1(data_Harman, corkind='pearson', display=TRUE)
# test on the Rosenberg Self-Esteem Scale (RSE) using Pearson correlations
NEVALSGT1(data_RSE, corkind='pearson', display=TRUE)
# test on the Rosenberg Self-Esteem Scale (RSE) using polychoric correlations
NEVALSGT1(data_RSE, corkind='polychoric', display=TRUE)
## End(Not run)

PARALLEL

parallel analysis of eigenvalues (random data only)

Description
This function generates eigenvalues for random data sets with specified numbers of variables and
cases. Typically, the eigenvalues derived from an actual data set are compared to the eigenvalues
derived from the random data. In Horn’s original description of this procedure, the mean eigenvalues from the random data served as the comparison baseline, whereas the more common current
practice is to use the eigenvalues that correspond to the desired percentile (typically the 95th) of
the distribution of random data eigenvalues. Factors or components are retained as long as the ith
eigenvalue from the actual data is greater than the ith eigenvalue from the random data. This function produces only random data eigenvalues and it does not take real data as input. See the rawpar
function in this package for parallel analyses that also involve real data.
Usage
PARALLEL(Nvars, Ncases, Ndatasets=100, extract='PCA', percentile='95',
corkind='pearson', display=TRUE)

PARALLEL

Arguments
Nvars

The number of variables.

Ncases

The number of cases.

Ndatasets

An integer indicating the # of random data sets for parallel analyses.

extract

The factor extraction method. The options are: ’PAF’ for principal axis / common factor analysis; ’PCA’ for principal components analysis. ’image’ for image analysis.

percentile

An integer indicating the percentile from the distribution of parallel analysis
random eigenvalues. Suggested value: 95

corkind

The kind of correlation matrix to be used for the random data. The options are
’pearson’, ’kendall’, and ’spearman’.

display

Display the results? The options are TRUE or FALSE.

Details
Although the PARALLEL function permits users to specify PCA or PAF or image as the factor
extraction method, users should be aware of an unresolved issue in the literature. Principal components eigenvalues are often used to determine the number of common factors. This is the default
in most statistical software packages, and it is the primary practice in the literature. It is also
the method used by many factor analysis experts, including Cattell, who often examined principal
components eigenvalues in his scree plots to determine the number of common factors. But others
believe that this common practice is wrong. Principal components eigenvalues are based on all
of the variance in correlation matrices, including both the variance that is shared among variables
and the variances that are unique to the variables. In contrast, principal axis eigenvalues are based
solely on the shared variance among the variables. The procedures are qualitatively different. Some
therefore claim that the eigenvalues from one extraction method should not be used to determine
the number of factors for another extraction method. The issue remains neglected and unsettled.
The PAF option in the extract argument for this function was included for research purposes. It is
otherwise probably best to use PCA as the extraction method for regular data analyses. The MAP
test (also in this package) is probably more suitable for determining the number of common factors.
Value
The random data eigenvalues
Author(s)
Brian P. O’Connor
References
Horn, J. L. (1965). A rationale and test for the number of factors in factor analysis. Psychometrika,
30, 179-185.
Zwick, W. R., & Velicer, W. F. (1986). Comparison of five rules for determining the number of
components to retain. Psychological Bulletin, 99, 432-442.
O’Connor, B. P. (2000). SPSS and SAS programs for determining the number of components
using parallel analysis and Velicer’s MAP test. Behavior Research Methods, Instrumentation, and
Computers, 32, 396-402.

PA_FA

Examples
## Not run:
PARALLEL(Nvars=15, Ncases=250, Ndatasets=100, extract='PCA', percentile=95,
corkind='pearson', display=TRUE)
## End(Not run)

PA_FA

principal axis (common) factor analysis

Description
principal axis (common) factor analysis with squared multiple correlations as the initial communality estimates
Usage
PA_FA(data, corkind, nfactors, iterpaf, rotate, ppower, display)
Arguments
data

An all-numeric dataframe where the rows are cases & the columns are the variables, or a correlation matrix with ones on the diagonal.The function internally
determines whether the data are a correlation matrix.

corkind

nfactors

The number of factors to extract.

iterpaf

The maximum number of iterations.

rotate

The factor rotation method. The options are: ’promax’, ’varimax’, and ’none’.

ppower

The power value to be used in a promax rotation (required only if rotate = ’promax’). Suggested value: 3

display

Display the results? The options are TRUE or FALSE.

Value
A list with the following elements:
eigenvalues

The eigenvalues

loadingsNOROT

The unrotated factor loadings

loadingsROT

The rotated factor loadings (for varimax rotation)

structure

The structure matrix (for promax rotation)

pattern

The pattern matrix (for promax rotation)

correls

The correlations between the factors (for promax rotation)

Author(s)
Brian P. O’Connor

PCA

Examples
PA_FA(data_RSE, corkind="pearson", nfactors = 2, iterpaf = 50,
rotate='promax', ppower=3, display="yes")

PCA

principal components analysis

Description
principal components analysis
Usage
PCA(data, corkind, nfactors, rotate, ppower, display)
Arguments
data

An all-numeric dataframe where the rows are cases & the columns are the variables, or a correlation matrix with ones on the diagonal.The function internally
determines whether the data are a correlation matrix.

corkind

nfactors

The number of components to extract.

rotate

The factor rotation method. The options are: ’promax’, ’varimax’, and ’none’.

ppower

The power value to be used in a promax rotation (required only if rotate = ’promax’). Suggested value: 3

display

Display the results? The options are TRUE or FALSE.

Value
A list with the following elements:
eigenvalues

The eigenvalues

loadingsNOROT

The unrotated factor loadings

loadingsROT

The rotated factor loadings (for varimax rotation)

structure

The structure matrix (for promax rotation)

pattern

The pattern matrix (for promax rotation)

correls

The correlations between the factors (for promax rotation)

Author(s)
Brian P. O’Connor
Examples
PCA(data_RSE, corkind='pearson', nfactors=2, rotate='promax', ppower=3, display=TRUE)

POLYCHORIC_R

polychoric correlation matrix

Description
This function produces a polychoric correlation matrix
Usage
POLYCHORIC_R(data, method)
Arguments
data

An all-numeric dataframe where the rows are cases & the columns are the variables. All values should be integers, as in the values for Likert rating scales.

method

(optional) The source package used to estimate the polychoric correlations: ’Revelle’ for the psych package (the default); ’Fox’ for the polycor package.

Details
Applying familiar factor analysis procedures to item-level data can produce misleading or uninterpretable results. Common factor analysis, maximum likelihood factor analysis, and principal
components analysis produce meaningful results only if the data are continuous and multivariate
normal. Item-level data almost never meet these requirements.
The correlation between any two items is affected by both their substantive (content-based) similarity and by the similarities of their statistical distributions. Items with similar distributions tend
to correlate more strongly with one another than do with items with dissimilar distributions. Easy
or commonly endorsed items tend to form factors that are distinct from difficult or less commonly
endorsed items, even when all of the items measure the same unidimensional latent variable. Itemlevel factor analyses using traditional methods are almost guaranteed to produce at least some factors that are based solely on item distribution similarity. The items may appear multidimensional
when in fact they are not. Conceptual interpretations of the nature of item-based factors will often
be erroneous.
A common, expert recommendation is that factor analyses of item-level data (e.g., for binary response options or for ordered response option categories) or should be conducted on matrices of
polychoric correlations. Factor analyses of polychoric correlation matrices are essentially factor
analyses of the relations among latent response variables that are assumed to underlie the data and
that are assumed to be continuous and normally distributed.
This is a cpu-intensive function. It is probably not necessary when there are > 8 item response
categories.
By default, the function uses the polychoric function from William Revelle’s’ psych package to
produce a full matrix of polychoric correlations. The function uses John Fox’s hetcor function from
the polycor package when requested or when the number of item response categories is > 8.
The hetcor function from the polycor package requires a dataframe as input. It also "computes a heterogenous correlation matrix, consisting of Pearson product-moment correlations between numeric
variables, polyserial correlations between numeric and ordinal variables, and polychoric correlations between ordinal variables." This means that polychoric correlations will not be computed if
a variable is numeric. A numeric variable must first be converted to an ordinal variable (ordered
factor), by the user, for the function to produce polychoric correlations for that variable.’

PROCRUSTES

Value
The polychoric correlation matrix
Author(s)
Brian P. O’Connor
Examples
## Not run:
# polychoric correlation matrix for the Rosenberg Self-Esteem Scale (RSE)
Rpoly <- POLYCHORIC_R(data_RSE, method = 'Fox')
Rpoly
## End(Not run)

PROCRUSTES

factor solution congruence

Description
This function conducts Procrustes rotations of a factor loading matrix to a target factor matrix, and it
computes the factor solution congruence and the root mean square residual (based on comparisons
of the entered factor loading matrix with the Procrustes-rotated matrix).
Usage
PROCRUSTES(loadings, target, type, display)
Arguments
loadings

The loading matrix that will be aligned with the target.

target

The target loading matrix.

type

The options are ’orthogonal’ or ’oblique’ rotation.

display

Display the results? The options are TRUE or FALSE.

Details
This function conducts Procrustes rotations of a factor loading matrix to a target factor matrix,
and it computes the factor solution congruence and the root mean square residual (based on comparisons of the entered factor loading matrix with the Procrustes-rotated matrix). The orthogonal
Procrustes rotation is based on Schonemann (1966; see also McCrae et al., 1996). The oblique
Procrustes rotation is based on Hurley and Cattell (1962). The factor solution congruence is the
Tucker-Wrigley-Neuhaus factor solution congruence coefficient (see Guadagnoli & Velicer, 1991;
and ten Berge, 1986, for reviews).

PROMAX

Value
A list with the following elements:
loadingsPROC

The Procrustes-rotated loadings

congruence

The factor solution congruence after factor Procrustes rotation

rmsr

The root mean square residual

residmat

The residual matrix after factor Procrustes rotation

Author(s)
Brian P. O’Connor
References
Guadagnoli, E., & Velicer, W. (1991). A comparison of pattern matching indices. Multivariate
Behavior Research, 26, 323-343.
Hurley, J. R., & Cattell, R. B. (1962). The Procrustes program: Producing direct rotation to test a
hypothesized factor structure. Behavioral Science, 7, 258-262.
McCrae, R. R., Zonderman, A. B., Costa, P. T. Jr., Bond, M. H., & Paunonen, S. V. (1996). Evaluating replicability of factors in the revised NEO personality inventory: Confirmatory factor analysis
versus Procrustes rotation. Journal of Personality and Social Psychology, 70, 552-566.
Schonemann, P. H. (1966). A generalized solution of the orthogonal Procrustes problem. Psychometrika, 31, 1-10.
ten Berge, J. M. F. (1986). Some relationships between descriptive comparisons of components
from different studies. Multivariate Behavioral Research, 21, 29-40.
Examples
# RSE data
loadings <- PCA(data_RSE[1:150,],
nfactors = 2, rotate='varimax', display=FALSE)
target
<- PCA(data_RSE[151:300,], nfactors = 2, rotate='varimax', display=FALSE)
PROCRUSTES(loadings$loadingsROT, target$loadingsROT, type = 'orthogonal', display=TRUE)

PROMAX

promax rotation

Description
promax rotation
Usage
PROMAX(loadings, ppower, display)

RAWPAR

Arguments
loadings

A loading matrix.

ppower

The exponent for the promax target matrix. ’ppower’ must be 1 or greater. ’4’
is a conventional value.

display

Display the results? The options are TRUE or FALSE.

Value
A list with the following elements:
structure

The structure matrix (for promax rotation)

pattern

The pattern matrix (for promax rotation)

correls

The correlations between the factors (for promax rotation)

Author(s)
Brian P. O’Connor
Examples
## Not run:
loadings <- PCA(data_NEOPIR, corkind='pearson', nfactors = 5, rotate='none', display=TRUE)
PROMAX(loadings, ppower = 3, display=TRUE)
## End(Not run)

RAWPAR

parallel analysis of eigenvalues (with real data as input)

Description
The parallel analysis procedure for deciding on the number of components or factors involves extracting eigenvalues from random data sets that parallel the actual data set with regard to the number
of cases and variables. For example, if the original data set consists of 305 observations for each
of 8 variables, then a series of random data matrices of this size (305 by 8) would be generated,
and eigenvalues would be computed for the correlation matrices for the original, real data and for
each of the random data sets. The eigenvalues derived from the actual data are then compared to
the eigenvalues derived from the random data. In Horn’s original description of this procedure, the
mean eigenvalues from the random data served as the comparison baseline, whereas the more common current practice is to use the eigenvalues that correspond to the desired percentile (typically
the 95th) of the distribution of random data eigenvalues. Factors or components are retained as long
as the ith eigenvalue from the actual data is greater than the ith eigenvalue from the random data.
Usage
RAWPAR(data, randtype, extract, Ndatasets, percentile,
corkind, corkindRAND, Ncases, display)

RAWPAR

Arguments
data

randtype

The kind of random data to be used in the parallel analysis: ’generated’ for
random normal data generation; ’permuted’ for perumatations of the raw data
matrix.

extract

The factor extraction method. The options are: ’PAF’ for principal axis / common factor analysis; ’PCA’ for principal components analysis. ’image’ for image analysis.

Ndatasets

An integer indicating the # of random data sets for parallel analyses.

percentile

An integer indicating the percentile from the distribution of parallel analysis
random eigenvalues to be used in determining the # of factors. Suggested value:
95

corkind

corkindRAND

The kind of correlation matrix to be used for the random data analyses. The
options are ’pearson’, ’kendall’, ’spearman’, and ’polychoric’. The default is
’pearson’.

Ncases

The number of cases upon which a correlation matrix is based. Required only if
data is a correlation matrix.

display

Display the results? The options are TRUE or FALSE.

Details
Although the RAWPAR function permits users to specify PCA or PAF as the factor extraction
method, users should be aware of an unresolved issue in the literature. Principal components eigenvalues are often used to determine the number of common factors. This is the default in most
statistical software packages, and it is the primary practice in the literature. It is also the method
used by many factor analysis experts, including Cattell, who often examined principal components
eigenvalues in his scree plots to determine the number of common factors. But others believe that
this common practice is wrong. Principal components eigenvalues are based on all of the variance
in correlation matrices, including both the variance that is shared among variables and the variances
that are unique to the variables. In contrast, principal axis eigenvalues are based solely on the shared
variance among the variables. The two procedures are qualitatively different. Some therefore claim
that the eigenvalues from one extraction method should not be used to determine the number of
factors for the other extraction method. The issue remains neglected and unsettled. The PAF option
in the extract argument for this function was included for research purposes. It is otherwise probably best to use PCA as the extraction method for regular data analyses. The MAP test (also in this
package) is probably more suitable for determining the number of common factors.
Polychoric correlations are time-consuming to compute. While polychoric correlations should probably be specified for the real data eigenvalues when data consists of item-level responses, polychoric
correlations should probably not be specified for the random data computations, even for item-level
data. The procedure would take much time and it is unnecessary. Polychoric correlations are estimates of what the Pearson correlations would be had the real data been continuous. For item-level
data, specify polychoric correlations for the real data eigenvalues (corkind=’polychoric’) and use
the default for the random data eigenvalues (corkindRAND=’pearson’). The option for using polychoric correlations for the random data computations (corkindRAND=’polychoric’) was provided
for research purposes.

ROOTFIT

Value
A list with:
eigenvalues

the eigenvalues for the real and random data

nfPA

the number of factors based on the parallel analysis

Author(s)
Brian P. O’Connor
References
Horn, J. L. (1965). A rationale and test for the number of factors in factor analysis. Psychometrika,
30, 179-185.
Zwick, W. R., & Velicer, W. F. (1986). Comparison of five rules for determining the number of
components to retain. Psychological Bulletin, 99, 432-442.
O’Connor, B. P. (2000). SPSS and SAS programs for determining the number of components
using parallel analysis and Velicer’s MAP test. Behavior Research Methods, Instrumentation, and
Computers, 32, 396-402.
Examples
## Not run:
# parallel analysis of the WISC data
RAWPAR(data_WISC, randtype='generated', extract='PCA', Ndatasets=100,
percentile=95, corkind='pearson', display=TRUE)
# parallel analysis of the Harman correlation matrix
RAWPAR(data_Harman, randtype='generated', extract='PCA', Ndatasets=100,
percentile=95, corkind='pearson', Ncases=305, display=TRUE)
# parallel analysis of the Rosenberg Self-Esteem Scale (RSE)
RAWPAR(data_RSE, randtype='permuted', extract='PCA', Ndatasets=100,
percentile=95, corkind='pearson', corkindRAND='pearson', display=TRUE)
# parallel analysis of the Rosenberg Self-Esteem Scale (RSE) using polychoric correlations
RAWPAR(data_RSE, randtype='generated', extract='PCA', Ndatasets=100,
percentile=95, corkind='polychoric', display=TRUE)
# parallel analysis of the NEO-PI-R data
RAWPAR(data_NEOPIR, randtype='generated', extract='PCA', Ndatasets=100,
percentile=95, corkind='pearson', Ncases=305, display=TRUE)
## End(Not run)

ROOTFIT

factor fit coefficients

Description
A variety of fit coefficients for the possible N-factor solutions in exploratory factor analysis

SALIENT

Usage
ROOTFIT(data, corkind, Ncases, extract, verbose)
Arguments
data

An all-numeric dataframe where the rows are cases & the columns are the variables, or a correlation matrix with ones on the diagonal.The function internally
determines whether the data are a correlation matrix.

corkind

Ncases

The number of cases upon which a correlation matrix is based. Required only if
data is a correlation matrix.

extract

The factor extraction method. The options are: ’PAF’ for principal axis / common factor analysis; ’PCA’ for principal components analysis. ’ML’ for maximum likelihood estimation.

verbose

Display descriptions of the fit coefficients? The options are ’TRUE’ (default) or
’FALSE’.

Value
A list with eigenvalues & fit coefficients.
Author(s)
Brian P. O’Connor
Examples
# RSE data
ROOTFIT(data_RSE, corkind='pearson', extract='ML')
ROOTFIT(data_RSE, corkind='pearson', extract='PCA', verbose = 'FALSE')
## Not run:
# NEO-PI-R data
ROOTFIT(data_NEOPIR, corkind='pearson', extract='ML')
ROOTFIT(data_NEOPIR, corkind='pearson', extract='PCA', verbose = 'FALSE')
## End(Not run)

SALIENT

The salient loadings criterion for determing the number of factors.

Description
This is a procedure for determining the number of factors recommended by Gorsuch. Factors are
retained when they consist of a specified minimum number (or more) variables that have a specified
minimum (or higher) loading value.
Usage
SALIENT(data, salvalue, numsals, corkind, display)

SESCREE

Arguments
data

salvalue

The loading value that is considered salient. Default = .40

numsals

The required number of salient loadings for a factor. Default = 3

corkind

display

Display the loadings? The options are TRUE or FALSE.

Value
The number of factors according to the salient loadings criterion.
Author(s)
Brian P. O’Connor
References
Gorsuch, R. L. (1997a). Exploratory factor analysis: Its role in item analysis. Journal of Personality
Assessment, 68, 532-560.
Boyd, K. C. (2011). Factor analysis. In M. Stausberg & S. Engler (Eds.), The Routledge Handbook of Research Methods in the Study of Religion (pp. 204-216). New York: Routledge.
Examples
# test on the Harman correlation matrix
SALIENT(data_Harman, salvalue=.4, numsals=3, corkind='pearson', display=TRUE)
## Not run:
# test on the Rosenberg Self-Esteem Scale (RSE) using Pearson correlations
SALIENT(data_RSE, salvalue=.4, numsals=3, corkind='pearson', display=TRUE)
# test on the Rosenberg Self-Esteem Scale (RSE) using polychoric correlations
SALIENT(data_RSE, salvalue=.4, numsals=3, corkind='polychoric', display=TRUE)
## End(Not run)

SESCREE

Standard Error Scree test for the number of components.

Description
This is a linear regression operationalization of the scree test for determining the number of components. The results are purportedly identical to those from the visual scree test. The test is based
on the standard error of estimate values that are computed for the set of eigenvalues in a scree plot.
The number of components to retain is the point where the standard error exceeds 1/m, where m is
the numbers of variables.

VARIMAX

Usage
SESCREE(data, corkind, display)
Arguments
data

corkind

display

Display the eigenvalues, slopes, SE estimates, & the number of components?
The options are TRUE or FALSE.

Value
The number of components according to the Standard Error Scree test.
Author(s)
Brian P. O’Connor
References
Zoski, K., & Jurs, S. (1996). An objective counterpart to the visual scree test for factor analysis:
the standard error scree test. Educational and Psychological Measurement, 56(3), 443-451.
Examples
# test on the Harman correlation matrix
SESCREE(data_Harman, corkind='pearson', display=TRUE)
## Not run:
# test on the Rosenberg Self-Esteem Scale (RSE) using Pearson correlations
SESCREE(data_RSE, corkind='pearson', display=TRUE)
# test on the Rosenberg Self-Esteem Scale (RSE) using polychoric correlations
SESCREE(data_RSE, corkind='polychoric', display=TRUE)
## End(Not run)

VARIMAX

varimax rotation

Description
varimax rotation
Usage
VARIMAX(loadings, display)

VARIMAX

Arguments
loadings

A loading matrix.

display

Display the results? The options are TRUE or FALSE.

Value
The varimax-rotated loadings
Author(s)
Brian P. O’Connor
Examples
## Not run:
loadings <- PCA(data_NEOPIR, corkind='pearson', nfactors = 5, rotate='none', display=TRUE)
VARIMAX(loadings, display=TRUE)
## End(Not run)

Index
CONGRUENCE, 2
data_Harman, 3
data_NEOPIR, 4
data_RSE, 5
data_WISC, 5
EXTENSION_FA, 6
FACTORABILITY, 9
IMAGE_FA, 10
LOCALDEP, 11
MAP, 12
MAXLIKE_FA, 13
NEVALSGT1, 15
PA_FA, 18
PARALLEL, 16
paramap-package, 2
PCA, 19
POLYCHORIC_R, 20
PROCRUSTES, 21
PROMAX, 22
RAWPAR, 23
ROOTFIT, 25
SALIENT, 26
SESCREE, 27
VARIMAX, 28

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