EFA: Exploratory factor analysis
In EFA.dimensions: Exploratory Factor Analysis Functions for Assessing Dimensionality
EFA R Documentation
Exploratory factor analysis
Description
Exploratory factor analysis with multiple options for factor extraction and rotation
Usage
EFA(data, Nfactors=NULL, extraction = 'paf', rotation='promax', corkind='pearson',
Ncases=NULL, iterpaf=100, ppower = 3,
GPA_options = list(delta = .01,
epsilon = .00001,
normalize = FALSE,
maxit = 1000,
randomStarts = 50),
verbose=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.
Nfactors
The number of factors to extract. If not specified, then the EMPKC procedure
will be used to determine the number of factors.
extraction
The factor extraction method for the analysis.
The options are 'paf' (the default), 'alpha', 'fullinfo', 'gls', 'image', 'minres',
'ml', 'ols', 'uls', and 'wls'.
rotation
The factor rotation method for the analysis.
The orthogonal rotation options are:
'varimax' (the default), 'bentlerT', 'entropy',
'equamax', 'geominT', 'quartimax', and 'none'.
The oblique rotation options are: 'promax' (the default), 'bentlerQ',
'geominQ', 'oblimin', 'oblimax', 'quartimin',
'simplimax', and 'none'.
corkind
The kind of correlation matrix to be used if data is not a correlation matrix.
The options are 'pearson', 'kendall', 'spearman', 'gamma', and 'polychoric'. Required
only if the entered data is not a correlation matrix.
Ncases
The number of cases. Required only if data is a correlation matrix.
iterpaf
The maximum number of iterations for paf.
ppower
The power value to be used in a promax rotation (required only if
rotation = 'promax'). Suggested value: 3
GPA_options
(optional) A list with options for when a GPArotation is used.
See the Details section below.
verbose
Should detailed results be displayed in console? TRUE (default) or FALSE
Details
The factor extraction computations for the following methods are conducted
using the psych package (Revelle, 2026): 'alpha', 'gls', 'minres', 'ols', 'uls', and 'wls'.
The factor extraction computations for 'fullinfo' are conducted
using the mirt package (Chalmers, 2012). Full-information methods are considered more
appropriate for item-level data than other factor extraction methods (Wirth & Edwards, 2007).
The factor rotation computations for the following methods are conducted
using the GPArotation package (Bernaards & Jennrich, 2005, 2026): 'bentlerQ',
'bentlerT', 'entropy', 'geominQ', 'geominT', 'oblimax', 'oblimin', 'quartimax',
'quartimin', and 'simplimax'.
For factor extraction (see Mulaik, 2010, for a review):
alpha is for an alpha factor analysis
gls is for a generalized weighted least squares factor analysis
image is for image factor analysis
minres is for a minimum residual factor analysis
ml is for maximum likelihood factor analysis
ols is for an ordinary least squares factor analysis
paf is for principal axis factor analysis
uls is for an unweighted least squares factor analysis
wls is for a weighted least squares factor analysis
For factor rotation (see Jennrich, 2018, for a review):
bentlerQ is an oblique rotation based on Bentler's invariant
pattern simplicity criterion.
bentlerT is an orthogonal rotation based on Bentler's
invariant pattern simplicity criterion
entropy entropy is an orthogonal factor rotation
that pushes factor loadings to be either
strongly dominant (close to 1.0) or cleanly absent (close to 0.0),
reducing the overall informational "noise" of the matrix.
equamax is an orthogonal rotation designed as a
mathematical compromise between varimax and quartimax.
geominQ is an oblique factor rotation method that
find a clean "simple structure"
even when the data contains highly complex variables (variables that
naturally load on multiple factors).
geominT is an orthogonal factor rotation that
combines the strict geometric constraint of uncorrelated factors with
Brownes flexible, product-based geomin complexity criterion.
oblimax is an oblique factor rotation method that maximizes
the number of very high and very low (near-zero) factor loadings.
oblimin is an oblique factor rotation that allows
factors to correlate freely to achieve the simplest possible loading structure.
promax is a two-stage oblique factor rotation method.
quartimax is an orthogonal factor rotation
designed to spread the explained variance evenly across the factors,
actively preventing a single general factor from dominating.
simplimax is an oblique factor rotation that attempts to
recover complex or messy "simple structures"
where traditional continuous methods like oblimin or geomin fail.
quartimin is an oblique factor rotation that focuses
on simplifying the rows of the loading matrix.
varimax is an orthogonal
factor rotation that maximizes the variance of the squared factor loadings within
each individual factor column.
For GPA_options:
A list with options for when a GPArotation is used. The list
elements are 'delta', 'epsilon', 'normalize', 'maxit', and 'randomStarts'.
delta is a small positive number that is added to the squared
factor loadings in the (geomin) optimization denominator to prevent division by
zero and control the sharpness of the penalty near zero.
(Note: In MPLus, delta is called "epsilon".)
epsilon is the numeric rotational convergence tolerance threshold.
It controls how precise or strict the stopping rule is for the rotation algorithm.
The gradient projection algorithm stops optimizing and assumes the
rotation has converged when the norm of the gradient falls below this value.
normalize specifies whether and how the loadings
matrix is normalized prior to rotation and denormalized after rotation.
The possibilities are FALSE (the default) for no normalization, or
TRUE, for the Kaiser/Horst normalization.
maxit is the maximum number of iterations.
randomStarts is an integer that specifies the number of
random starting configurations used to optimize the factor rotation.
It helps avoid getting trapped in local minima during factor rotation
optimization. It runs the gradient projection algorithm multiple timeseach
using a different random orthogonal or oblique starting rotation
matrix (Tmat) generated via QR decomposition.
It returns the rotated factor loadings matrix and results corresponding
to the run with the lowest objective criterion value among all attempts.
Run one of the following commands for more detailed descriptions of the
above extraction and rotation methods:
RShowDoc("EFA_BIFACTOR_vignettes", package = "EFA.dimensions")
vignette("EFA_BIFACTOR_vignettes")
Value
A list with the following elements:
loadingsNOROT
The unrotated factor loadings
loadingsROT
The rotated factor loadings
pattern
The pattern matrix
structure
The structure matrix
phi
The correlations between the factors
varexplNOROT1
The initial eigenvalues and total variance explained
varexplNOROT2
The eigenvalues and total variance explained after factor extraction (no rotation)
varexplROT
The rotation sums of squared loadings and total variance explained for the rotated loadings
cormat_reprod
The reproduced correlation matrix, based on the rotated loadings
fit_coefs
Model fit coefficients
chisqMODEL
The model chi squared
dfMODEL
The model degrees of freedom
pvalue
The model p-value
chisqNULL
The null model chi squared
dfNULL
The null model degrees of freedom
communalities
The unrotated factor solution communalities
uniquenesses
The unrotated factor solution uniquenesses
Author(s)
Brian P. O'Connor
References
Bernaards, C. A., & Jennrich, R. I. (2005). Gradient Projection Algorithms
and Software for Arbitrary Rotation Criteria in Factor Analysis.
Educational and Psychological Measurement, 65(5), 676-696.
https://doi.org/10.1177/0013164404272507
Bernaards, C. A., & Jennrich, R. I. (2026). GPArotation: Gradient Projection Factor Rotation.
R package version 2026.4-1, https://CRAN.R-project.org/package=GPArotation
Chalmers, R. P. (2012). mirt: A Multidimensional Item Response Theory Package for the R Environment.
Journal of Statistical Software, 48(6), 129. doi:10.18637/jss.v048.i06.
Jennrich, R. I. (2018). Rotation. In P. Irwing, T. Booth, & D. J. Hughes (Eds.), The Wiley handbook
of psychometric testing: A multidisciplinary reference on survey, scale and test development (pp. 279304).
Wiley Blackwell. https://doi.org/10.1002/9781118489772.ch10
Mulaik, S. A. (2010). Foundations of factor analysis (2nd ed.). Boca Raton, FL: Chapman
and Hall/CRC Press, Taylor & Francis Group.
Revelle, W. (2026). psych: Procedures for Psychological, Psychometric, and Personality Research.
R package version 2.6.5, https://CRAN.R-project.org/package=psych
Sellbom, M., & Tellegen, A. (2019). Factor analysis in psychological assessment research:
Common pitfalls and recommendations.
Psychological Assessment, 31(12), 1428-1441. https://doi.org/10.1037/pas0000623
Watts, A. L., Greene, A. L., Ringwald, W., Forbes, M. K., Brandes, C. M., Levin-Aspenson,
H. F., & Delawalla, C. (2023). Factor analysis in personality disorders research: Modern issues
and illustrations of practical recommendations.
Personality Disorders: Theory, Research, and Treatment, 14(1), 105-117.
https://doi.org/10.1037/per0000581
Wirth, R. J., & Edwards, M. C. (2007). Item factor analysis: current approaches and future directions.
Psychological methods, 12(1), 58-79. https://doi.org/10.1037/1082-989X.12.1.58
Examples
# the Harman (1967) correlation matrix
EFA(data=data_Harman, extraction = 'paf', Nfactors=2, Ncases=305, rotation='oblimin')
# Rosenberg Self-Esteem scale items, using ml extraction & polychoric correlations
EFA(data=data_RSE, Nfactors=2, extraction = 'ml', corkind='polychoric')
# NEO-PI-R scales
EFA(data=data_NEOPIR, Nfactors=5, extraction='minres', rotation='promax')
EFA.dimensions documentation built on Sept. 14, 2026, 9:08 a.m.
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| EFA | R Documentation |
Exploratory factor analysis
Description
Exploratory factor analysis with multiple options for factor extraction and rotation
Usage
EFA(data, Nfactors=NULL, extraction = 'paf', rotation='promax', corkind='pearson',
Ncases=NULL, iterpaf=100, ppower = 3,
GPA_options = list(delta = .01,
epsilon = .00001,
normalize = FALSE,
maxit = 1000,
randomStarts = 50),
verbose=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. |
Nfactors |
The number of factors to extract. If not specified, then the EMPKC procedure will be used to determine the number of factors. |
extraction |
The factor extraction method for the analysis. The options are 'paf' (the default), 'alpha', 'fullinfo', 'gls', 'image', 'minres', 'ml', 'ols', 'uls', and 'wls'. |
rotation |
The factor rotation method for the analysis.
|
corkind |
The kind of correlation matrix to be used if data is not a correlation matrix. The options are 'pearson', 'kendall', 'spearman', 'gamma', and 'polychoric'. Required only if the entered data is not a correlation matrix. |
Ncases |
The number of cases. Required only if data is a correlation matrix. |
iterpaf |
The maximum number of iterations for paf. |
ppower |
The power value to be used in a promax rotation (required only if rotation = 'promax'). Suggested value: 3 |
GPA_options |
(optional) A list with options for when a GPArotation is used. See the Details section below. |
verbose |
Should detailed results be displayed in console? TRUE (default) or FALSE |
Details
The factor extraction computations for the following methods are conducted using the psych package (Revelle, 2026): 'alpha', 'gls', 'minres', 'ols', 'uls', and 'wls'.
The factor extraction computations for 'fullinfo' are conducted using the mirt package (Chalmers, 2012). Full-information methods are considered more appropriate for item-level data than other factor extraction methods (Wirth & Edwards, 2007).
The factor rotation computations for the following methods are conducted using the GPArotation package (Bernaards & Jennrich, 2005, 2026): 'bentlerQ', 'bentlerT', 'entropy', 'geominQ', 'geominT', 'oblimax', 'oblimin', 'quartimax', 'quartimin', and 'simplimax'.
For factor extraction (see Mulaik, 2010, for a review):
alpha is for an alpha factor analysis
gls is for a generalized weighted least squares factor analysis
image is for image factor analysis
minres is for a minimum residual factor analysis
ml is for maximum likelihood factor analysis
ols is for an ordinary least squares factor analysis
paf is for principal axis factor analysis
uls is for an unweighted least squares factor analysis
wls is for a weighted least squares factor analysis
For factor rotation (see Jennrich, 2018, for a review):
bentlerQ is an oblique rotation based on Bentler's invariant pattern simplicity criterion.
bentlerT is an orthogonal rotation based on Bentler's invariant pattern simplicity criterion
entropy entropy is an orthogonal factor rotation that pushes factor loadings to be either strongly dominant (close to 1.0) or cleanly absent (close to 0.0), reducing the overall informational "noise" of the matrix.
equamax is an orthogonal rotation designed as a mathematical compromise between varimax and quartimax.
geominQ is an oblique factor rotation method that find a clean "simple structure" even when the data contains highly complex variables (variables that naturally load on multiple factors).
geominT is an orthogonal factor rotation that combines the strict geometric constraint of uncorrelated factors with Brownes flexible, product-based geomin complexity criterion.
oblimax is an oblique factor rotation method that maximizes the number of very high and very low (near-zero) factor loadings.
oblimin is an oblique factor rotation that allows factors to correlate freely to achieve the simplest possible loading structure.
promax is a two-stage oblique factor rotation method.
quartimax is an orthogonal factor rotation designed to spread the explained variance evenly across the factors, actively preventing a single general factor from dominating.
simplimax is an oblique factor rotation that attempts to recover complex or messy "simple structures" where traditional continuous methods like oblimin or geomin fail.
quartimin is an oblique factor rotation that focuses on simplifying the rows of the loading matrix.
varimax is an orthogonal factor rotation that maximizes the variance of the squared factor loadings within each individual factor column.
For GPA_options:
A list with options for when a GPArotation is used. The list elements are 'delta', 'epsilon', 'normalize', 'maxit', and 'randomStarts'.
delta is a small positive number that is added to the squared factor loadings in the (geomin) optimization denominator to prevent division by zero and control the sharpness of the penalty near zero. (Note: In MPLus, delta is called "epsilon".)
epsilon is the numeric rotational convergence tolerance threshold. It controls how precise or strict the stopping rule is for the rotation algorithm. The gradient projection algorithm stops optimizing and assumes the rotation has converged when the norm of the gradient falls below this value.
normalize specifies whether and how the loadings matrix is normalized prior to rotation and denormalized after rotation. The possibilities are FALSE (the default) for no normalization, or TRUE, for the Kaiser/Horst normalization.
maxit is the maximum number of iterations.
randomStarts is an integer that specifies the number of random starting configurations used to optimize the factor rotation. It helps avoid getting trapped in local minima during factor rotation optimization. It runs the gradient projection algorithm multiple timeseach using a different random orthogonal or oblique starting rotation matrix (Tmat) generated via QR decomposition. It returns the rotated factor loadings matrix and results corresponding to the run with the lowest objective criterion value among all attempts.
Run one of the following commands for more detailed descriptions of the above extraction and rotation methods:
RShowDoc("EFA_BIFACTOR_vignettes", package = "EFA.dimensions")
vignette("EFA_BIFACTOR_vignettes")
Value
A list with the following elements:
loadingsNOROT |
The unrotated factor loadings |
loadingsROT |
The rotated factor loadings |
pattern |
The pattern matrix |
structure |
The structure matrix |
phi |
The correlations between the factors |
varexplNOROT1 |
The initial eigenvalues and total variance explained |
varexplNOROT2 |
The eigenvalues and total variance explained after factor extraction (no rotation) |
varexplROT |
The rotation sums of squared loadings and total variance explained for the rotated loadings |
cormat_reprod |
The reproduced correlation matrix, based on the rotated loadings |
fit_coefs |
Model fit coefficients |
chisqMODEL |
The model chi squared |
dfMODEL |
The model degrees of freedom |
pvalue |
The model p-value |
chisqNULL |
The null model chi squared |
dfNULL |
The null model degrees of freedom |
communalities |
The unrotated factor solution communalities |
uniquenesses |
The unrotated factor solution uniquenesses |
Author(s)
Brian P. O'Connor
References
Bernaards, C. A., & Jennrich, R. I. (2005). Gradient Projection Algorithms
and Software for Arbitrary Rotation Criteria in Factor Analysis.
Educational and Psychological Measurement, 65(5), 676-696.
https://doi.org/10.1177/0013164404272507
Bernaards, C. A., & Jennrich, R. I. (2026). GPArotation: Gradient Projection Factor Rotation.
R package version 2026.4-1, https://CRAN.R-project.org/package=GPArotation
Chalmers, R. P. (2012). mirt: A Multidimensional Item Response Theory Package for the R Environment.
Journal of Statistical Software, 48(6), 129. doi:10.18637/jss.v048.i06.
Jennrich, R. I. (2018). Rotation. In P. Irwing, T. Booth, & D. J. Hughes (Eds.), The Wiley handbook
of psychometric testing: A multidisciplinary reference on survey, scale and test development (pp. 279304).
Wiley Blackwell. https://doi.org/10.1002/9781118489772.ch10
Mulaik, S. A. (2010). Foundations of factor analysis (2nd ed.). Boca Raton, FL: Chapman
and Hall/CRC Press, Taylor & Francis Group.
Revelle, W. (2026). psych: Procedures for Psychological, Psychometric, and Personality Research.
R package version 2.6.5, https://CRAN.R-project.org/package=psych
Sellbom, M., & Tellegen, A. (2019). Factor analysis in psychological assessment research:
Common pitfalls and recommendations.
Psychological Assessment, 31(12), 1428-1441. https://doi.org/10.1037/pas0000623
Watts, A. L., Greene, A. L., Ringwald, W., Forbes, M. K., Brandes, C. M., Levin-Aspenson,
H. F., & Delawalla, C. (2023). Factor analysis in personality disorders research: Modern issues
and illustrations of practical recommendations.
Personality Disorders: Theory, Research, and Treatment, 14(1), 105-117.
https://doi.org/10.1037/per0000581
Wirth, R. J., & Edwards, M. C. (2007). Item factor analysis: current approaches and future directions.
Psychological methods, 12(1), 58-79. https://doi.org/10.1037/1082-989X.12.1.58
Examples
# the Harman (1967) correlation matrix
EFA(data=data_Harman, extraction = 'paf', Nfactors=2, Ncases=305, rotation='oblimin')
# Rosenberg Self-Esteem scale items, using ml extraction & polychoric correlations
EFA(data=data_RSE, Nfactors=2, extraction = 'ml', corkind='polychoric')
# NEO-PI-R scales
EFA(data=data_NEOPIR, Nfactors=5, extraction='minres', rotation='promax')
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