00.GPArotation: Gradient Projection Algorithms for Factor Rotation

In GPArotation: Gradient Projection Factor Rotation

Gradient Projection Algorithms for Factor Rotation

Description

GPA Rotation for Factor Analysis

The GPArotation package contains functions for the rotation of factor loadings matrices. The functions implement Gradient Projection (GP) algorithms for orthogonal and oblique rotation. Additionally, a number of rotation criteria are provided. The GP algorithms minimize the rotation criterion function and provide the corresponding rotation matrix. For oblique rotation, the covariance/correlation matrix of the factors is also provided. The rotation criteria implemented in this package are described in Bernaards and Jennrich (2005) in addition to a number of others. Theory of the GP algorithm is described in Jennrich (2001, 2002).

Vignettes are provided covering general usage, local minima diagnostics, bifactor rotation and reliability, and a visual walkthrough of the gradient projection algorithm. Access them via browseVignettes("GPArotation").

Package:	GPArotation
Depends:	R (>= 3.5.0)
License:	GPL Version 2.

Index of functions:

Rotations using gradient projection algorithms

oblimin	Oblimin rotation
quartimin	Quartimin rotation
targetT	Orthogonal target rotation
targetQ	Oblique target rotation
pstT	Orthogonal partially specified target rotation
pstQ	Oblique partially specified target rotation
oblimax	Oblimax rotation
entropy	Minimum entropy rotation
quartimax	Quartimax rotation
Varimax	Varimax rotation
simplimax	Simplimax rotation
bentlerT	Orthogonal Bentler invariant pattern simplicity rotation
bentlerQ	Oblique Bentler invariant pattern simplicity rotation
tandemI	Tandem criteria principle I rotation
tandemII	Tandem criteria principle II rotation
geominT	Orthogonal Geomin rotation
geominQ	Oblique Geomin rotation
bigeominT	Orthogonal Bi-Geomin rotation
bigeominQ	Oblique Bi-Geomin rotation
cfT	Orthogonal Crawford-Ferguson family rotation
cfQ	Oblique Crawford-Ferguson family rotation
equamax	Equamax rotation
parsimax	Parsimax rotation
infomaxT	Orthogonal Infomax rotation
infomaxQ	Oblique Infomax rotation
mccammon	McCammon minimum entropy ratio rotation
varimin	Varimin rotation
bifactorT	Orthogonal bifactor rotation
bifactorQ	Oblique bifactor rotation
lpT	Orthogonal L^p rotation
lpQ	Oblique L^p rotation

Core gradient projection algorithms

GPForth	Orthogonal rotation function
GPFoblq	Oblique rotation function

Random-start wrappers and internal engine

GPFRSorth	Random-start wrapper for orthogonal rotation
GPFRSoblq	Random-start wrapper for oblique rotation

2D orthogonal trajectory plot

plot2fOrthComparison	2D trajectory plot for 2 factor orthogonal rotation

S3 methods

print.GPArotation	Print rotated solution with simple structure measures
summary.GPArotation	Summary with pattern, structure, and diagnostics
plot.GPArotation	Plot: salient, pairs, profile, vector, target, heatmap, trajectory, diagnostics, residuals
residuals.GPArotation	Observed minus modeled residual of correlation matrix
update	update [The R Stats Package]

Data sets

Harman8	Harman's 8 physical variables; centroid loadings
NetherlandsTV	Wansbeek and Meijer Netherlands TV viewership; correlation matrix
box26	Thurstone's 26 box variables; unrotated factor loadings
CCAI	CCAI Climate-Friendly Purchasing Choices domain; correlation matrix, pattern matrix, and factor intercorrelations
GriffithMulaik	Griffith and Mulaik interpersonal personality traits; 24-variable correlation matrix

Other rotations not using gradient projection algorithms

eiv	Errors-in-variables rotation
echelon	Echelon rotation
varimax	varimax [The R Stats Package]
promax	promax [The R Stats Package]

Legacy gradient projection algorithms (code unchanged since 2008)

GPForth.legacy	Orthogonal rotation, original implementation (not exported)
GPFoblq.legacy	Oblique rotation, original implementation (not exported)

Utility functions

Random.Start	Random starting matrix for factor rotation
GPForth.lp	Single-start L^p orthogonal rotation
GPFoblq.lp	Single-start L^p oblique rotation

Utility functions (not exported)

.GPA_RS_engine	Internal random-start engine
.sortGPALoadings	Sort and sign-correct factors
NormalizingWeight	Normalizing weights utility
calc_AUC	AUC simple structure measure
calc_FSI	Factor Simplicity Index
calc_simplicity	Hoffman, Gini, Bentler simplicity indices
calc_hyperplane	Hyperplane count
calc_fitstats	ML fit statistics: RMSEA, SRMR
plot_trajectory	Helper function for plot2fOrthComparison
plot_gpa_diagnostics	Helper function for plot2fOrthComparison
plot_algorithm_comparison	Helper function for plot2fOrthComparison
.reconstruct_trajectory	Helper function for plot2fOrthComparison
plotRotationLandscape	Helper function for plot2fOrthComparison
plot_landscape_trajectory	Helper function for plot2fOrthComparison

Value, gradient, rotation criterion functions (not exported)

vgQ.oblimin	Oblimin
vgQ.quartimin	Quartimin
vgQ.target	Target
vgQ.pst	Partially specified target
vgQ.oblimax	Oblimax
vgQ.entropy	Minimum entropy
vgQ.quartimax	Quartimax
vgQ.varimax	Varimax
vgQ.simplimax	Simplimax
vgQ.bentler	Bentler invariant pattern simplicity
vgQ.tandemI	Tandem criteria principle I
vgQ.tandemII	Tandem criteria principle II
vgQ.geomin	Geomin
vgQ.bigeomin	Bi-Geomin
vgQ.cf	Crawford-Ferguson family
vgQ.infomax	Infomax
vgQ.mccammon	McCammon minimum entropy ratio
vgQ.varimin	Varimin
vgQ.bifactor	Bifactor
vgQ.lp.wls	Weighted least squares for L^p rotation

Vignettes

GPA1guide	Gradient Projection Factor Rotation (main guide)
GPA2local	Assessing Local Minima in Factor Rotation
GPA3bifactor	Bifactor Rotation and Reliability Coefficients
GPA4fitstats	Factor Model Fit and Simple Structure Diagnostics

Author(s)

Coen A. Bernaards and Robert I. Jennrich with some R modifications by Paul Gilbert.

References

The software reference is:

Bernaards, C.A. and Jennrich, R.I. (2005). Gradient projection algorithms and software for arbitrary rotation criteria in factor analysis. Educational and Psychological Measurement, 65, 676–696. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1177/0013164404272507")}

Theory of gradient projection algorithms:

Jennrich, R.I. (2001). A simple general procedure for orthogonal rotation. Psychometrika, 66, 289–306. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/BF02294840")}

Jennrich, R.I. (2002). A simple general method for oblique rotation. Psychometrika, 67, 7–19. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/BF02294706")}

A clear and accessible introduction to gradient projection algorithms for factor rotation is provided in:

Mansolf, M. and Reise, S.P. (2016). Exploratory bifactor analysis: The Schmid-Leiman orthogonalization and Jennrich-Bentler analytic rotations. Multivariate Behavioral Research, 51(5), 698–717. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1080/00273171.2016.1215898")}

Barzilai-Borwein step size:

Barzilai, J. and Borwein, J.M. (1988). Two-point step size gradient methods. IMA Journal of Numerical Analysis, 8, 141–148. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1093/imanum/8.1.141")}

Cayley transform retraction:

Wen, Z. and Yin, W. (2013). A feasible method for optimization with orthogonality constraints. Mathematical Programming, 142, 397–434. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s10107-012-0584-1")}

Non-monotone line search:

Grippo, L., Lampariello, F., and Lucidi, S. (1986). A nonmonotone line search technique for Newton's method. SIAM Journal on Numerical Analysis, 23(4), 707–716. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1137/0723046")}

Zhang, H. and Hager, W.W. (2004). A nonmonotone line search technique and its application to unconstrained optimization. SIAM Journal on Optimization, 14(4), 1043–1056. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1137/S1052623403428208")}

Simple structure measures:

Liu, X., Wallin, G., Chen, Y., and Moustaki, I. (2023). Rotation to sparse loadings using L^p losses and related inference problems. Psychometrika, 88(2), 527–553. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/s11336-023-09911-y")}

Lorenzo-Seva, U. (2003). A factor simplicity index. Psychometrika, 68(1), 49–60. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1007/BF02296652")}

See Also

GPFRSorth, GPFRSoblq, rotations, vgQ, simple_structure, plot.GPArotation browseVignettes("GPArotation")

GPArotation documentation built on June 18, 2026, 9:06 a.m.