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). Theory of the GP algorithm is described in Jennrich (2001, 2002).

Additionally, two rotation methods are provided that do not rely on GP (eiv and echelon).

Four vignettes are provided covering general usage, local minima diagnostics, bifactor rotation and reliability, and derivative-free gradient projection. Access them via 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

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]

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
.GPA_RS_engine	Internal random-start engine (not exported)

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

print.GPArotation	Print results (S3 method)
summary.GPArotation	Summary of results (S3 method)
.sortGPALoadings	Sort and sign-correct factors (not exported)
Random.Start	Random starting matrix for factor rotation
NormalizingWeight	Normalizing weights utility (not exported)
GPForth.lp	Single-start L^p orthogonal rotation
GPFoblq.lp	Single-start L^p oblique rotation

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

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
box20	Thurstone's 20 box variables (deprecated, use box26)
CCAI	CCAI Climate-Friendly Purchasing Choices domain; correlation matrix, pattern matrix, and factor intercorrelations

Vignettes

GPA1guide	Gradient Projection Factor Rotation (main guide)
GPA2local	Assessing Local Minima in Factor Rotation
GPA3bifactor	Bifactor Rotation and Reliability Coefficients

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. doi: 10.1177/0013164404272507

Theory of gradient projection algorithms:

Jennrich, R.I. (2001). A simple general procedure for orthogonal rotation. Psychometrika, 66, 289–306. doi: 10.1007/BF02294840

Jennrich, R.I. (2002). A simple general method for oblique rotation. Psychometrika, 67, 7–19. 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. doi: 10.1080/00273171.2016.1215898

See Also

GPFRSorth, GPFRSoblq, rotations, vgQ browseVignettes("GPArotation")

GPArotation documentation built on April 29, 2026, 9:08 a.m.