Gradient Projection Algorithms for Factor Rotation
00.GPArotation.RdGPA 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 |
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