# rotations: Rotations In GPArotation: Gradient Projection Factor Rotation

 rotations R Documentation

## Rotations

### Usage

``````    oblimin(A, Tmat=diag(ncol(A)), gam=0, normalize=FALSE, eps=1e-5,
maxit=1000, randomStarts=0)
quartimin(A, Tmat=diag(ncol(A)), normalize=FALSE, eps=1e-5,
maxit=1000, randomStarts=0)
targetT(A, Tmat=diag(ncol(A)), Target=NULL, normalize=FALSE, eps=1e-5,
maxit=1000, randomStarts=0, L=NULL)
targetQ(A, Tmat=diag(ncol(A)), Target=NULL, normalize=FALSE, eps=1e-5,
maxit=1000, randomStarts=0, L=NULL)
pstT(A, Tmat=diag(ncol(A)), W=NULL, Target=NULL, normalize=FALSE, eps=1e-5,
maxit=1000, randomStarts=0, L=NULL)
pstQ(A, Tmat=diag(ncol(A)), W=NULL, Target=NULL, normalize=FALSE, eps=1e-5,
maxit=1000, randomStarts=0, L=NULL)
oblimax(A, Tmat=diag(ncol(A)), normalize=FALSE, eps=1e-5, maxit=1000, randomStarts=0)
entropy(A, Tmat=diag(ncol(A)), normalize=FALSE, eps=1e-5, maxit=1000, randomStarts=0)
quartimax(A, Tmat=diag(ncol(A)), normalize=FALSE, eps=1e-5,maxit=1000,randomStarts=0)
Varimax(A, Tmat=diag(ncol(A)), normalize=FALSE, eps=1e-5, maxit=1000, randomStarts=0)
simplimax(A, Tmat=diag(ncol(A)), k=nrow(A), normalize=FALSE, eps=1e-5,
maxit=1000, randomStarts=0)
bentlerT(A, Tmat=diag(ncol(A)), normalize=FALSE, eps=1e-5, maxit=1000, randomStarts=0)
bentlerQ(A, Tmat=diag(ncol(A)), normalize=FALSE, eps=1e-5, maxit=1000, randomStarts=0)
tandemI(A, Tmat=diag(ncol(A)), normalize=FALSE, eps=1e-5, maxit=1000, randomStarts=0)
tandemII(A, Tmat=diag(ncol(A)), normalize=FALSE, eps=1e-5, maxit=1000, randomStarts=0)
geominT(A, Tmat=diag(ncol(A)), delta=.01, normalize=FALSE, eps=1e-5,
maxit=1000, randomStarts=0)
geominQ(A, Tmat=diag(ncol(A)), delta=.01, normalize=FALSE, eps=1e-5,
maxit=1000, randomStarts=0)
bigeominT(A, Tmat=diag(ncol(A)), delta=.01, normalize=FALSE, eps=1e-5,
maxit=1000, randomStarts=0)
bigeominQ(A, Tmat=diag(ncol(A)), delta=.01, normalize=FALSE, eps=1e-5,
maxit=1000, randomStarts=0)
cfT(A, Tmat=diag(ncol(A)), kappa=0, normalize=FALSE, eps=1e-5,
maxit=1000, randomStarts=0)
cfQ(A, Tmat=diag(ncol(A)), kappa=0, normalize=FALSE, eps=1e-5,
maxit=1000, randomStarts=0)
equamax(A, Tmat=diag(ncol(A)), kappa=ncol(A)/(2*nrow(A)), normalize=FALSE,
eps=1e-5, maxit=1000, randomStarts = 0)
parsimax(A, Tmat=diag(ncol(A)), kappa=(ncol(A)-1)/(ncol(A)+nrow(A)-2),
normalize=FALSE, eps=1e-5, maxit=1000, randomStarts = 0)
infomaxT(A, Tmat=diag(ncol(A)), normalize=FALSE, eps=1e-5, maxit=1000, randomStarts=0)
infomaxQ(A, Tmat=diag(ncol(A)), normalize=FALSE, eps=1e-5, maxit=1000, randomStarts=0)
mccammon(A, Tmat=diag(ncol(A)), normalize=FALSE, eps=1e-5, maxit=1000, randomStarts=0)
varimin(A, Tmat=diag(ncol(A)), normalize=FALSE, eps=1e-5, maxit=1000, randomStarts=0)
bifactorT(A, Tmat=diag(ncol(A)), normalize=FALSE, eps=1e-5, maxit=1000,randomStarts=0)
bifactorQ(A, Tmat=diag(ncol(A)), normalize=FALSE, eps=1e-5, maxit=1000,randomStarts=0)
``````

### Arguments

 `A` an initial loadings matrix to be rotated. `Tmat` initial rotation matrix. `gam` 0=Quartimin, .5=Biquartimin, 1=Covarimin. `Target` rotation target for objective calculation. `W` weighting of each element in target. `k` number of close to zero loadings. `delta` constant added to Lambda^2 in objective calculation. `kappa` see details. `normalize` parameter passed to optimization routine (GPForth or GPFoblq). `eps` parameter passed to optimization routine (GPForth or GPFoblq). `maxit` parameter passed to optimization routine (GPForth or GPFoblq). `randomStarts` parameter passed to optimization routine (GPFRSorth or GPFRSoblq). `L` provided for backward compatibility in target rotations only. Use A going forward.

### Details

These functions optimize a rotation objective. They can be used directly or the function name can be passed to factor analysis functions like `factanal`. Several of the function names end in T or Q, which indicates if they are orthogonal or oblique rotations (using `GPFRSorth` or `GPFRSoblq` respectively).

Rotations which are available are

 `oblimin` oblique oblimin family `quartimin` oblique `targetT` orthogonal target rotation `targetQ` oblique target rotation `pstT` orthogonal partially specified target rotation `pstQ` oblique partially specified target rotation `oblimax` oblique `entropy` orthogonal minimum entropy `quartimax` orthogonal `varimax` orthogonal `simplimax` oblique `bentlerT` orthogonal Bentler's invariant pattern simplicity criterion `bentlerQ` oblique Bentler's invariant pattern simplicity criterion `tandemI` orthogonal Tandem principle I criterion `tandemII` orthogonal Tandem principle II criterion `geominT` orthogonal `geominQ` oblique `bigeominT` orthogonal `bigeominQ` oblique `cfT` orthogonal Crawford-Ferguson family `cfQ` oblique Crawford-Ferguson family `equamax` orthogonal Crawford-Ferguson family `parsimax` orthogonal Crawford-Ferguson family `infomaxT` orthogonal `infomaxQ` oblique `mccammon` orthogonal McCammon minimum entropy ratio `varimin` orthogonal `bifactorT` orthogonal Jennrich and Bentler bifactor rotation `bifactorQ` oblique Jennrich and Bentler biquartimin rotation

Note that `Varimax` defined here uses `vgQ.varimax` and is not `varimax` defined in the `stats` package. `stats:::varimax` does Kaiser normalization by default whereas `Varimax` defined here does not.

The argument `kappa` parameterizes the family for the Crawford-Ferguson method. If `m` is the number of factors and `p` is the number of indicators then `kappa` values having special names are 0=Quartimax, 1/p=Varimax, m/(2*p)=Equamax, (m-1)/(p+m-2)=Parsimax, 1=Factor parsimony.

### Value

A list (which includes elements used by `factanal`) with:

 `loadings` Lh from `GPFRSorth` or `GPFRSoblq`. `Th` Th from `GPFRSorth` or `GPFRSoblq`. `Table` Table from `GPForth` or `GPFoblq`. `method` A string indicating the rotation objective function. `orthogonal` A logical indicating if the rotation is orthogonal. `convergence` Convergence indicator from `GPFRSorth` or `GPFRSoblq`. `Phi` t(Th) %*% Th. The covariance matrix of the rotated factors. This will be the identity matrix for orthogonal rotations so is omitted (NULL) for the result from GPFRSorth and GPForth. `randStartChar` Vector indicating results from random starts from `GPFRSorth` or `GPFRSoblq`

### Author(s)

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

### References

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.

Bifactor rotation, bifactorT and bifactorQ are called bifactor and biquartimin in Jennrich, R.I. and Bentler, P.M. (2011) Exploratory bi-factor analysis. Psychometrika, 76.

`GPFRSorth`, `GPFRSoblq`, `vgQ`, `eiv`, `echelon`, `WansbeekMeijer`, `factanal`, `varimax`

### Examples

``````  # see GPFRSorth and GPFRSoblq for more examples

data("Harman", package="GPArotation")
qHarman  <- GPFRSorth(Harman8, Tmat=diag(2), method="quartimax")
qHarman <- quartimax(Harman8)

data("WansbeekMeijer", package="GPArotation")
fa.unrotated  <- factanal(factors = 2, covmat=NetherlandsTV, normalize=TRUE, rotation="none")

# passing arguments to factanal (See vignette for a caution)
# vignette("GPAguide", package = "GPArotation")
data(ability.cov)
factanal(factors = 2, covmat = ability.cov, rotation="infomaxT")
factanal(factors = 2, covmat = ability.cov, rotation="infomaxT",
control=list(rotate=list(normalize = TRUE, eps = 1e-6)))
# when using factanal for oblique rotation it is best to use the rotation command directly
# instead of including it in the factanal command (see Vignette).
fa.unrotated  <- factanal(factors = 3, covmat=NetherlandsTV, normalize=TRUE, rotation="none")

# oblique target rotation of 2 varimax rotated matrices towards each other
# See vignette for additional context and computation,
trBritain <- matrix( c(.783,-.163,.811,.202,.724,.209,.850,.064,
-.031,.592,-.028,.723,.388,.434,.141,.808,.215,.709), byrow=TRUE, ncol=2)
trGermany <- matrix( c(.778,-.066, .875,.081, .751,.079, .739,.092,
.195,.574, -.030,.807, -.135,.717, .125,.738, .060,.691), byrow=TRUE, ncol = 2)
trx <- targetQ(trGermany, Target = trBritain)

# partially specified target; See vignette for additional method
A <- matrix(c(.664, .688, .492, .837, .705, .82, .661, .457, .765, .322,
.248, .304, -0.291, -0.314, -0.377, .397, .294, .428, -0.075,.192,.224,
.037, .155,-.104,.077,-.488,.009), ncol=3)
SPA <- matrix(c(rep(NA, 6), .7,.0,.7, rep(0,3), rep(NA, 7), 0,0, NA, 0, rep(NA, 4)), ncol=3)
targetT(A, Target=SPA)

# using random starts
data("WansbeekMeijer", package="GPArotation")
fa.unrotated  <- factanal(factors = 3, covmat=NetherlandsTV, normalize=TRUE, rotation="none")
# single rotation with a random start
# multiple random starts

# assessing local minima for box26 data
data(Thurstone, package = "GPArotation")
infomaxQ(box26, normalize = TRUE, randomStarts = 150)
geominQ(box26, normalize = TRUE, randomStarts = 150)
# for detailed investigation of local minima, consult package 'fungible'
# library(fungible)