GPArotation (version 2022.10-2)

rotations: Rotations

Description

Optimize factor loading rotation objective.

Usage

oblimin(L, Tmat=diag(ncol(L)), gam=0, normalize=FALSE, eps=1e-5, maxit=1000)
    quartimin(L, Tmat=diag(ncol(L)), normalize=FALSE, eps=1e-5, maxit=1000)
    targetT(L, Tmat=diag(ncol(L)), Target=NULL, normalize=FALSE, eps=1e-5, maxit=1000)
    targetQ(L, Tmat=diag(ncol(L)), Target=NULL, normalize=FALSE, eps=1e-5, maxit=1000)
    pstT(L, Tmat=diag(ncol(L)), W=NULL, Target=NULL, 
               normalize=FALSE, eps=1e-5, maxit=1000)
    pstQ(L, Tmat=diag(ncol(L)), W=NULL, Target=NULL,
               normalize=FALSE, eps=1e-5, maxit=1000)
    oblimax(L, Tmat=diag(ncol(L)), normalize=FALSE, eps=1e-5, maxit=1000)
    entropy(L, Tmat=diag(ncol(L)), normalize=FALSE, eps=1e-5, maxit=1000)
    quartimax(L, Tmat=diag(ncol(L)), normalize=FALSE, eps=1e-5, maxit=1000)
    Varimax(L, Tmat=diag(ncol(L)), normalize=FALSE, eps=1e-5, maxit=1000)
    simplimax(L, Tmat=diag(ncol(L)), k=nrow(L), 
               normalize=FALSE, eps=1e-5, maxit=1000)
    bentlerT(L, Tmat=diag(ncol(L)), normalize=FALSE, eps=1e-5, maxit=1000)
    bentlerQ(L, Tmat=diag(ncol(L)), normalize=FALSE, eps=1e-5, maxit=1000)
    tandemI(L, Tmat=diag(ncol(L)), normalize=FALSE, eps=1e-5, maxit=1000)
    tandemII(L, Tmat=diag(ncol(L)), normalize=FALSE, eps=1e-5, maxit=1000)
    geominT(L, Tmat=diag(ncol(L)), delta=.01, 
               normalize=FALSE, eps=1e-5, maxit=1000)
    geominQ(L, Tmat=diag(ncol(L)), delta=.01, 
               normalize=FALSE, eps=1e-5, maxit=1000)
    cfT(L, Tmat=diag(ncol(L)), kappa=0, normalize=FALSE, eps=1e-5, maxit=1000)
    cfQ(L, Tmat=diag(ncol(L)), kappa=0, normalize=FALSE, eps=1e-5, maxit=1000)
    infomaxT(L, Tmat=diag(ncol(L)), normalize=FALSE, eps=1e-5, maxit=1000)
    infomaxQ(L, Tmat=diag(ncol(L)), normalize=FALSE, eps=1e-5, maxit=1000)
    mccammon(L, Tmat=diag(ncol(L)), normalize=FALSE, eps=1e-5, maxit=1000)
    bifactorT(L, Tmat=diag(ncol(L)), normalize=FALSE, eps=1e-5, maxit=1000)
    bifactorQ(L, Tmat=diag(ncol(L)), normalize=FALSE, eps=1e-5, maxit=1000)
    
    vgQ.oblimin(L, gam=0)
    vgQ.quartimin(L)
    vgQ.target(L, Target=NULL)
    vgQ.pst(L, W=NULL, Target=NULL)
    vgQ.oblimax(L)
    vgQ.entropy(L)
    vgQ.quartimax(L)
    vgQ.varimax(L)
    vgQ.simplimax(L, k=nrow(L))
    vgQ.bentler(L)
    vgQ.tandemI(L)
    vgQ.tandemII(L)
    vgQ.geomin(L, delta=.01)
    vgQ.cf(L, kappa=0)
    vgQ.infomax(L)
    vgQ.mccammon(L)
    vgQ.bifactor(L)

Value

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

loadings

Lh from GPForth or GPFoblq.

Th

Th from GPForth or GPFoblq.

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 GPForth or GPFoblq.

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 GPForth.

Arguments

L

a factor loading matrix

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).

Author

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

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 GPForth or GPFoblq respectively.

The vgQ.* versions of the code are called by the optimization routine and would typically not be used directly, so these methods are not exported from the package namespace. (They simply return the function value and gradient for a given rotation matrix.) You can print these functions, but the package name needs to be specified since they are not exported. For example, use GPArotation:::vgQ.oblimin to view the function vgQ.oblimin. The T or Q ending on function names should be omitted for the vgQ.* versions of the code so, for example, use GPArotation:::vgQ.target to view the target criterion calculation.

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 Criterion
tandemII	orthogonal	Tandem Criterion
geominT	orthogonal
geominQ	oblique
cfT	orthogonal	Crawford-Ferguson family
cfQ	oblique	Crawford-Ferguson family
infomaxT	orthogonal
infomaxQ	oblique
mccammon	orthogonal	McCammon minimum entropy ratio
bifactorT	orthogonal	Jennrich and Bentler bifactor rotation
bifactorQ	oblique	Jennrich and Bentler biquartimin rotation

Also included for convenience are two analytic rotations eiv and echelon which do not require GPForth or GPFoblq.

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.

New rotation methods can be programmed with a name "vgQ.newmethod". The inputs are the matrix L, and optionally any additional arguments. The output should be a list with elements

f	the value of the criterion at L.
Gq	the gradient at L.
Method	a string indicating the criterion.

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.

A discussion of rotation objectives can be found in many references, for example,

Tom Wansbeek and Erik Meijer (2000) Measurement Error and Latent Variables in Econometrics, Amsterdam: North-Holland.

See Also

GPForth, GPFoblq, WansbeekMeijer, eiv, echelon, factanal, varimax, promax

Examples

  data(ability.cov)
  factanal(factors = 2, covmat = ability.cov, rotation="oblimin")

  data("Harman", package="GPArotation")
  qHarman  <- GPForth(Harman8, Tmat=diag(2), method="quartimax")
  qHarman2 <- quartimax(Harman8) 

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

  fa.varimax <- factanal(factors = 2, covmat=NetherlandsTV, 
                rotation="varimax", control=list(rotate=list(normalize=TRUE)))
  fa.oblimin <- factanal(factors = 2, covmat=NetherlandsTV,
                rotation="oblimin", control=list(rotate=list(normalize=TRUE)))
  
  cbind(loadings(fa.unrotated), loadings(fa.varimax), loadings(fa.oblimin))