rotations() R function from [GPArotation]

Rotations

Optimize factor loading rotation objective.


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)
    lpT(A, Tmat=diag(ncol(A)), p=1, normalize=FALSE, eps=1e-05, maxit=1000, 
            randomStarts=0, gpaiter=5) 
    lpQ(A, Tmat=diag(ncol(A)), p=1, normalize=FALSE, eps=1e-05, maxit=1000, 
                randomStarts=0, gpaiter=5)

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 the 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.
p: Component-wise $Lp$ , where 0 < p $=<$ 1.
gpaiter: Maximum iterations for GPA rotation loop in $Lp$ rotation.

Returns

A GPArotation object which is a list with elements (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

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
`lpT`	orthogonal	$Lp$ rotation
`lpQ`	oblique	$Lp$ 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.

Bifactor rotations, bifactorT and bifactorQ are called bifactor and biquartimin in Jennrich, R.I. and Bentler, P.M. (2011).

The argument p is needed for $Lp$ rotation. See Lp rotation for details on the rotation method.

Examples


# see GPFRSorth and GPFRSoblq for more examples
  
  # getting loadings matrices
  data("Harman", package="GPArotation")
  qHarman  <- GPFRSorth(Harman8, Tmat=diag(2), method="quartimax")
  qHarman <- quartimax(Harman8) 
  loadings(qHarman) - qHarman$loadings   #2 ways to get the loadings

  # factanal loadings used in GPArotation
  data("WansbeekMeijer", package="GPArotation")
  fa.unrotated  <- factanal(factors = 2, covmat=NetherlandsTV, normalize=TRUE, rotation="none")
  quartimax(loadings(fa.unrotated), normalize=TRUE)
  geominQ(loadings(fa.unrotated), normalize=TRUE, randomStarts=100)

  # 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")
  quartimin(loadings(fa.unrotated), normalize=TRUE)

  # 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)
  # Difference between rotated loadings matrix and target matrix 
  y <- trx$loadings - 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
  oblimin(loadings(fa.unrotated), Tmat=Random.Start(3))
  oblimin(loadings(fa.unrotated), randomStarts=1)
  # multiple random starts
  oblimin(loadings(fa.unrotated), randomStarts=100)

  # 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)
  # faMain(urLoadings=box26, rotate="geominQ", rotateControl=list(numberStarts=150))
  # library(psych) # package 'psych' with random starts:
  # faRotations(box26, rotate = "geominQ", hyper = 0.15, n.rotations = 150)

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.

Jennrich, R.I. and Bentler, P.M. (2011) Exploratory bi-factor analysis. Psychometrika, 76 .

Author(s)

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

GPArotation package Read PDF manual

Maintainer: Coen Bernaards
License: GPL (>= 2)
Last published: 2025-04-12
https://optimizer.r-forge.r-project.org/GPArotation_www/

rotations function