GPArotation (version 2025.3-1)

rotations: Rotations

Description

Optimize factor loading rotation objective.

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)
    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)

Value

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

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 \(L^p\), where 0 < p \(=<\) 1.

gpaiter

Maximum iterations for GPA rotation loop in \(L^p\) rotation.

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 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	\(L^p\) rotation
lpQ	oblique	\(L^p\) 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 \(L^p\) rotation. See Lp rotation for details on the rotation method.

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.

See Also

factanal, GPFRSorth, GPFRSoblq, vgQ, Harman, box26, WansbeekMeijer,

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)