rotations {GPArotation}R Documentation

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)

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.

See Also

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

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)
[Package GPArotation version 2024.3-1 Index]