rotations.RdOptimize 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)an initial loadings matrix to be rotated.
initial rotation matrix.
0=Quartimin, .5=Biquartimin, 1=Covarimin.
rotation target for objective calculation.
weighting of each element in target.
number of close to zero loadings.
constant added to \(\Lambda^2\) in the objective calculation.
see details.
parameter passed to optimization routine (GPForth or GPFoblq).
parameter passed to optimization routine (GPForth or GPFoblq).
parameter passed to optimization routine (GPForth or GPFoblq).
parameter passed to optimization routine (GPFRSorth or GPFRSoblq).
provided for backward compatibility in target rotations only. Use A going forward.
Component-wise \(L^p\), where 0 < p \(=<\) 1.
Maximum iterations for GPA rotation loop in \(L^p\) rotation.
A GPArotation object which is a list with elements
(includes elements used by factanal) with:
Lh from GPFRSorth or GPFRSoblq.
Th from GPFRSorth or GPFRSoblq.
Table from GPForth or GPFoblq.
A string indicating the rotation objective function.
A logical indicating if the rotation is orthogonal.
Convergence indicator from GPFRSorth or GPFRSoblq.
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.
Vector indicating results from random starts
from GPFRSorth or GPFRSoblq
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.
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 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
#> [,1] [,2]
#> [1,] 0 0
#> [2,] 0 0
#> [3,] 0 0
#> [4,] 0 0
#> [5,] 0 0
#> [6,] 0 0
#> [7,] 0 0
#> [8,] 0 0
# 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)
#> Orthogonal rotation method Quartimax converged.
#> Loadings:
#> Factor1 Factor2
#> NL1 0.327 0.720
#> TV2 0.424 0.725
#> NL3 0.323 0.703
#> RTL4 0.669 0.214
#> RTL5 0.738 0.183
#> Veronica 0.811 0.109
#> SBS6 0.719 0.106
#>
#> Factor1 Factor2
#> SS loadings 2.556 1.641
#> Proportion Var 0.365 0.234
#> Cumulative Var 0.365 0.600
geominQ(loadings(fa.unrotated), normalize=TRUE, randomStarts=100)
#> Oblique rotation method Geomin converged at lowest minimum.
#> Of 100 random starts 100% converged, 100% at the same lowest minimum.
#> Loadings at lowest minimum:
#> Factor1 Factor2
#> NL1 -0.0159 0.8002
#> TV2 0.0897 0.7848
#> NL3 -0.0111 0.7796
#> RTL4 0.6306 0.1126
#> RTL5 0.7235 0.0601
#> Veronica 0.8437 -0.0459
#> SBS6 0.7435 -0.0295
#>
#> Factor1 Factor2
#> SS loadings 2.254 1.943
#> Proportion Var 0.322 0.278
#> Cumulative Var 0.322 0.600
#>
#> Phi:
#> Factor1 Factor2
#> Factor1 1.000 0.582
#> Factor2 0.582 1.000
# passing arguments to factanal (See vignette for a caution)
# vignette("GPAguide", package = "GPArotation")
data(ability.cov)
factanal(factors = 2, covmat = ability.cov, rotation="infomaxT")
#>
#> Call:
#> factanal(factors = 2, covmat = ability.cov, rotation = "infomaxT")
#>
#> Uniquenesses:
#> general picture blocks maze reading vocab
#> 0.455 0.589 0.218 0.769 0.052 0.334
#>
#> Loadings:
#> Factor1 Factor2
#> general 0.495 0.547
#> picture 0.151 0.623
#> blocks 0.199 0.861
#> maze 0.105 0.469
#> reading 0.955 0.189
#> vocab 0.783 0.231
#>
#> Factor1 Factor2
#> SS loadings 1.844 1.738
#> Proportion Var 0.307 0.290
#> Cumulative Var 0.307 0.597
#>
#> Test of the hypothesis that 2 factors are sufficient.
#> The chi square statistic is 6.11 on 4 degrees of freedom.
#> The p-value is 0.191
factanal(factors = 2, covmat = ability.cov, rotation="infomaxT",
control=list(rotate=list(normalize = TRUE, eps = 1e-6)))
#>
#> Call:
#> factanal(factors = 2, covmat = ability.cov, rotation = "infomaxT", control = list(rotate = list(normalize = TRUE, eps = 1e-06)))
#>
#> Uniquenesses:
#> general picture blocks maze reading vocab
#> 0.455 0.589 0.218 0.769 0.052 0.334
#>
#> Loadings:
#> Factor1 Factor2
#> general 0.568 0.471
#> picture 0.629 0.124
#> blocks 0.869 0.161
#> maze 0.473
#> reading 0.231 0.946
#> vocab 0.265 0.772
#>
#> Factor1 Factor2
#> SS loadings 1.821 1.761
#> Proportion Var 0.304 0.293
#> Cumulative Var 0.304 0.597
#>
#> Test of the hypothesis that 2 factors are sufficient.
#> The chi square statistic is 6.11 on 4 degrees of freedom.
#> The p-value is 0.191
# 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 rotation method Quartimin converged.
#> Loadings:
#> Factor1 Factor2 Factor3
#> NL1 0.8085 -0.03829 0.0166
#> TV2 0.7866 0.07540 0.0116
#> NL3 0.7801 -0.00504 -0.0113
#> RTL4 0.0533 0.74602 -0.0768
#> RTL5 0.0483 0.65405 0.0903
#> Veronica -0.0721 0.81431 0.0553
#> SBS6 0.0205 0.02007 0.9755
#>
#> Factor1 Factor2 Factor3
#> SS loadings 1.926 1.724 1.026
#> Proportion Var 0.275 0.246 0.147
#> Cumulative Var 0.275 0.521 0.668
#>
#> Phi:
#> Factor1 Factor2 Factor3
#> Factor1 1.000 0.615 0.381
#> Factor2 0.615 1.000 0.687
#> Factor3 0.381 0.687 1.000
# 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)
#> Orthogonal rotation method Target rotation converged.
#> Loadings:
#> [,1] [,2] [,3]
#> [1,] 0.611 0.0231 0.4203
#> [2,] 0.734 0.0554 0.1731
#> [3,] 0.607 -0.0861 0.0926
#> [4,] 0.608 0.6218 0.1756
#> [5,] 0.549 0.5642 0.0172
#> [6,] 0.499 0.7130 0.2614
#> [7,] 0.705 -0.0691 0.3139
#> [8,] 0.235 0.0234 0.6910
#> [9,] 0.768 -0.0392 0.4216
#>
#> [,1] [,2] [,3]
#> SS loadings 3.341 1.231 1.068
#> Proportion Var 0.371 0.137 0.119
#> Cumulative Var 0.371 0.508 0.627
# 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))
#> Oblique rotation method Oblimin Quartimin converged.
#> Loadings:
#> Factor1 Factor2 Factor3
#> NL1 0.81156 -0.04402 0.01607
#> TV2 0.79137 0.06961 0.00987
#> NL3 0.78396 -0.01027 -0.01260
#> RTL4 0.06646 0.74487 -0.08584
#> RTL5 0.05754 0.65088 0.08480
#> Veronica -0.06034 0.81206 0.04782
#> SBS6 0.00682 0.00651 0.99012
#>
#> Factor1 Factor2 Factor3
#> SS loadings 1.947 1.701 1.029
#> Proportion Var 0.278 0.243 0.147
#> Cumulative Var 0.278 0.521 0.668
#>
#> Phi:
#> Factor1 Factor2 Factor3
#> Factor1 1.000 0.610 0.402
#> Factor2 0.610 1.000 0.704
#> Factor3 0.402 0.704 1.000
oblimin(loadings(fa.unrotated), randomStarts=1)
#> Oblique rotation method Oblimin Quartimin converged.
#> Loadings:
#> Factor1 Factor2 Factor3
#> NL1 0.81155 -0.04402 0.01607
#> TV2 0.79137 0.06961 0.00987
#> NL3 0.78395 -0.01027 -0.01260
#> RTL4 0.06646 0.74487 -0.08584
#> RTL5 0.05754 0.65088 0.08480
#> Veronica -0.06034 0.81206 0.04782
#> SBS6 0.00682 0.00651 0.99012
#>
#> Factor1 Factor2 Factor3
#> SS loadings 1.947 1.701 1.029
#> Proportion Var 0.278 0.243 0.147
#> Cumulative Var 0.278 0.521 0.668
#>
#> Phi:
#> Factor1 Factor2 Factor3
#> Factor1 1.000 0.610 0.402
#> Factor2 0.610 1.000 0.704
#> Factor3 0.402 0.704 1.000
# multiple random starts
oblimin(loadings(fa.unrotated), randomStarts=100)
#> Oblique rotation method Oblimin Quartimin converged at lowest minimum.
#> Of 100 random starts 100% converged, 100% at the same lowest minimum.
#> Loadings at lowest minimum:
#> Factor1 Factor2 Factor3
#> NL1 0.81156 -0.04402 0.01607
#> TV2 0.79137 0.06961 0.00987
#> NL3 0.78396 -0.01027 -0.01260
#> RTL4 0.06646 0.74487 -0.08584
#> RTL5 0.05754 0.65088 0.08480
#> Veronica -0.06034 0.81206 0.04782
#> SBS6 0.00682 0.00651 0.99012
#>
#> Factor1 Factor2 Factor3
#> SS loadings 1.947 1.701 1.029
#> Proportion Var 0.278 0.243 0.147
#> Cumulative Var 0.278 0.521 0.668
#>
#> Phi:
#> Factor1 Factor2 Factor3
#> Factor1 1.000 0.610 0.402
#> Factor2 0.610 1.000 0.704
#> Factor3 0.402 0.704 1.000
# assessing local minima for box26 data
data(Thurstone, package = "GPArotation")
infomaxQ(box26, normalize = TRUE, randomStarts = 150)
#> Oblique rotation method Infomax converged at lowest minimum.
#> Of 150 random starts 100% converged, 33% at the same lowest minimum.
#> Random starts converged to 9 different local minima
#> Loadings at lowest minimum:
#> [,1] [,2] [,3]
#> [1,] -0.6979 0.733147 0.7412
#> [2,] 0.6320 -0.445892 0.6788
#> [3,] 0.7337 0.664612 -0.4890
#> [4,] -0.0121 0.102265 0.9371
#> [5,] 0.0554 0.906390 0.0942
#> [6,] 0.8940 0.107168 0.0571
#> [7,] -0.3082 0.407153 0.8923
#> [8,] 0.3103 -0.144540 0.8407
#> [9,] -0.2544 0.895291 0.3483
#> [10,] 0.3288 0.933984 -0.1188
#> [11,] 0.8560 -0.112855 0.2744
#> [12,] 0.8807 0.301273 -0.1431
#> [13,] -1.0675 1.018758 -0.0251
#> [14,] 1.0675 -1.018758 0.0251
#> [15,] -1.1473 0.000242 1.0721
#> [16,] 1.1473 -0.000242 -1.0721
#> [17,] 0.0080 -1.057398 1.0279
#> [18,] -0.0080 1.057398 -1.0279
#> [19,] 0.0904 -0.005925 0.9215
#> [20,] 0.0962 0.922332 0.0229
#> [21,] 0.8755 0.108684 0.0816
#> [22,] 0.1000 0.000592 0.8995
#> [23,] 0.1219 0.886078 0.0131
#> [24,] 0.8371 0.101842 0.1155
#> [25,] 0.3478 0.405154 0.4190
#> [26,] 0.4730 0.309918 0.3582
#>
#> [,1] [,2] [,3]
#> SS loadings 8.668 8.559 8.183
#> Proportion Var 0.333 0.329 0.315
#> Cumulative Var 0.333 0.663 0.977
#>
#> Phi:
#> [,1] [,2] [,3]
#> [1,] 1.000 0.548 0.607
#> [2,] 0.548 1.000 0.543
#> [3,] 0.607 0.543 1.000
geominQ(box26, normalize = TRUE, randomStarts = 150)
#> Oblique rotation method Geomin converged at lowest minimum.
#> Of 150 random starts 100% converged, 21% at the same lowest minimum.
#> Random starts converged to 4 different local minima
#> Loadings at lowest minimum:
#> [,1] [,2] [,3]
#> [1,] -0.01480 -0.00966 0.992934
#> [2,] 0.94194 0.04837 0.059132
#> [3,] 0.06094 0.96469 -0.000941
#> [4,] 0.64216 -0.01368 0.637319
#> [5,] 0.00200 0.64560 0.596311
#> [6,] 0.61311 0.64401 -0.024703
#> [7,] 0.38294 0.00698 0.834795
#> [8,] 0.81242 0.03353 0.384702
#> [9,] -0.02006 0.41809 0.788315
#> [10,] 0.02744 0.85762 0.444477
#> [11,] 0.76608 0.45213 -0.018941
#> [12,] 0.44155 0.78388 -0.028514
#> [13,] -0.82910 0.00876 0.747107
#> [14,] 0.82910 -0.00876 -0.747107
#> [15,] 0.00630 -0.82506 0.816393
#> [16,] -0.00630 0.82506 -0.816393
#> [17,] 0.84823 -0.79818 -0.008815
#> [18,] -0.84823 0.79818 0.008815
#> [19,] 0.71018 -0.02017 0.548317
#> [20,] -0.02338 0.68851 0.556578
#> [21,] 0.61810 0.63120 -0.006276
#> [22,] 0.70019 -0.00779 0.537441
#> [23,] -0.00946 0.68113 0.525157
#> [24,] 0.61769 0.59906 0.015742
#> [25,] 0.47824 0.46620 0.452515
#> [26,] 0.52777 0.48684 0.340444
#>
#> [,1] [,2] [,3]
#> SS loadings 8.824 8.795 7.790
#> Proportion Var 0.339 0.338 0.300
#> Cumulative Var 0.339 0.678 0.977
#>
#> Phi:
#> [,1] [,2] [,3]
#> [1,] 1.000 0.268 0.206
#> [2,] 0.268 1.000 0.278
#> [3,] 0.206 0.278 1.000
# 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)