# Rotations Optimize factor loading rotation objective. ```r 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 ```r # 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) ``` ## See Also `factanal`, `GPFRSorth`, `GPFRSoblq`, `vgQ`, `Harman`, `box26`, `WansbeekMeijer`, ## 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.

rotations function