# I have a question about how factanal is calculating the regression factor
# scores based on an oblique rotation (promax) of the factors.
#
# As is explained in the help file, regression factor scores are
# obtained as
#
# hat f = Lambda' Sigma^-1 x
#
# However, according to Harman's "Modern Factor Analysis" (e.g.
second
# edition, pp. 351-352) the formula is
#
# hat f = Phi Lambda' Sigma^-1 x,
#
# where Phi is the correlation between factors. Of course, for orthogonal
# rotations the formulas are identical because in this case Phi=I.
#
# Let me illustrate the difference between the formulas numerically:
# X is a data (10x6) data matrix of standardized variables.
"X" <-
structure(c(-0.697, -1.787, 0.206, -0.191, -0.606, 0.171, 1.46,
-0.639, 0.779, 1.304, -0.7, -1.538, -0.913, -0.43, -0.225, -0.417,
1.038, 0.888, 1.595, 0.702, -1.268, -2.018, 0.079, 1.074, 0.296,
-0.62, 0.532, 0.306, 0.775, 0.844, -2.245, 0.486, 0.801, 0.002,
-0.602, -0.519, 1.261, -0.372, 0.499, 0.688, -1.973, -0.163,
0.964, -0.071, -0.99, 0.694, -0.364, -0.305, 1.215, 0.992, -1.674,
-0.065, 1.043, -0.159, -1.174, 0.648, 0.848, 1.101, -1.055, 0.488
), .Dim = c(10, 6))
library(mva)
res <- factanal(X, factors=2, rotation="promax",
scores="regression")
round(res$scores, 3)
# yields the factor scores:
# Factor1 Factor2
# [1,] 0.031 -2.137
# [2,] -2.319 1.141
# [3,] -0.899 1.375
# [4,] 0.098 -0.058
# [5,] 0.215 -0.943
# [6,] -0.402 0.218
# [7,] 0.828 0.488
# [8,] 0.364 -0.364
# [9,] 1.313 -0.204
# [10,] 0.771 0.485
# Now, calculating the factor scores using the first formula I obtain
# (up to rounding differences) the same result
R <- cor(X)
L <- loadings(res)
round(t(t(L) %*% solve(R) %*% t(X)), 3)
# However, if I use the following SAS commands using the exactly the
# same data
# data x;
# input x1 x2 x3 x4 x5 x6;
# cards;
# -0.697 -0.700 -1.268 -2.245 -1.973 -1.674
# -1.787 -1.538 -2.018 0.486 -0.163 -0.065
# 0.206 -0.913 0.079 0.801 0.964 1.043
# -0.191 -0.430 1.074 0.002 -0.071 -0.159
# -0.606 -0.225 0.296 -0.602 -0.990 -1.174
# 0.171 -0.417 -0.620 -0.519 0.694 0.648
# 1.460 1.038 0.532 1.261 -0.364 0.848
# -0.639 0.888 0.306 -0.372 -0.305 1.101
# 0.779 1.595 0.775 0.499 1.215 -1.055
# 1.304 0.702 0.844 0.688 0.992 0.488
# ;
#
# proc factor data=x corr method=ml nfactors=2 prerotate=varimax rotate=promax
out=fscores score;
# run;
#
# proc print data=fscores;
# var factor1 factor2;
# run;
#
# The factor scores one obtains:
# Obs Factor1 Factor2
# 1 -1.00830 -2.12208
# 2 -1.76352 0.09283
# 3 -0.23031 0.96792
# 4 0.06982 -0.01378
# 5 -0.24412 -0.84552
# 6 -0.29643 0.03564
# 7 1.06502 0.86195
# 8 0.18723 -0.19927
# 9 1.21346 0.38882
# 10 1.00714 0.83347
# Clearly, these scores are very different from the ones factanal
# produced. However, because the promax-function does not calculate
# the Phi-matrix, one cannot readily calculate the factor scores
# according to the second formula from Harman's book.
# However, one may calculate Phi in the following way using varimax as
# the prerotation:
res <- factanal(X, factors=2, rotation="none")
vm <- varimax(loadings(res))
pm <- promax(loadings(vm), m=3) # m=3 is default in SAS
L <- loadings(pm)
# Calculate the "normalized oblique transformation matrix" (label from
# SAS output)
A <- vm$rotmat %*% pm$rotmat
# Calculate the "inter-factor correlation matrix" (label from SAS
output
Phi <- solve(t(A) %*% A)
# Calculating regression factor scores according to the second formula
# as
t(Phi %*% t(L) %*% solve(R) %*% t(X))
# yields scores that are pretty close, although not identical to the
# SAS factor scores. Specifically the scores one obtains are
#
# [1,] -0.95635745 -2.12230012
# [2,] -1.79135903 0.08454732
# [3,] -0.26372828 0.96485151
# [4,] 0.07127957 -0.01351224
# [5,] -0.22118873 -0.84499215
# [6,] -0.30174316 0.03427963
# [7,] 1.05315509 0.86483663
# [8,] 0.19603657 -0.19803312
# [9,] 1.21844347 0.39342954
#[10,] 0.99537610 0.83626200
# So, I guess, I am wondering does factanal calculate the
# regression factor scores correctly? Maybe I am missing
# something. Any comments will be highly appreciated.
# Thanks,
# Christof Schuster
Christof Schuster
University of Notre Dame
Department of Psychology
103 Haggar Hall
Notre Dame, IN 46556
Tel: (574) 631-5473 email: cschuste at nd.edu
Fax: (574) 631-8883 www.nd.edu/~cschuste
