st: PCA and rotation

["Michael I. Lichter" <mlichter@buffalo.edu>]

I recently found that when I extracted components using -pca-, rotated 
them using an orthogonal rotation (e.g., -rotate, varimax-), and scored 
them using -predict-, the correlations between what I presumed were 
uncorrelated factors were actually as high as 0.6. I know that component 
scores may be correlated, but this seemed a bit much. Somebody else 
noted the same thing a few months ago 
(http://www.stata.com/statalist/archive/2009-08/msg00793.html). On the 
other hand, I found that factor scores (produced with -factor, pcf-) for 
the same data remained virtually uncorrelated after orthogonal rotation.

I therefore assumed that the behavior of rotated PCs was a bug. I 
contacted Stata. Isabel Canette told me that I was mistaken. She 
referred me to "Methods of Multivariate Analysis" by A. Rencher, Second 
Edition,Wiley, 2002, page 403, where Rencher says:

 "...If the resulting components do not have satisfactory interpretation,
  they can be further rotated, seeking dimensions in which many of the
 coefficients of the linear combination and near zero to simplify 
 interpretation. 
   However, the new rotated components are correlated, and they do not
 successively account for maximum variance. They are, therefore, no 
 longer principal components in the usual sense, and their routine use 
 is questionable".

In other words, it's not a bug, it's ... something else. Isabel said 
that for this reason Stata discourages the use of rotation after -pca-.

What's odd is that I've seen a number of articles that use varimax 
rotations  (with Kaiser normalization) of principal components in scale 
development. The authors only use the PCA to guide scale development; 
they perform further analysis with Cronbach's alpha and create summative 
scales rather than using factor scores. Still, their interpretation of 
the components are based on rotated component loadings that, at least 
from Rencher's perspective, are "questionable".

--
Michael I. Lichter, Ph.D. <mlichter@buffalo.edu>
Research Assistant Professor & NRSA Fellow
UB Department of Family Medicine / Primary Care Research Institute
UB Clinical Center, 462 Grider Street, Buffalo, NY 14215
Office: CC 126 / Phone: 716-898-4751 / FAX: 716-898-3536

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