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st: PCA and rotation


From   "Michael I. Lichter" <mlichter@buffalo.edu>
To   statalist@hsphsun2.harvard.edu
Subject   st: PCA and rotation
Date   Mon, 21 Dec 2009 16:33:37 -0500

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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