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st: RE: RE: RE: RE: PCA and rotation


From   "Verkuilen, Jay" <JVerkuilen@gc.cuny.edu>
To   "'statalist@hsphsun2.harvard.edu'" <statalist@hsphsun2.harvard.edu>
Subject   st: RE: RE: RE: RE: PCA and rotation
Date   Tue, 22 Dec 2009 14:25:12 -0500

A bit more on this:

I suspect part of the confusion is that PCA is often a semi-decent approximation to the common factor model (which does have a rotational indeterminacy), when the factor reliabilities are high. Older methods of factor analysis such as Principal Axis were, essentially, PCA applied to a modified correlation matrix with the diagonal shrunk by replacing the 1s with communalities (often estimated as 1 - R^2, with the R^2 being from the regression of a given variable on all others). "Modern" factor analysis methods such as ML, GLS or ULS/MINRES are estimated in a rather different way and are created to make the main diagonal fit perfectly, which in turn implies that only information from the off-diagonals (covariances or correlations) contributes to the solution. Of course, nearly everything comes down to relationships among the spectral decomposition of the matrix being fit, but the different fitting functions consider different aspects of that information. (Recent research sugge!
 sts that ULS/MINRES seems to be the best option for most problems. ML's optimality means if fits better on the data you have, but it generally has poor out of sample performance.) However, the various issues with *orthogonal* rotation still hold, in particular the fact that orthogonality is empirically unlikely between meaningful variables. It might hold in one sample but is unlikely to do so upon replication. 

All that said, PCA is often a superb tool for making low dimensional pictures from a multivariate dataset and also as a preprocessor for other data reduction tools. It's also a lot less brittle than the common factor model (not estimating anything helps with that!) and can often be fruitfully applied to small datasets that would cause factor analysis to blow up. So don't take my criticism of it above as being total---it's really useful, it's just not a common factor model, even if some of my colleagues seem to think so. 



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