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st: RE: PCA vs. Factor Loadings


From   "Verkuilen, Jay" <JVerkuilen@gc.cuny.edu>
To   "'statalist@hsphsun2.harvard.edu'" <statalist@hsphsun2.harvard.edu>
Subject   st: RE: PCA vs. Factor Loadings
Date   Wed, 16 Dec 2009 12:00:34 -0500

Michael I. Lichter wrote:

>>Hello. Why are component loadings from -pca- so much smaller than factor 
loadings from -factor-? Is there something about the procedure used by 
Stata that makes them systematically smaller? I get the sense (which may 
be mistaken; I don't have any evidence in my hand) that in other 
packages -pca- and -factor- loadings are more similar.<<

As Nick already pointed out, the normalization you choose will affect the coefficients you get. This is very much analogous to coding variables in regression. What is preserved by the procedure is the objective function's measure of discrepancy between the input matrix and the parameters. The resulting coefficients can end up looking VERY different for statistically equivalent models. (Yes, PCA isn't properly a statistical model but it's still the result of an optimization process, or could be thought of that way.) 

In general, comparing loadings from factor analysis and PCA is more than a bit dangerous, though it often works, roughly speaking, especially when the variables input into factor analysis have relatively small uniquenesses. 

JV

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