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RE: st: RE: Aren't distinct factors from factor analysis or PCA orthogonal to each other?


From   "Verkuilen, Jay" <JVerkuilen@gc.cuny.edu>
To   "'statalist@hsphsun2.harvard.edu'" <statalist@hsphsun2.harvard.edu>
Subject   RE: st: RE: Aren't distinct factors from factor analysis or PCA orthogonal to each other?
Date   Wed, 19 Aug 2009 13:39:32 -0400

Nick Cox wrote:

>I guess Cameron does not mean quite what he says, which is that factor
analysis can only be used on psychometric measures. In principle I can
readily imagine fruitful applications on quite different kinds of data.<

It can be used on many sets of data but I think that the *model* that underlies factor analysis should reasonably apply, for which see Edwards, J.R., Bagozzi, R.P. 2000. On the nature and direction of relationships between constructs and measures. Psych. Methods, 5, 155-174. 

In a nutshell, the regression model

     Y_{ij} = \mu_{j} + \sum_{k=1}^{K} \lambda_k F_{ik} + E_{ij}

where the F_{ik} term is a missing independent variable for subject i on the kth factor. Once these variables are conditioned on, the Y should be conditionally independent. (Apologies for the LaTeX but it's the clearest way to write an equation in ASCII.) 

This was called the "congeneric test model" by Karl Joreskog. It applies when the indicators are considered exchangeable, in the sense that, say, items on a math test are. Thus it makes perfect sense for a situation of different error laden physical measurements but not when the measures essentially define the construct as, say, the items on the Apgar scale used to assess a newborn. 


>>But I largely agree with the spirit of his comment, which I take to be
-- my words not his -- that expecting factor analysis to see structure
in a mess independently of some understanding is likely to be expecting
far too much. However, my impression is that is exactly what almost all
users of factor analysis seem to expect! <<

Alas the "old skool" factor analysis that many people do is like this. As my SEM professor Rod McDonald liked to say, "exploratory factor analysis is much better done *after* confirmatory factor analysis, to make sure you didn't miss anything." This is easily illustrated by the fact that students pick up CFA very readily and EFA becomes an afterthought, while EFA first makes no sense whatsoever. Far too many people using factor analysis (a) aren't (because they're actually using PCA!) and (b) have no earthly idea what they are doing either way. 

Old skool make good R&B, but it's terrible here.

Jay

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