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st: RE: Negative eigen values in factor, pf command?


From   "Verkuilen, Jay" <JVerkuilen@gc.cuny.edu>
To   <statalist@hsphsun2.harvard.edu>
Subject   st: RE: Negative eigen values in factor, pf command?
Date   Tue, 28 Apr 2009 12:39:35 -0400

Jean-Gael Collomb wrote:

<<I am having trouble interpreting the results of a principle factor  
analysis I am conducting. <snip> >>

This has nothing to do with Stata whatsoever and has to do with the
structure of the problem. Stata is honest about its reporting, unlike
far too many other stats programs, and generally doesn't hide issues
from you when there are problems. Anyway, confusing principal factor PF
with principal components is a common mistake, because the names are
similar. In fact, PF is PCA applied not to the correlation matrix, R,
but to 

     C = R - U

where U is an estimate of the variables' uniquenesses (unreliability, a
measure of the variables' error variances). The usual estimate used in
this kind of a procedure is 

     U = diag(1 - rsq_jj), 

where rsq_jj is the multiple R-squared from the regression of variable j
on all other variables. R is guaranteed to be positive semi-definite. C
is not, and often isn't. Slight violations are no big deal. Substantial
ones, signaled by big negative eigenvalues, are a sign that the model
does not apply. 

In a sense you can think about removing the uniquenesses as the opposite
of ridging. Ridging adds positive value to the main diagonal of a matrix
relative to the off-diagonal to push it towards being positive definite.
This deemphasizes the off-diagonal. Because the goal of a factor
analysis is to analyze what the variables have in common, which is
measured by the covariances (or correlations), removing the influence of
the diagonal helps focus on this in a way that PCA does not. See, e.g.,
Chapter 5 in 

     Lattin, J., Carroll, J. D. and Green, P. (2003). Analyzing
Multivariate Data. Duxbury Press. 

(Aside: PF is an antiquated method that belongs on the dustbin of
history as far as I am concerned.)



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