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Re: st: RE: PCA vs. Factor Loadings


From   Michael Lichter <lichter@uclalumni.net>
To   statalist@hsphsun2.harvard.edu
Subject   Re: st: RE: PCA vs. Factor Loadings
Date   Thu, 17 Dec 2009 09:56:18 -0500

Thanks again, Nick. I've downloaded -cpcorr-. It will be helpful for 
this project and elsewhere.

I now have a different but related problem that could be a bug but could 
also represent a gap in my understanding of PCA (which would not be a 
surprise).

Checking the correlations between the observed variables and the 
extracted components requires the estimation of component scores using 
-predict-. Once the component scores are estimated, it's easy to check 
the correlation between components. There should be no correlation 
between the initially-extracted components, and none between components 
after orthogonal rotation. I found the former but not the latter; in 
fact, my rotated components are all moderately correlated! The 
correlations between components for a three-component model extracted 
from the auto.dta data (program at end of this message) look like this:

             |       c1       c2       c3
-------------+---------------------------
          c1 |   1.0000
          c2 |   0.6248   1.0000
          c3 |  -0.4571  -0.4338   1.0000

What's going on? When I the same (give or take) factors using -factor, 
pcf- and do the same orthogonal (varimax) rotation, the rotated factors 
are, as expected, uncorrelated.

Does this indicate a bug in one or more of the relevant programs (-pca-, 
-rotate-, or -predict-), or is there some aspect of PCA that I'm not 
taking into account?

Thanks.

Re. Jay's response: Thanks to you also. I wasn't suggesting that the 
results of -pca- and -factor- should look the same; I am/was only 
looking to have results from the two be on comparable scales, which, as 
both of you observed, is a function of the normalization used.

Michael


// ----- program to
//     - extract components/factors
//     - show correlations between vars and components/factors
//     - show correlations between components/factors
//     - compare -pca- results to -factor- results

sysuse auto

pca price mpg rep78 headroom trunk weight length displacement, mineigen(1)
predict c1 c2 c3
cpcorr price mpg rep78 headroom trunk weight length displacement \ c1-c3
corr c1-c3
rotate, varimax
drop c1-c3
predict c1 c2 c3
cpcorr price mpg rep78 headroom trunk weight length displacement \ c1-c3
corr c1-c3

factor price mpg rep78 headroom trunk weight length displacement, pcf
predict f1 f2 f3
cpcorr price mpg rep78 headroom trunk weight length displacement \ f1-f3
corr f1-f3
rotate, varimax
drop f1-f3
predict f1 f2 f3
cpcorr price mpg rep78 headroom trunk weight length displacement \ f1-f3
corr f1-f3


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