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

[khigbee@stata.com]

Jean-Gael Collomb <jg@ufl.edu> asks:

> I am having trouble interpreting the results of a principle factor  
> analysis I am conducting. The command and results are shown below.  
> Several things puzzle me about the results table. Why are some  
> eigenvalues < 0? Why are some of the proportions <0? Why are most of  
> the cumulative values >1. I suspect the answer to one of these  
> questions applies to all three. My understanding of factor analysis is  
> that I would interpret the results table as retaining all factor with  
> an eigen value >1 because they explain more of the variance than the  
> original variable and that the set of retained factors explains the  
> "cumulative" percent of the variance in the dataset. I thought that  
> all the variance (100%) would be explained by all the factors, but  
> that a subset of those factors would therefor only explain less than  
> 100%. In my case, I would retain factor 1 and by itself it would  
> explain 133% of the variance, which does not make much sense to me.  
> When I run a principle component analysis on the same data, I get a  
> two component solution explaining 52% of the variance. That result  
> table is more similar to what I have seen elsewhere, but I am puzzled  
> as to why there seems to be such a difference between procedures on  
> the same data (and the single factor solution of the pfa also makes  
> more theoretical sense as this point)
> 
> I am not a statistician but would like to understand in general terms  
> what is happening with the factor command and how to interpret its  
> results. I have spoken with two statisticians I work with and they are  
> surprised to see eigen values<0 and cumulative values >1, but they are  
> not STATA users. Maybe we are misinterpreting the results or maybe I  
> am doing something wrong with the software. If the results were not  
> valid, I would have expected STATA to give me some sort of error  
> message rather than an aberrant result.

Take a look at pages 421-423 of Rencher (2002), especially the
top half of page 423.  For the principal factor method you are
examining the eigenvalues of R - Psi_hat.  Rencher says "... are
not necessarily positive semidefinite and will often have some
small negative eigenvalues.  In such a case, the cumulative
proportion of variance will exceed 1 and then decline to 1 as the
negative eigenvalues are added."

If this property/behavior of the default -pf- option for -factor-
is not something you want, consider using one of the other method
options (such as -pcf-).

Rencher, A. C.  2002.  Methods of Multivariate Analysis. 2nd ed.
    New York: Wiley.


Ken Higbee    khigbee@stata.com
StataCorp     1-800-STATAPC

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