Can i use the results from a PCA if the matrix is 'not positive definite'

#1
Hi,

I have a 'not positive definite' correlation matrix having done a principal component analysis (PCA) on SPSS. The data i have used is from a questionnaire i did using a 7 point likert type scale. There were 36 questions (36 variables) i got 16 responses (n=16). The questionnaire was very specific and not many people are qualified to answer it thats why this number is so low. I know that this sample size is not enough to accurately compute a PCA for the number of variables present and is most likely to be the cause of the 'not positive definite' but there is no more time to try do another round or anyone else to ask.

I set out with the intention to use this procedure and made the questionnaire accordingly and written the methodology of my dissertation so am very hesitant to change as my deadline is soon. I do have results from this computation and would like to present the results even if i was to say that these results are tentative or something similar. The question(s) i want to ask is:

Could i keep the results i have even if the matrix is 'not positive definite'?

What does 'not positive definite mean'? What would be my other options ? Could this be fixed somehow?

I don't know what other procedure i can use, I've read a little about latent component analysis but still not sure if that is correct for what I am trying to do and is not available on spss.

Thank you in advance
 

spunky

King of all Drama
#2
I know that this sample size is not enough to accurately compute a PCA for the number of variables present and is most likely to be the cause of the 'not positive definite'
Most likely.

Could i keep the results i have even if the matrix is 'not positive definite'?
I would not advise that. Covariance matrices are, by definition, positive (semi-) definite matrices and for the purposes you want to use PCA (which I'm assuming is some sort of substitute for Factor Analysis) you need that property. Otherwise your results are quite suspect.


What does 'not positive definite mean'?
One or more eigenvalues are negative and for true covariance matrices they are always positive (or 0, but that's another issue).

What would be my other options ?
Not use PCA comes to mind. Perhaps doing more basic analyses using Classical Test Theory and then simply acknowledge you don't have the sample size to do anything else.

Could this be fixed somehow?
Yes. But nothing within SPSS will let you do that. You'd need to use a syntax-based statistical language (R, SAS, STATA, etc.) and learn some extra statistical methods so for your own and specific practical purposes I'd say no.