I am doing a principal components analysis on a 4 item psychological scale (response format is a 0-10 point Likert scale for each item). As I hoped, an exploratory factor analysis yielded one factor. The result are looking good, one factor explains 83% of variance, the correlation coefficients are high but not too high (0.7-0.8), the KMO and Bartlett's tests are all fine.

My problem is that 4 residuals (66%) of the reproduced correlations are over an absolute value of 0.05. I know this is a problem, but am trying to establish how much of problem this is to the model. Also what (if anything) can be done to fix it? Any answers or advice on where to find answers would be appreciated.

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    $\begingroup$ Nice question, I bet it would help to state your sample size to interpret the residuals for the reproduced correlations. In smaller samples we might expect such deviations with more regularity. $\endgroup$ – Andy W Aug 15 '12 at 13:08
  • $\begingroup$ The sample size for the analysis described above is 218 subjects. I have done a second PCA using on an alternate language version of our scale with almost 2000 participants and the same problem remains although the residuals are somewhat smaller. $\endgroup$ – user13313 Aug 16 '12 at 10:17

When you speak of reproduced correlations in factor analysis it is very important to remember the difference between factor analysis in proper sense (FA) and principal component analysis (PCA). FA aims to reproduce correlations (or covariances) by means of m latent variables (m<p, where p is the number of variables). PCA, on the other hand, aims to account for multivariate variance as much as possible by m its latent variables; it does not pursue to explain correlations, and often m components reproduce them poorly (all p components reproduce them perfectly though). So there's no wonder if your single component fails to reproduce correlations precisely enough. Try FA instead of PCA, to see, and try to extract 2 factors in place of one, too.


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