# Testing significant difference of correlation matrices

I'm interested in exploring whether responses to a series of attitudinal questions differ across various countries. I could check significant difference in mean scores but am more interested in the difference between the relationships (i.e. correlations) between attitudes, if there happens to be any noticeable difference in the correlation structure across the different countries.

I was wondering if there were any formal test(s) for this?

One unorthodox (or perhaps orthodox, I don’t know) idea I had was to take the correlations for each country and then calculate the sum of squared difference between all the correlations. A small sum of squared difference value would suggest high similarity between the two correlation matrices. Conversely a high sum of squared value would suggest differing correlation values between the two countries.

This could be used as a yard stick but could this sum of squared difference be significant tested also (after normalizing for size of correlation matrix)? Or is this completely wrong to do...

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• You can do this in a sem framework. I will try to write an answer later if no one else does. – Jeremy Miles Apr 24 '15 at 14:59
• Perhaps you should think in terms of distributions of the correlation matrices. You could try to derive an expression how a difference between two correlation matrices should look under the null hypothesis of no difference between the matrices in population. One keyword is Wishart distribution. – Richard Hardy Apr 24 '15 at 18:29

I'm looking at the same issue - I just came across the functions cortest.normal and cortest.jennrich, in the excellent psych package for R by William Revelle (see http://www.personality-project.org/r/html/cortest.mat.html). That page also contains references to articles on these tests. In you (our) case, the Jennrich test seems most appropriate (a quick scan of the Steiger paper (see http://ww.w.statpower.net/Steiger%20Biblio/Steiger80b.pdf) gives me the impression that that test, cortest.normal, also tests whether all correlations are zero - but I haven't looked into it more thoroughly yet).