An analysis of correlation matrices: Equal correlations

Abstract

SUMMARY We study the large-sample joint distribution of Z, the -fp(p - 1) Fisher z-transforms of the elements in a p variable correlation matrix. Under the null hypothesis of equal population correlations the variance matrix of Z has just three projector matrices in its spectral decomposition. These define three mutually orthogonal invariant subspaces of sample space, or 'error strata' as they would be called in the analysis of variance. The squared lengths of the projections of the sample vector onto each of these subspaces, when divided by the stratum variance, provide a natural partition for the large-sample chi-squared test for equality of correlations. A linear model is given which provides a statistical interpretation for the error strata, and hence the components in the partition. As well as providing a simple test for equality of correlations, the procedure indicates how familiar techniques in the spirit of analysis of variance can be used to investigate correlation matrices.