Further aspects of the tests of the equality of correlation matrices

A common problem in social, educational, behavioral and biological research is to investigate relationships between variables across several groups. Factor analysis is one such procedure with a long history of use. A more modern graphical procedure is the biplot of Gabriel (1971). Biplots result in a graphical display of correlation structure. We present results that extend an existing test of equality of correlation matrices. A new test statistic is proposed and is shown to be asymptotically distributed as a linear combination of independent $\chi\sp2$ random variables. This new formulation allows us to find the power of the existing test and our extensions by deriving the distribution under the alternative as that of a linear combination of independent non-central $\chi\sp2$ random variables. We also investigate the null and the alternative distributions of two related statistics. The first test statistic is defined as a quadratic form in deviations from a control group with the remaining K-1 groups to be compared. The second test is designed for comparing adjacent groups. In order to find a simple and accurate approximation for our statistics, we discuss several methods of approximating the distribution function of a definite quadratic form in normal variates. In particular, we investigate the two moment, the three moment, and the normal approximations. We also consider an approximation technique due to Imhof (1961). A Monte Carlo study is presented to assess the robustness of our procedure and the likelihood ratio test. We show that aside from the computational difficulties associated with the likelihood ratio test, this test is very sensitive to the assumption of multivariate normality. We also show that our test statistic performs well under non-normal distributions. To illustrate the techniques and assess the applicability of our procedure, we present several examples.