An Asymptotic χ²Test for the Equality of Two Correlation Matrices

An asymptotic χ2 test for the equality of two correlation matrices is derived. The key result is a simple representation for the inverse of the asymptotic covariance matrix of a sample correlation matrix. The test statistic has the form of a standard normal theory statistic for testing the equality of two covariance matrices with a correction term added. The applicability of asymptotic theory is demonstrated by two simulation studies and the statistic is used to test the difference in the factor patterns resulting from a set of tests given to retarded and non-retarded children. Two related tests are presented: a test for a specified correlation matrix and a test for equality of correlation matrices in two or more populations.