The multisample block-diagonal equicorrelation and equivariance test

The equicorrelation and equivariance test, also known as compound symmetry, or intraclass correlation test, was introduced in [9] and is of great importance in different fields in multivariate statistics like inAnalysisof Variance, ProfileAnalysisand Growth Curveanalysis.In this paper we consider an extension of this test based on the composition of three tests; the equality of covariance matrices test, the independence of several groups of variables test and the equicorrelation and equivariance test. Our objective is to derive a procedure that allows us to test whether in different populations we have equal covariance matrices all with a block-diagonal equicorrelation and equivariance structure, i.e. a block-diagonal matrix where each diagonal block has an equicorrelation and equivariance structure. We designate this test by the multisample block-diagonal equicorrelation and equivariance test. Taking this test as the composition of the three tests mentioned above we show that it is possible to obtain the likelihood ratio test statistic, the expression of its null moments and thecharacteristicfunction of its logarithm. This approach also allows us to write thecharacteristicfunction of the logarithm of likelihood ratio test statistic in a way that enables the development of new and highly accurate near-exact distributions for that statistic. These distributions have been applied with considerable success to various test statistics used inmultivariate analysis.Furthermore they are easy to implement computationally and will allow us to carry out the test with a high precision.