A Closed Testing Procedure for Canonical Correlations

The problem of deciding on dimensionality in the canonical correlation model is considered. It is related to the investigation of significant empirical departures from the overall hypothesis on the nullity of the population canonical correlations. A closed testing procedure for a sequence of relevant dimensionality hypotheses is proposed, following an earlier suggestion made for testing dimensionality in a MANOVA model. Unlike the classical procedures based on asymptotic distributions, the proposed method ensures that the Type I familywise error rate does not exceed the nominal α-level. In the derivation of the procedure, the problem is considered in the framework of a multivariate linear regression model, which allows the hypotheses on canonical correlations to be treated as linear hypotheses under such a model. Two examples are given to illustrate application of the proposed procedure and to compare it with some other methods.