A Multivariate Extension of the Correlation Ratio

A measure of the magnitude of the effect in a one-factor multivariate analysis of variance design is considered. Cooley and Lohnes have proposed the use of the quantity (1 — | W|/| T|) as a multivariate extension of the correlation ratio, where | W| is the determinant of the within-groups cross-products matrix and | T| is the determinant of the total cross-products matrix. The measure is based on the use of | W| as the estimate of a generalized measure of within-groups variation and | T| as the estimate of a generalized measure of total variation. If a multivariate correlation ratio is defined as the proportion of variance in the multivariate domain predictable from the factor, it is argued that cr M = 1 - Tr( WW -1 )/ Tr( TW -1 ) is a more suitable multivariate generalization of the univariate correlation ratio.