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 crM = 1 - Tr( WW -1)/ Tr( TW -1) is a more suitable multivariate generalization of the univariate correlation ratio.