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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.RECENTLY, there has been considerable interest in measuring the magnitude of effects in analysis of variance designs, as well as their significance. A simple measure for this purpose is the correlation ratio, which is usually designated ~2. In this paper, 1}2 will be used to refer to a population value, while the sample estimates will be designated cr. The different measures considered in this paper will be presented in terms of their sample estimates; the corresponding population values should be clear. Just as there are a variety of plausible generalizations of univariate analysis of variance techniques for measuring significance to the multivariate situation, there appears to be no single obvious way to generalize measures of magnitude. Yet, such measures should be as useful in the multivariate situation as in the univariate situation.

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