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Educational and Psychological Measurement
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A Multivariate Extension of the Correlation Ratio

Juliet Popper Shaffer

University of Kansas

Martin W. Gillo

University of Washington

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.

Educational and Psychological Measurement, Vol. 34, No. 3, 521-524 (1974)
DOI: 10.1177/001316447403400305
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