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January 2023

Correlation Between Two Vector Variables

Journal/Book: Reprinted from THE JOURNAL OF THE ROYAL STATISTICAL SOCIETY SERIES B (METHODOLOGICAL) Volume 31 No. 3 1969 (pp. 477-485). 1969;

Abstract: Southern Methodist University Dallas Texas [Received February 1969. Revised June 1969] SUMMARY Ruben (1966) has suggested a simple approximate normalization for the correlation coefficient in normal samples by representing it as the ratio of a linear combination of a standard normal variable and a X variable to an independent X variable and then using Fisher's approximation to a X variable. This result is extended in this paper to a matrix which in a sense is the correlation coefficient between two vector variables x and y. The result is then used to obtain large sample null and non-null (but in the linear case) distributions of the Hotelling-Lawley criterion and the Pillai criterion in multivariate analysis. Williams (1955) and Bartlett (1951) have derived some exact tests for the goodness of fit of a single hypothetical function to bring out adequately the entire relationship between two vectors x and y by factorizing Wilks's suitably. These factors are known as "direction" and "collinearity" factors as they refer to the direction and collinearity aspects of the null hypothesis. In this paper the other two criteria viz. the Hotelling-Lawley and Pillai criteria are partitioned into direction and collinearity parts and large sample tests corresponding to them are derived for testing the goodness of fit of an assigned function. schö

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