A GENERALIZATION OF VECTOR CORRELATION AND ITS RELATION TO CANONICAL CORRELATION

A symmetric generalization of vector correlation is given which applies to matrices with different numbers of variables and with linear dependencies among the variables, and suggests that the vector correlation may be used as a measure of linear relationship between sets of variables.

Abstract

Chow (1966) has shown that the least-squares estimate of the regression coefficient matrix in multivariate linear regression maximizes the squared vector correlation coefficient between the dependent variables and a linear transformation of the independent variables. This paper shows that the problem is closely related to canonical correlation, and that the correlation involved is the product of the canonical correlations between the independent and dependent variables. The paper gives a symmetric generalization of vector correlation which applies to matrices with different numbers of variables and with linear dependencies among the variables. It is shown to be also related to canonical correlation, as well as to a measure of correlation between sets of variables proposed by Rozeboom (1965). This provides a test of significance for both measures and suggests that the vector correlation may be used as a measure of linear relationship between sets of variables.