Maximum eccentricity as a union-intersection test statistic in multivariate analysis

Union-intersection is a heuristic method of test construction developed by S. N. Roy. Among the well-known applications of this principle is the test for independence between two sets of variates which leads directly to the concept of canonical correlation. Another multivariate application of considerable importance is the test of internal independence. In this article we consider the structure of a correlation matrix and derive a union-intersection test statistic for internal independence. This statistic will be shown to be a function of the maximum eccentricity of the p-dimensional correlation ellipsoid x'R-1x = 1. The statistic will be applied to problems in factor analysis and categorical scaling.