An Informational Measure of Association and Dimension Reduction for Multiple Sets and Groups With Applications in Morphometric Analysis

Abstract

In this article we propose a new general index that measures relationships between multiple sets of random vectors. This index is based on Kullback–Leibler (KL) information, which measures linear or nonlinear dependence between multiple sets using joint and marginal densities of affine transformations of the random vectors. Estimates of the matrixes are obtained by maximizing the KL information and are shown to be consistent. The motivation for introducing such an index comes from morphological integration studies, a topic in biological science. As a special case of this index, we define an overall measure of association and two other measures for dimension reduction. The use of these measures is illustrated through real data analysis in morphometric studies and extensive simulations, and their performance is compared with that of approaches based on canonical correlation analysis. Extensions of the aforementioned measures to multiple groups are also discussed. In contrast to canonical correlation analysis, our general index not only provides an overall measure of association, but also determines nonlinear relationships, thereby making it useful in many other applications.