STATISTICAL INFERENCE IN A CANONICAL CORRELATION ANALYSIS WITH LINEAR CONSTRAINTS

Abstract

This paper discusses the statistical inference problems seen in a canonical correlation analysis with linear constraints (CCALC), previously formulated by Yanai and Takane [15]. Linear constraints are imposed on the coefficient vectors of canonical variates, and canonical correlations with the constraints are not greater than the usual canonical correlations without the constraints. It is thus important to evaluate the effect of the linear constraints on the canonical correlations. We consider here in the test of the hypothesis that the canonical correlations are invariant by imposing the linear constraints. This testing problem can be treated as an additional information test in the usual (without any constraints) canonical correlation analysis (CCA) discussed by McKay [9] and Fujikoshi [4], and the likelihood ratio criterion and an asymptotic expansion formula of its null distribution are derived. An asymptotic distribution of the sample canonical correlation decrease by imposing the linear constraints is also obtained. Using this, an approximate confidence interval for the decrease can be constructed. Another problem discussed in this paper is the test of a hypothesis concerning a number of useful pairs canonical variates in CCALC, an important test in obtaining the number of useful canonical variates. The likelihood ratio criterion for this testing problem and an asymptotic expansion formula of its null distribution are obtained.