Statistical Inference in Redundancy Analysis: A Direct Covariance Structure Modeling Approach

Abstract

Redundancy analysis (RA) is a multivariate method that maximizes the mean variance of a set of criterion variables explained by a small number of redundancy variates (i.e., linear combinations of a set of predictor variables). However, two challenges exist in RA. First, inferential information for the RA estimates might not be readily available. Second, the existing methods addressing the dimensionality problem in RA are limited for various reasons. To aid the applications of RA, we propose a direct covariance structure modeling approach to RA. The proposed approach (1) provides inferential information for the RA estimates, and (2) allows the researcher to use a simple yet practical criterion to address the dimensionality problem in RA. We illustrate our approach with an artificial example, validate some standard error estimates by simulations, and demonstrate our new criterion in a real example. Finally, we conclude with future research topics.

Keywords:

• Redundancy analysis
• standard error estimate
• covariance structure analysis

Correction Statement

This article has been corrected with minor changes. These changes do not impact the academic content of the article.

Notes

1 For p > q, define d = p − q as a positive integer. If d ≥ 2, the last d eigenvectors w_i (i = q + 1, …, p) cannot be uniquely determined. To identify these eigenvectors, one has to arbitrarily fix d(d − 1)/2 elements from the last d eigenvectors. For example, if p = 5 and q = 3, then d = p − q = 2. To identify the last 2 eigenvectors w₄ and w₅, one must fix 1 element from either w₄ or w₅.