Comparing Covariance Matrices by Relative Eigenanalysis, with Applications to Organismal Biology

Most biologists are familiar with principal component analysis as an ordination tool for questions about within-group or between-group variation in systems of quantitative traits, and with multivariate analysis of variance as a tool for one useful description of the latter in the context of the former. Less familiar is the mathematical approach of relative eigenanalysis of which both of these are special cases: computing linear combinations for which two variance-covariance patterns have maximal ratios of variance. After reviewing this common algebraic-geometric core, we demonstrate the effectiveness of this exploratory approach in studies of developmental canalization and the identification of divergent and stabilizing selection. We further outline a strategy for statistical classification when group differences in variance dominate over differences in group averages.