Multivariate comparison of variance in R

Abstract

Abstract Comparisons of pattern and magnitude of phenotypic variation are central to many studies in evolution and ecology, but a meaningful comparison of multivariate variance patterns can be challenging. Here, we review an effective exploratory strategy, relative principal component analysis (relative PCA), for the comparison of variance–covariance matrices based on their relative eigenvalues and eigenvectors. Relative PCA allows for the identification of multivariate traits (linear combinations of variables) with maximal or minimal variance ratios between two groups. It can be used to explore the generation and canalization of phenotypic variance throughout ontogeny and phylogeny. Relative PCA also gives rise to a natural metric for the ordination of a sample of covariance matrices. We present a novel biometric justification of these approaches and discuss numerical difficulties as well as strategies for statistical inference, along with a new implementation of these methods in R. In an application of relative PCA to geometric morphometric data on cichlid body shape, we found that the phenotypic variance–covariance structure differs between males and females as well as between allopatric and sympatric populations of Tropheus moorii. Divergent selection in these populations mainly affected shape features related to swimming ability, whereas tropic morphology appears to be under stabilizing selection. In biology and biomedicine, individual variation is a key signal. Standard ordination methods as well as scalar summary statistics of multivariate variation often pool different biological factors with heterogeneous variational dynamics and thus conceal differences in variance–covariance pattern among groups. Relative PCA complements existing tests for the proportionality of covariance matrices and is an effective exploratory method to identify multivariate traits that differ in variance between populations, age groups, or treatment groups. By comparing within‐ and between‐population covariance matrices, relative PCA can reveal traits under divergent or stabilizing selection.