Quantifying and Comparing Phylogenetic Evolutionary Rates for Shape and Other High-Dimensional Phenotypic Data

Many questions in evolutionary biology require the quantification and comparison of rates of phenotypic evolution. Recently, phylogenetic comparative methods have been developed for comparing evolutionary rates on a phylogeny for single, univariate traits (σ²), and evolutionary rate matrices (R) for sets of traits treated simultaneously. However, high-dimensional traits like shape remain under-examined with this framework, because methods suited for such data have not been fully developed. In this article, I describe a method to quantify phylogenetic evolutionary rates for high-dimensional multivariate data $\left( {\sigma _{mult}^2} \right)$, found from the equivalency between statistical methods based on covariance matrices and those based on distance matrices (R-mode and Q-mode methods). I then use simulations to evaluate the statistical performance of hypothesis-testing procedures that compare $\sigma _{mult}^1$ for two or more groups of species on a phylogeny. Under both isotropic and non-isotropic conditions, and for differing numbers of trait dimensions, the proposed method displays appropriate Type I error and high statistical power for detecting known differences in $\sigma _{mult}^1$ among groups. In contrast, the Type I error rate of likelihood tests based on the evolutionary rate matrix (R) increases as the number of trait dimensions (p) increases, and becomes unacceptably large when only a few trait dimensions are considered. Further, likelihood tests based on R cannot be computed when the number of trait dimensions equals or exceeds the number of taxa in the phylogeny (i.e., when p> N). These results demonstrate that tests based on $\sigma _{mult}^1$ provide a useful means of comparing evolutionary rates for high-dimensional data that are otherwise not analytically accessible to methods based on the evolutionary rate matrix. This advance thus expands the phylogenetic comparative toolkit for high-dimensional phenotypic traits like shape. Finally, I illustrate the utility of the new approach by evaluating rates of head shape evolution in a lineage of Plethodon salamanders.