On the covariance of phylogenetic quantitative trait evolution models and their matrix condition

Abstract

Phylogenetic comparative methods (PCMs) use phylogenetic tree and trait data to explore the evolutionary information among a group of related species. Given a phylogenetic tree of extant species, the evolutionary evidence among a group of species can be represented by a squared matrix C obtained by an isomorphic transformation using the shared branch lengths in the tree. The quantitative trait evolution for species along the branch can be described by stochastic processes. Currently, most statistical models for trait evolution are built under the assumption of Gaussian processes, where the trait vector Y follows a multivariate normal distribution with covariance matrix V , transformed from the C matrix and model parameters. This study investigates the effects on parameter estimation in phylogenetic comparative methods that are caused by ill-conditioned phylogenetic tree matrices ( C matrix) and their associated model-adjusted variance–covariance matrices ( V matrix). Several popular models—the Brownian motion model, the Ornstein–Uhlenbeck model, the early burst model, and the phylogenetic mixed model (Pagel’s λ)—are evaluated and compared through an empirical dataset and extensive simulations. Suggestions are provided to users in the community for identifying and overcoming potential issues with their own data.