Multi-response phylogenetic mixed models: concepts and application.

The scale and resolution of trait databases and molecular phylogenies is increasing rapidly. These resources permit many open questions in comparative biology to be addressed with the right statistical tools. Multi-response (MR) phylogenetic mixed models (PMMs) offer great potential for multivariate analyses of trait evolution. While flexible and powerful, these methods are not often employed by researchers in ecology and evolution, reflecting a specialised and technical literature that creates barriers to usage for many biologists. Here we present a practical and accessible guide to MR-PMMs. We begin with a review of single-response (SR) PMMs to introduce key concepts and outline the limitations of this approach for characterising patterns of trait coevolution. We emphasise MR-PMMs as a preferable approach for analyses involving multiple species traits, due to the explicit decomposition of trait covariances. We discuss multilevel models, multivariate models of evolution, and extensions to non-Gaussian response traits. We highlight techniques for causal inference using graphical models, as well as advanced topics including prior specification and latent factor models. Using simulated data and visual examples, we discuss interpretation, prediction, and model validation. We implement many of the techniques discussed in example analyses of plant functional traits to demonstrate the general utility of MR-PMMs in handling complex real-world data sets. Finally, we discuss the emerging synthesis of comparative techniques made possible by MR-PMMs, highlight strengths and weaknesses, and offer practical recommendations to analysts. To complement this material, we provide online tutorials including side-by-side model implementations in two popular R packages, MCMCglmm and brms.