A General Method for Simultaneously Accounting for Phylogenetic and Species Sampling Uncertainty via Rubin's Rules in Comparative Analysis.

Phylogenetic comparative methods (PCMs), especially ones based on linear models, have played a central role in understanding species' trait evolution. These methods, however, usually assume that phylogenetic trees are known without error or uncertainty, but this assumption is most likely incorrect. So far, Markov chain Monte Carlo (MCMC)-based Bayesian methods have mainly been deployed to account for such "phylogenetic uncertainty" in PCMs. Herein, we propose an approach with which phylogenetic uncertainty is incorporated in a simple, readily implementable and reliable manner. Our approach uses Rubin's rules, which are an integral part of a standard multiple imputation procedure, often employed to recover missing data. We see true phylogenetic trees as missing data under this approach. Further, unmeasured species in comparative data (i.e., missing trait data) can be seen as another source of uncertainty in PCMs because arbitrary sampling of species in a given taxon or "species sampling uncertainty" can affect estimation in PCMs. Using two simulation studies, we show our method can account for phylogenetic uncertainty under many different scenarios (e.g., uncertainty in topology and branch lengths) and, at the same time, it can handle missing trait data (i.e., species sampling uncertainty). A unique property of the multiple imputation procedure is that an index, named "relative efficiency," could be used to quantify the number of trees required for incorporating phylogenetic uncertainty. Thus, by using the relative efficiency, we show the required tree number is surprisingly small ($\sim$50 trees). However, the most notable advantage of our method is that it could be combined seamlessly with PCMs that utilize multiple imputation to handle simultaneously phylogenetic uncertainty (i.e., missing true trees) and species sampling uncertainty (i.e., missing trait data) in PCMs.