Fast and Robust Inference of Phylogenetic Ornstein-Uhlenbeck Models Using Parallel Likelihood Calculation

A bstract Phylogenetic comparative methods have been used to model trait evolution, to test selection versus neutral hypotheses, to estimate optimal trait-values, and to quantify the rate of adaptation towards these optima. Several authors have proposed algorithms calculating the likelihood for trait evolution models, such as the Ornstein-Uhlenbeck (OU) process, in time proportional to the number of tips in the tree. Combined with gradient-based optimization, these algorithms enable maximum likelihood (ML) inference within seconds, even for trees exceeding 10,000 tips. Despite its useful statistical properties, ML has been criticised for being a point estimator prone to getting stuck in local optima. As an elegant alternative, Bayesian inference explores the entire information in the data and compares it to prior knowledge but, usually, runs in much longer time, even for small trees. Here, we propose an approach to use the full potential of ML and Bayesian inference, while keeping the runtime within minutes. Our approach combines (i) a new algorithm for parallel likelihood calculation; (ii) a previously published method for adaptive Metropolis sampling. In principle, the strategy of (i) and (ii) can be applied to any likelihood calculation on a tree which proceeds in a pruning-like fashion leading to enormous speed improvements. As a showcase, we implement the phylogenetic Ornstein-Uhlenbeck mixed model (POUMM) in the form of an easy-to-use and highly configurable R-package. In addition to the above-mentioned usage of comparative methods, the POUMM allows to estimate non-heritable variance and phylogenetic heritability. Using simulations and empirical data from 487 mammal species, we show that the POUMM is far more reliable in terms of unbiased estimates and false positive rate for stabilizing selection, compared to its alternative - the non-mixed Ornstein-Uhlenbeck model, which assumes a fully heritable and perfectly measurable trait. Further, our analysis reveals that the phylogenetic mixed model (PMM), which assumes neutral evolution (Brownian motion) can be a very unstable estimator of phylogenetic heritability, even if the Brownian motion assumption is only weakly violated. Our results prove the need for a simultaneous account for selection and non-heritable variance in phylogenetic evolutionary models and challenge stabilizing selection hypotheses stated in numerous macro-evolutionary studies.