Phylogenetic modelling of compositional and exchange rate changes over time

Changes in the process of evolution occurs over time, including compositional tree heterogeneity (CTH) and exchange rate tree heterogeneity (ERTH). Models that can accommodate CTH and ERTH in molecular evolution are described. Fit of these models was compared using a likelihood ratio test in maximum likelihood, and in Bayesian analysis using the conditional predictive ordinate (CPO)-based log pseudomarginal likelihood (LPML), also leave-one-out cross-validation (LOO-CV). CTH and ERTH can be flexibly modelled in a Bayesian framework with tree-heterogeneous models that tune themselves to the amount of heterogeneity in the data being analysed. Since phylogenetic analysis is usually done using tree-homogeneous models, effects of CTH and ERTH on subsequent phylogenetic analysis using such models were described. Compositional effects due to CTH were seen as expected, for example where unrelated taxa with similar compositions would group together in homogeneous analysis. Similar effects were also demonstrated due to ERTH. Detection of CTH and ERTH by modelling is compared to detection using matched pairs tests (MPTs) that have been used to test molecular sequences for stationarity, reversibility, and homogeneity (SRH). Comparisons between modelling and MPTs on data simulated on very simple trees showed that the two approaches were equivalent, but simulations on larger trees showed that the two approaches differed greatly. Modelling showed greater power, especially in detection of ERTH, and some ERTH was completely invisible to MPTs but was decisively detected by modelling. Detection and modelling of CTH and ERTH is shown in two empirical examples.