Testing for heterogeneous rates of discrete character evolution on phylogenies

Many hypotheses in the field of phylogenetic comparative biology involve specific changes in the rate or process of trait evolution. This is particularly true of approaches designed to connect macroevolutionary pattern to microevolutionary process. We present a method to test whether the rate of evolution of a discrete character has changed in one or more clades, lineages, or time periods. This method differs from other related approaches (such as the "covarion" model) in that the "regimes" in which the rate or process is postulated to have changed are specified a priori by the user, rather than inferred from the data. Similarly, it differs from methods designed to model a correlation between two binary traits in that the regimes mapped onto the tree are fixed. We apply our method to investigate the rate of dewlap colour and/or caudal vertebra number evolution in Caribbean and mainland clades of the diverse lizard genus Anolis. We find little evidence to support any difference in the evolutionary process between mainland and island evolution for either character. We also examine the statistical properties of the method more generally and show that it has acceptable type I error, parameter estimation, and power. Finally, we discuss some general issues of frequentist hypothesis testing and model adequacy, as well as the relationship of our method to existing models of heterogeneity in the rate of discrete character evolution on phylogenies. Graphical Abstract