A discrete character evolution model for phylogenetic comparative biology with Γ-distributed rate heterogeneity among branches of the tree

Phylogenetic comparative methods are now widely used to measure trait evolution on the tree of life. Often these methods involve fitting an explicit model of character evolution to trait data and then comparing the explanatory power of this model to alternative scenarios. In this article, we present a new model for discrete trait evolution in which the rate of character change in the tree varies from edge (i.e., “branch”) to edge of the phylogeny according to a discretized Γ distribution. When the edge-wise rates of evolution are, in fact, Γ-distributed, we show via simulation that this model can be used to reliably estimate the shape parameter ( α ) of the distribution of rate variation among edges. We also describe how our model can be employed in ancestral state reconstruction, and demonstrate via simulation how doing so will tend to increase the accuracy of our estimated states when the generating edge rates are Γ-distributed. We discuss how marginal edge rates are estimated under the model, and apply our method to a real dataset of digit number in squamate reptiles, modified from Brandley et al. (2008).