A Bayesian extension of phylogenetic generalized least squares (PGLS): incorporating uncertainty in the comparative study of trait relationships and evolutionary rates

Phylogenetic comparative methods use tree topology, branch lengths, and
models of phenotypic change to take into account non-independence in
statistical analysis. However, these methods normally assume that trees
and models are known without error. Approaches relying on evolutionary
regimes also assume specific distributions of character states across a
tree, which often result from ancestral state reconstructions that are
subject to uncertainty. Several methods have been proposed to deal with
some of these sources of uncertainty, but approaches accounting for all of
them are less common. Here we show how Bayesian statistics facilitates
this task while relaxing the homogeneous rate assumption of the well-known
phylogenetic generalized least squares (PGLS) framework.  This
Bayesian formulation allows uncertainty about phylogeny, evolutionary
regimes, or other statistical parameters to be taken into account for
studies as simple as testing for coevolution in two traits or as complex
as testing whether bursts of phenotypic change are associated with
evolutionary shifts in inter-trait correlations. A mixture of validation
approaches indicate that the approach has good inferential properties and
predictive performance. We provide suggestions for implementation and show
its usefulness by exploring the coevolution of ankle posture and forefoot
proportions in Carnivora.