Increasing the efficiency of MCMC for hierarchical phylogenetic models of categorical traits using reduced mixed models

Integrating out the random effects in generalised linear mixed models (GLMM) cannot be done analytically unless the response is Gaussian. Many stochastic, deterministic or hybrid algorithms have been developed to perform the integration. With categorical data and probit link (aka the threshold model), the random effect structure can be partitioned into a part that can be easily integrated deterministically (the R-structure) and a part that cannot (the G-structure). We show that in the context of phylogenetic mixed models, part of the G-structure (the phylogenetic effects at the tips) can be moved into the R-structure and integrated out deterministically. This result follows directly from the concept of the reduced animal model from quantitative genetics (Journal of Animal Science, 51, 1980, 1277) and its implications for discrete data (Genetics Selection Evolution, 42, 2010, 1). Although the conditional distribution of the phylogenetic variance is no longer in standard from, it does provide a stable and efficient 2-block MCMC algorithm for situations when the phylogenetic heritability is assumed to be one. We show that a GLMM with such an assumption is equivalent to the model proposed by Felsenstein (American Naturalist, 179, 2005, 145). Extensions to multivariate models are straightforward and a 3-block algorithm can be constructed when there is only a single categorical trait but multiple Gaussian traits. With ≥2 categorical traits, an additional non-Gibbs update is required for the correlation (sub)matrix. An implementation of these algorithms is distributed in the r package MCMCglmm and is up to several orders of magnitude faster than published alternatives.