Estimation of Bayes Factors in a Class of Hierarchical Random Effects Models Using a Geometrically Ergodic MCMC Algorithm

Abstract

We consider a Ba yesian random effects model that is commonly used in meta-analysis, in which the random effects have a t distribution, with degrees of freedom parameter to be estimated. We develop a Markov chain Monte Carlo algorithm for estimating the posterior distribution in this model, and establish geometric convergence of the algorithm. The geometric convergence rate has important theoretical and practical ramifications. Indeed, it implies that, under standard moment conditions, the ergodic averages used to estimate posterior quantities of interest satisfy central limit theorems. Moreover, it guarantees the consistency of a batch means estimate of the asymptotic variance in the CLT, which in turn allows for the construction of asymptotically valid standard errors. We show how our Markov chain can be used, in conjunction with an importance sampling method, to carry out an empirical Bayes approach for estimating the degrees of freedom parameter. To illustrate our methodology we consider a meta-analysis of studies that link intake of nonsteroidal anti-inflammatory drugs to a reduction in colon cancer risk, in which some of the studies are outliers. To model the distribution of the study effects we consider the family of t distributions, as well as a family of mixtures of Dirichlet process priors centered at the t distributions, and show how our methodology can be used to make a choice of model. Supplemental materials are available online.