A Bayesian Semi-Parametric Model for Random Effects Meta-Analysis

In meta-analysis there is an increasing trend to explicitly acknowledge the presence of study variability through random effects models. That is, one assumes that for each study, there is a study-specific effect and one is observing an estimate of this latent variable. In a random effects model, one assumes that these study-specific effects come from some distribution, and one can estimate the parameters of this distribution, as well as the studyspecific effects themselves. This distribution is most often modelled through a parametric family, usually a family of normal distributions. The advantage of using a normal distribution is that the mean parameter plays an important role, and much of the focus is on determining whether or not this mean is 0. For example, it may be easier to justify funding further studies if it is determined that this mean is not 0. Typically, this normality assumption is made for the sake of convenience, rather than from some theoretical justification, and may not actually hold. We present a Bayesian model in which the distribution of the study-specific effects is modelled through a certain class of nonparametric priors. These priors can be designed to concentrate most of their mass around the family of normal distributions, but still allow for any other distribution. The priors involve a univariate parameter that plays the role of the mean parameter in the normal model, and they give rise to robust inference about this parameter. We present a Markov chain algorithm for estimating the posterior distributions under the model. Finally, we give two illustrations of the use of the model.