Bayesian heterogeneity in a meta–analysis with two studies and binary data

The meta–analysis of two trials is valuable in many practical situations, such as studies of rare and/or orphan diseases focussed on a single intervention. In this context, additional concerns, like small sample size and/or heterogeneity in the results obtained, might make standard frequentist and Bayesian techniques inappropriate. In a meta–analysis, moreover, the presence of between–sample heterogeneity adds model uncertainty, which must be taken into consideration when drawing inferences. We suggest that the most appropriate way to measure this heterogeneity is by clustering the samples and then determining the posterior probability of the cluster models. The meta–inference is obtained as a mixture of all the meta–inferences for the cluster models, where the mixing distribution is the posterior model probability. We present a simple two–component form of Bayesian model averaging that is unaffected by characteristics such as small study size or zero–cell counts, and which is capable of incorporating uncertainties into the estimation process. Illustrative examples are given and analysed, using real sparse binomial data.