Bayesian bivariate meta-analysis of correlated effects: impact of the prior distributions on the between-study correlation, borrowing of strength, and joint inferences

Multivariate random-effects meta-analysis allows the joint synthesis of correlated results from multiple studies, for example for multiple outcomes or multiple treatment groups. In a Bayesian univariate meta-analysis of one endpoint, the importance of specifying a sensible prior distribution for the between-study variance is well-understood. However, in multivariate meta-analysis there is little guidance about the choice of prior distributions for the variances or, crucially, the between-study correlation, ρB; for the latter, researchers often use a Uniform(-1,1) distribution assuming it is vague. In this article, an extensive simulation study and a real illustrative example is used to examine the impact of various (realistically) vague prior distributions for ρB and the between-study variances within a Bayesian bivariate random-effects meta-analysis of two correlated treatment effects. A range of diverse scenarios are considered, including complete and missing data, to examine the impact of the prior distributions on posterior results (for treatment effect and between-study correlation), amount of borrowing of strength, and joint predictive distributions of treatment effectiveness in new studies. Two key recommendations are identified to improve the robustness of multivariate meta-analysis results. Firstly, the routine use of a Uniform(-1,1) prior distribution for ρB should be avoided, if possible, as it is not necessarily vague. Instead, researchers should identify a sensible prior distribution, for example by restricting values to be positive or negative as indicated by prior knowledge. Secondly, it remains critical to use sensible (e.g. empirically-based) prior distributions for the between-study variances, as an inappropriate choice can adversely impact the posterior distribution for ρB, which may then adversely affect inferences such as joint predictive probabilities. These recommendations are especially important with a small number of studies and missing data.