Bayesian multivariate meta-analysis with multiple outcomes

There has been a recent growth in developments of multivariate meta-analysis. We extend the methodology of Bayesian multivariate meta-analysis to the situation when there are more than two outcomes of interest, which is underexplored in the current literature. Our objective is to meta-analyse summary data from multiple outcomes simultaneously, accounting for potential dependencies among the data. One common issue is that studies do not all report all of the outcomes of interests, and we take an approach relying on marginal modelling of only the reported data. We employ a separation prior for the between-study variance-covariance matrix, which offers an improvement on the conventional inverse-Wishart prior, showing robustness in estimation and flexibility in incorporating prior information. Particular challenges arise when the number of outcomes is large relative to the number of studies because the number of parameters in the variance-covariance matrix can become substantial and there can be very little information with which to estimate between-study correlation coefficients. We explore assumptions that reduce the number of parameters in this matrix, including assumptions of homogenous variances, homogenous correlations for certain outcomes and positive correlation coefficients. We illustrate the methods with an example data set from the Cochrane Database of Systematic Reviews.