Bayesian multivariate meta-analysis by using the Birge ratio method

In the paper, we develop Bayesian inference procedures for the model parameters of the multivariate location-scale model connected to the multivariate Birge ratio method, a novel approach for pooling multivariate measurements together which extends the widely-used univariate Birge ratio method. In particular, the expressions of the joint posterior, the marginal posterior and the conditional posterior distributions are derived. These findings lead to the introduction of the Metropolis–Hastings algorithm and the Gibbs sampler approach for drawing samples from the joint posterior distribution and for conducting Bayesian inference procedures based on the simulated samples. The theoretical findings of the paper are implemented in an empirical illustration by studying the effectiveness of the hypertension treatment. It is found that the anti-hypertension drugs lead to the statistically significant reduction of the systolic and diastolic blood pressure as well as to the reduction of the risk of cardiovascular disease and stroke.