Testing the Overall Effect Using the Savage–Dickey Bayes Factor in Binary Random-Effects Meta-Analyses

Meta-analyses of clinical trials with binary outcomes frequently rely on normal approximations of effect measures. However, these approximations may become unreliable when trials are small or when event probabilities are close to 0 or 1. In this study, we propose a Bayesian random-effects meta-analysis approach based directly on the binomial likelihood. Three measures of association are considered: the Risk Difference (RD), the Relative Risk (RR), and the Odds Ratio (OR). The existence of an overall effect is assessed using the Bayes factor calculated through the Savage–Dickey identity, while posterior distributions are obtained using Markov chain Monte Carlo (MCMC) methods. A simulation study examines the performance of the procedure under different configurations of the number of studies, sample size, heterogeneity, and effect magnitude. The results show that the ability to detect an effect increases with the amount of available information and the magnitude of the effect, whereas greater heterogeneity generally reduces this ability. The approach is then illustrated using 22 randomized trials evaluating the effect of intravenous streptokinase on mortality after myocardial infarction. All three measures indicate an effect direction favorable to treatment, but the strength of the Bayesian evidence differs according to the effect scale considered. These results demonstrate the value of the Savage–Dickey ratio for complementing effect estimation with an explicit quantification of evidence in Bayesian meta-analyses of binary data.