Meta-analysis with Jeffreys priors: Empirical frequentist properties

In small meta-analyses (e.g., up to 20 studies), the best-performing frequentist approaches for estimation and inference can yield very wide confidence intervals for the meta-analytic mean, as well as biased and imprecise estimates of the heterogeneity. We investigate the frequentist performance of alternative Bayesian methods that use the invariant Jeffreys prior. This prior can be motivated from the usual Bayesian perspective, but can alternatively be motivated from a purely frequentist perspective: the resulting posterior modes correspond to the established Firth bias correction of the maximum likelihood estimator. We consider two forms of the Jeffreys prior for random-effects meta-analysis: the previously established “Jeffreys1” prior treats the heterogeneity as a nuisance parameter, whereas the “Jeffreys2” prior treats both the mean and the heterogeneity as estimands of interest. In a large simulation study, we assess the performance of both Jeffreys priors, considering different types of Bayesian point estimates and intervals. We assess the performance of estimation and inference for both the mean and the heterogeneity parameters, comparing to the best-performing frequentist methods. We conclude that for small meta-analyses of binary outcomes, the Jeffreys2 prior may offer advantages over standard frequentist methods for estimation and inference of the mean parameter. In these cases, Jeffreys2 can substantially improve efficiency while more often showing nominal frequentist coverage. However, for small meta-analyses of continuous outcomes, standard frequentist methods seem to remain the best choices. The best-performing method for estimating the heterogeneity varied according to the heterogeneity itself.