A Likelihood Bridge for Bayesian Meta-Analysis With Dependent Effect Sizes

Primary studies in education and the behavioral sciences often report multiple, statistically dependent effect sizes, and the correlations needed to model that dependence are rarely reported. Frequentist meta-analysis handles this problem with robust variance estimation, which pairs a working model for estimation with a cluster-robust standard error that remains valid when the model is wrong. A Bayesian analyst who fits the same working model has no comparable option, because the posterior covariance reflects the prior and the assumed working correlation rather than the robust sandwich. A robust report therefore requires calibrating the posterior covariance to an external target without moving the point estimates, and the meta-analytic literature has never named that target. In this article, we show that Bayesian survey inference solved the same problem, and we prove the likelihood bridge that carries its solution into meta-analysis. Under matched conditions, a dependent-effects meta-analysis and a clustered Bayesian survey analysis are the same model under two interpretations, so the covariance to report is essentially the object the robust sandwich already targets. A center-preserving adjustment moves the posterior covariance onto this target while leaving every estimate unchanged, and we illustrate the framework on two published syntheses.