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The validity of conclusions from meta-analysis is potentially threatened bypublication bias. Most existing procedures for correcting publication biasassume normality of the study-specific effects that account for between-studyheterogeneity. However, this assumption may not be valid, and the performanceof these bias correction procedures can be highly sensitive to departures fromnormality. Further, there exist few measures to quantify the magnitude ofpublication bias based on selection models. In this paper, we address both ofthese issues. First, we explore the use of heavy-tailed distributions for thestudy-specific effects within a Bayesian hierarchical framework. The devianceinformation criterion (DIC) is used to determine the appropriate distributionto use for conducting the final analysis. Second, we develop a new measure toquantify the magnitude of publication bias based on Hellinger distance. Ourmeasure is easy to interpret and takes advantage of the estimation uncertaintyafforded naturally by the posterior distribution. We illustrate our proposedapproach through simulation studies and meta-analyses on lung cancer andantidepressants. To assess the prevalence of publication bias, we apply ourmethod to 1500 meta-analyses of dichotomous outcomes in the Cochrane Databaseof Systematic Reviews. Our methods are implemented in the publicly available Rpackage RobustBayesianCopas.

A Robust Bayesian Copas Selection Model for Quantifying and Correcting Publication Bias