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Bayesian meta-analysis using skewed elliptical distributions

Meta-analysis refers to a quantitative method for combining results from independent studies in order to draw overall conclusions. We consider hierarchical models including selection models under a skewed heavy tailed error distribution proposed originally by Chen, Dey, and Shao [M. H. Chen, D. K. Dey, Q. M. Shao, A new skewed link model for dichotomous quantal response data, J. Amer. Statist. Assoc. 94 (1983), pp. 1172–1186.] and Branco and Dey [D. Branco and D.K. Dey, A general class of multivariate skew-elliptical distributions, J. Multivariate Anal. 79, pp. 99–113.]. These rich classes of models combine the information of independent studies, allowing investigation of variability both between and within studies and incorporating weight functions. We constructed a detailed computational scheme under skewed normal and skewed Student's t distribution using the MCMC method. Bayesian model selection was conducted by Bayes factor under a different skewed error. Finally, we illustrated our methodology using a real data example taken from Johnson [M.F. Johnson, Comparative efficacy of Naf and SMFP dentifrices in caries prevention: a meta-analysis overview, J Eur. Organ. Caries Res. 27 (1993), pp. 328–336.].