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Effective Health Care Program

mmeta: Exact Bayesian Inference for Multivariate Meta-analysis of Binary Outcomes


Topic Abstract

Multivariate meta-analysis is useful in combining evidence from independent studies which involve several comparisons among groups based on a single outcome. We proposed a novel model for multivariate meta-analysis of binary outcomes where the risks are modeled by the multivariate beta distribution proposed by Sarmanov (1966). This model have several attractive features compared to the conventional multivariate generalized linear mixed effects models, including simplicity of likelihood function, no need to specify a link function, and has a closed-form expression of distribution functions for study-specific risk differences.


Standard statistical methods assume the effect sizes of multivariate outcomes are independent. This assumption may be violated when multiple treatment arms were compared with the same reference group, multivariate outcomes from the same subject were measured, or multiple groups in the same study share some common factors. Ignorance of such dependence may lead to invalid inference on the pooled effect sizes or functions of effect sizes. We provide an empirical Bayes method for evaluating overall and study-specific treatment effects in multivariate meta-analysis with binary outcomes. The exact posterior distributions of the study-specific comparative measures (e.g., odds ratio, relative risk, risk difference) are derived.


mmeta is a free, cross-platform, open-source program for performing meta-analysis for combining evidence on comparative measures of a binary outcome from independent studies. Specifically, the R package performs maximum likelihood inference on the overall comparative measures using Sarmanov Beta-Binomial model, and provides exact Bayesian inference on the posterior distribution of study-specific comparative measures via empirical Bayes methods. The former is important for pooling information from independent studies, and the latter is useful in quantifying the statistical evidence contributed from individual studies. This model has been used to study the association between the N-acetyltransferase 2 (NAT2) acetylation status and colorectal cancer (Chen et al. 2013 SMMR), quantify the increased risk of type 2 diabetes mellitus due to gestational diabetes mellitus (Chen et al. 2014 CIS-T&M), and comparing risks of adverse events between tricyclic antidepressants treatment arms in clinical trials (Chen et al. 2013 SBR). This software also provides an option to conduct exact Bayesian inference on a single 2×2 table. A detailed introduction of this package and its graphic features is provided in Luo et al. 2014 JSS at

Software Web Site

The manual and software can be downloaded from the project Web site at

The package is developed with partial support from AHRQ R03 HS020666 (H. Chu).