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Stat Med. 2016 Jan 15;35(1):21-40. doi: 10.1002/sim.6620. Epub 2015 Aug 24.

Meta-analysis of studies with bivariate binary outcomes: a marginal beta-binomial model approach.

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Department of Biostatistics and Epidemiology, University of Pennsylvania, Philadelphia, 19104, Pennsylvania, U.S.A.
Division of Biostatistics, University of Texas School of Public Health, 1200 Pressler St, Houston, 77030, Texas, U.S.A.
Department of Operations Research and Financial Engineering, Princeton University, Princeton, 08544, New Jersey, U.S.A.


When conducting a meta-analysis of studies with bivariate binary outcomes, challenges arise when the within-study correlation and between-study heterogeneity should be taken into account. In this paper, we propose a marginal beta-binomial model for the meta-analysis of studies with binary outcomes. This model is based on the composite likelihood approach and has several attractive features compared with the existing models such as bivariate generalized linear mixed model (Chu and Cole, 2006) and Sarmanov beta-binomial model (Chen et al., 2012). The advantages of the proposed marginal model include modeling the probabilities in the original scale, not requiring any transformation of probabilities or any link function, having closed-form expression of likelihood function, and no constraints on the correlation parameter. More importantly, because the marginal beta-binomial model is only based on the marginal distributions, it does not suffer from potential misspecification of the joint distribution of bivariate study-specific probabilities. Such misspecification is difficult to detect and can lead to biased inference using currents methods. We compare the performance of the marginal beta-binomial model with the bivariate generalized linear mixed model and the Sarmanov beta-binomial model by simulation studies. Interestingly, the results show that the marginal beta-binomial model performs better than the Sarmanov beta-binomial model, whether or not the true model is Sarmanov beta-binomial, and the marginal beta-binomial model is more robust than the bivariate generalized linear mixed model under model misspecifications. Two meta-analyses of diagnostic accuracy studies and a meta-analysis of case-control studies are conducted for illustration.


Sarmanov family; bivariate beta-binomial model; composite likelihood; marginal model; meta-analysis

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