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

Abstract

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.

On composite likelihood in bivariate meta-analysis of diagnostic test accuracy studies

“…This subsection summarizes the composite likelihood estimating equations and the asymptotic covariance matrix for the estimator that solves them in the context of diagnostic test accuracy studies. Chen et al (2014) and Chen et al (2016b) proposed the composite likelihood method for estimation of the copula mixed model with normal and beta margins, respectively. Composite likelihood is a surrogate likelihood which leads asymptotically to unbiased estimating equations obtained by the derivatives of the composite log-likelihoods.…”

“…The composite likelihood can be derived conveniently under the assumption of independence between the random effects. The CL method has been recommended by Chen et al (2014Chen et al ( , 2016b to overcome practical 'issues' in the joint likelihood inference such as computational difficulty caused by a double integral in the joint likelihood function, and restriction to bivariate normality.…”

“…However, (a) GLMM can only be unstable if there are too many parameters in the covariance matrix of the random effects or too many random effects for a small sample, which is not the case in this application domain. Furthermore, Chen et al (2014Chen et al ( , 2016b restrict themselves to SAS PROC NLMIXED which is a general routine for random effect models and thus gives limited capacity. The CL method is well established as a surrogate alternative of maximum likelihood when the joint likelihood is too difficult to compute (Varin et al, 2011), which is apparently not the case in the synthesis of diagnostic test accuracy studies.…”

Bayesian bivariate meta‐analysis of diagnostic test studies with interpretable priors

“…Zapf et al [8] propose a nonparametric frequentist model that shows good convergence properties but does not yet allow the incorporation of covariate or expert knowledge. Chen et al [37] propose a marginal beta-binomial model based on the composite likelihood approach and find a robust performance when the model is misspecified.…”

“…Chen et al. propose a marginal beta‐binomial model based on the composite likelihood approach and find a robust performance when the model is misspecified.…”