A unifying framework for marginalised random-intercept models of correlated binary outcomes

We demonstrate that many current approaches for marginal modelling of correlated binary outcomes produce likelihoods that are equivalent to the copula-based models herein. These general copula models of underlying latent threshold random variables yield likelihood-based models for marginal fixed effects estimation and interpretation in the analysis of correlated binary data with exchangeable correlation structures. Moreover, we propose a nomenclature and set of model relationships that substantially elucidates the complex area of marginalised random-intercept models for binary data. A diverse collection of didactic mathematical and numerical examples are given to illustrate concepts.