Deviations from the population-averaged versus cluster-specific relationship for clustered binary data

There has been much debate about the relative merits of mixed effects and population-averaged logistic models. We present a different perspective on this issue by noting that the investigation of the relationship between these models for a given dataset offers a type of sensitivity analysis that may reveal problems with assumptions of the mixed effects and/or population-averaged models for clustered binary response data in general and longitudinal binary outcomes in particular. We present several datasets in which the following violations of assumptions are associated with departures from the expected theoretical relationship between these two models: 1) negative intra-cluster correlations; 2) confounding of the response-covariate relationship by cluster effects; and 3) confounding of autoregressive relationships by the link between baseline outcomes and subject effects. Under each of these conditions, the expected theoretical attenuation of the population-averaged odds ratio relative to the cluster-specific odds ratio does not necessarily occur. In all cases, the naive fitting of a random intercept logistic model appears to lead to bias. In response, the random intercept model is modified to accommodate negative intra-cluster correlations, confounding due to clusters, or baseline correlations with random effects. Comparisons are made with GEE estimation of population-averaged models and conditional likelihood estimation of cluster-specific models. Several examples, including a cross-over trial, a multicentre nonrandomized treatment study, and a longitudinal observational study are used to illustrate these modifications.