Estimating marginal proportions and intraclass correlations with clustered binary data

A logistic regression with random effects model is commonly applied to analyze clustered binary data, and every cluster is assumed to have a different proportion of success. However, it could be of interest to obtain the proportion of success over clusters (i.e. the marginal proportion of success). Furthermore, the degree of correlation among data of the same cluster (intraclass correlation) is also a relevant concept to assess, but when using logistic regression with random effects it is not possible to get an analytical expression of the estimators for marginal proportion and intraclass correlation. In our paper, we assess and compare approaches using different kinds of approximations: based on the logistic-normal mixed effects model (LN), linear mixed model (LMM), and generalized estimating equations (GEE). The comparisons are completed by using two real data examples and a simulation study. The results show the performance of the approaches strongly depends on the magnitude of the marginal proportion, the intraclass correlation, and the sample size. In general, the reliability of the approaches get worsen with low marginal proportion and large intraclass correlation. LMM and GEE approaches arises as reliable approaches when the sample size is large.