Estimation efficiency and tests of covariate effects with clustered binary data

Several approaches have been proposed to analyze clustered binary data, which arise in fields such as teratology and ophthalmology. These methods include mixed-effects and quasi-likelihood approaches, as well as models that use cluster responses as covariates. The three approaches measure different effects of covariates on binary responses, but simple approximations relate the magnitudes of their parameters. In this article, we present approximations to relate the standard errors of model parameters and Wald tests for covariate effects obtained from the different approaches. These approximations show that Wald tests involving cluster-level covariates will be approximately equivalent using the different approaches. However, approaches that model intracluster correlation, such as the mixed-effects model, provide more powerful tests of within-cluster covariates than those that do not model the correlation. Simulations and example data illustrate these findings.