Waclawiw and Liang introduced an estimating function-based approach for estimating the parameters of the classical two-stage random effects model for longitudinal data. In the present paper, the authors conduct a case study of the general approach for the binary response setting, where the fully specified parametric two-stage model with fixed and univariate random effects has an analytically intractable likelihood. With successful convergence of the algorithm, the authors propose a fully parametric bootstrapping method for deriving empirical Bayes confidence intervals for all model parameters. The bootstrapping approach is a blend of the estimating function technique with the developments of Laird and Louis. The estimating function approach to estimation and inference provides a general framework for the analysis of a wide variety of medical data, including the setting of small and varying numbers of discrete repeated observations. An application of the methodology to the analysis of binary responses in a crossover clinical trial is presented.