A Contrasting Study of Likelihood Methods for the Analysis of Longitudinal Binary Data

Clinical trials often involve longitudinal binary endpoints, where the interest is to assess the effect of treatment over time on the response, possibly in the presence of time-dependent or time-independent covariates. These longitudinal binary endpoints can be viewed as short discrete time series, which poses specific analytic challenges that do not occur in Gaussian time series. In this manuscript, we contrast a transitional Markov chain (MC) model for binary time series with the multivariate probit (MP) model. The Markov model is used to develop a likelihood for serially correlated longitudinal binary observations, while the probit model an alternative likelihood method is constructed using latent variables. We discuss maximum likelihood estimation for both models, and estimate large- and small-sample efficiencies to compare the performance of each method in different scenarios. These calculations show that the MC method is more efficient in large samples, and the MP model is more efficient in small samples, especially in the presence of highly correlated responses, though the difference between the models depends upon the type of covariates under consideration. Both models are applied to several real-life data examples, where the parameter estimates are found similar.