An efficient MCEM algorithm for fitting generalized linear mixed models for correlated binary data

Abstract

Generalized linear mixed models have been widely used in the analysis of correlated binary data arisen in many research areas. Maximum likelihood fitting of these models remains to be a challenge because of the complexity of the likelihood function. Current approaches are primarily to either approximate the likelihood or use a sampling method to find the exact likelihood solution. The former results in biased estimates, and the latter uses Monte Carlo EM (MCEM) methods with a Markov chain Monte Carlo algorithm in each E-step, leading to problems of convergence and slow convergence. This paper develops a new MCEM algorithm to maximize the likelihood for generalized linear mixed probit-normal models for correlated binary data. At each E-step, utilizing the inverse Bayes formula, we propose a direct importance sampling approach (i.e. weighted Monte Carlo integration) to numerically evaluate the first- and the second-order moments of a truncated multivariate normal distribution, thus eliminating problems of convergence and slow convergence. To monitor the convergence of the proposed MCEM, we again employ importance sampling to directly calculate the log-likelihood values and then to plot the difference of the consecutive log-likelihood values against the MCEM iteration. Two real data sets from the children's wheeze study and a three-period crossover trial are analyzed to illustrate the proposed method and for comparison with existing methods. The results show that the new MCEM algorithm outperformed that of McCulloch [McCulloch, C.E., 1994, Maximum likelihood variance components estimation for binary data. Journal of the American Statistical Association, 89, 330–335.] substantially.