On pseudo-likelihood inference in the binary longitudinal mixed model

Binary logistic and Poisson mixed models are used to analyse over/under-dispersed proportion and count data, respectively. It is, however, well known that a full likelihood analysis for such mixed models is hampered by the need for numerical integrations. To overcome such integration problems, recently Sutradhar and Qu (On Approximate Likelihood Inference in Poisson Mixed Model. The Canadian Journal of Statistics 1998, 26, 169-186) has introduced a small variance component (for random effects) based likelihood approximation (LA) approach to estimate the parameters of the Poisson mixed models and have shown that their LA approach performs better as compared to other leading approaches. More recently, Sutradhar and Das (A Higher-Order Approximation to the Likelihood Inference in the Poisson Mixed Model. Statistics and Probability Letters 2001, 52, 59-67) further improved the LA approach of Sutradhar and Qu to accommodate larger values of the variance component. These likelihood approximation techniques developed for Poisson mixed models are however not applicable to the binary mixed models. In this paper, we propose a multivariate binary distribution based pseudo-likelihood approach for the estimation of the parameters of the binary mixed models. We, in fact, do this in a wider binary longitudinal mixed model set up, binary mixed model being a special case. More specifically, two types of binary longitudinal mixed models are considered. Under the first model, conditional on certain independent random effects, repeated binary responses are assumed to follow a Bahadur type multivariate binary distribution, so that, unconditionally, the responses in the cluster follow a longitudinal binary mixed model. Under the second model, however, the binary responses in the cluster are assumed to be conditionally independent, conditional on certain correlated random effects, so that, unconditionally, responses in the cluster also follow a binary longitudinal mixed model. It is of primary interest to estimate the regression and the variance component parameters of the binary longitudinal mixed model, longitudinal correlation parameters being nuisance. The performance of the proposed pseudo-likelihood based estimators is examined through a simulation study. A comparison is also made with a highly competitive generalized estimating equation (GEE) approach, especially for the estimation of the variance component of the random effects.