On Marginal Quasi-Likelihood Inference in Generalized Linear Mixed Models

In view of the cumbersome and often intractable numerical integrations required for a full likelihood analysis, several suggestions have been made recently for approximate inference in generalized linear mixed models (GLMMs). Two closely related approximate methods are the penalized quasi-likelihood (PQL) method and the marginal quasi-likelihood (MQL) method. The PQL approach generally produces biased estimates for the regression effects and the variance component of the random effects. Recently, some corrections have been proposed to remove these biases. But the corrections appear to be satisfactory only when the variance component of the random effects is small. The MQL approach has also been used for inference in the GLMMs. This approach requires the computations of the joint moments of the clustered observations, up to order four. But the derivation of these moments are not easy. Consequently, different "working" formulas have been used, especially for the mean and covariance matrix of the observations, which may not lead to desirable estimates. In this paper, we use a small variance component (of the random effects) approach and develop the MQL estimating equations for the parameters based on the joint moments of order up to four. The proposed approach thus avoids the use of the so-called "working" covariance and higher order moment matrices, leading to better estimates for the regression and the overdispersion parameters, in the sense of efficiency in particular.