Separating between- and within-cluster covariate effects by using conditional and partitioning methods

We consider the situation where the random effects in a generalized linear mixed model may be correlated with one of the predictors, which leads to inconsistent estimators. We show that conditional maximum likelihood can eliminate this bias. Conditional likelihood leads naturally to the partitioning of the covariate into between- and within-cluster components and models that include separate terms for these components also eliminate the source of the bias. Another viewpoint that we develop is the idea that many violations of the assumptions (including correlation between the random effects and a covariate) in a generalized linear mixed model may be cast as misspecified mixing distributions. We illustrate the results with two examples and simulations.