  This article concerns a class of generalized linear mixed models for
clustered data, where the random effects are mapped uniquely onto the grouping
structure and are independent between groups. We derive necessary and
sufficient conditions that enable the marginal likelihood of such class of
models to be expressed in closed-form. Illustrations are provided using the
Gaussian, Poisson, binomial and gamma distributions. These models are unified
under a single umbrella of conjugate generalized linear mixed models, where
"conjugate" refers to the fact that the marginal likelihood can be expressed in
closed-form, rather than implying inference via the Bayesian paradigm. Having
an explicit marginal likelihood means that these models are more
computationally convenient, which can be important in big data contexts. Except
for the binomial distribution, these models are able to achieve simultaneous
conjugacy, and thus able to accommodate both unit and group level covariates.
