Within- and between-cluster effects in generalized linear mixed models: A discussion of approaches and the xthybrid command

One typically analyzes clustered data using random- or fixed-effects models. Fixed-effects models allow consistent estimation of the effects of level-one variables, even if there is unobserved heterogeneity at level two. However, these models cannot estimate the effects of level-two variables. Hybrid and correlated random-effects models are flexible modeling specifications that separate within- and between-cluster effects and allow for both consistent estimation of level-one effects and inclusion of level-two variables. In this article, we elaborate on the sep- aration of within- and between-cluster effects in generalized linear mixed models. These models present a unifying framework for an entire class of models whose response variables follow a distribution from the exponential family (for exam- ple, linear, logit, probit, ordered probit and logit, Poisson, and negative binomial models). We introduce the user-written command xthybrid, a shell for the meglm command. xthybrid can fit a variety of hybrid and correlated random-effects models.