Abstract
The generalized linear mixed model (GLMM) is commonly used for the analysis of hierarchical non Gaussian data. It combines an exponential family model formulation with normally distributed random effects. A drawback is the difficulty of deriving convenient marginal mean functions with straightforward parametric interpretations. Several solutions have been proposed, including the marginalized multilevel model (directly formulating the marginal mean, together with a hierarchical association structure) and the bridging approach (choosing the random-effects distribution such that marginal and hierarchical mean functions share functional forms). Another approach, useful in both a Bayesian and a maximum-likelihood setting, is to choose a random-effects distribution that is conjugate to the outcome distribution. In this paper, we contrast the bridging and conjugate approaches. For binary outcomes, using characteristic functions and cumulant generating functions, it is shown that the bridge distribution is unique. Self-bridging is introduced as the situation in which the outcome and random-effects distributions are the same. It is shown that only the Gaussian and degenerate distributions have well-defined cumulant generating functions for which self-bridging holds.
Highlights
The class of generalized linear mixed models (GLMM; Breslow and Clayton 1993, Wolfinger andO’Connell 1993; Molenberghs and Verbeke 2005) is a standard framework for handling hierarchical data of a non-Gaussian nature
The above establishes the theorem: Proposition 1 Provided that an inverse link function f and a random-effects distribution h both admit a cumulant generating function and that they are in a bridge relationship, h is unique to f and vice versa
Note that the Cauchy distribution is a special case in the sense that it does not admit a cumulant generating function, which is necessary for the results of the previous section
Summary
The class of generalized linear mixed models For the binary case they established that marginalizing a GLMM, using the MMM framework, and using the bridge concept results in three different approaches They derived bridge distributions for several further links and/or for vector rather than scalar random effects. They formally established a relationship between the three operations mentioned above: (1) marginalizing a GLMM or a combined model; (2) finding the connector function for a MMM or a COMMM; and (3) deriving the bridge distribution They showed that, for the log and identity links, used when outcome variables have supports in the form of a half line and the real line, respectively, that the three specifications are identical in a number of situations and exhibit close connections in others.
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