On the Connections Between Bridge Distributions, Marginalized Multilevel Models, and Generalized Linear Mixed Models

Abstract

Generalized linear mixed models (GLMM) are commonly used to analyze hierarchical data. Unlike linear mixed models, they do not automatically provide parametric marginal regression functions, while such functions are needed for population-averaged inferences. This issue has received considerable attention and here three approaches to address it are reviewed, expanded, and compared: (1) the closed-form expressions of the marginal moments and distributions for a variety of GLMMs, derived by Molenberghs et al. (2010), as well as an extension that accommodates overdispersion; (2) the marginalized multilevel models of Heagerty (1999); (3) the bridge distribution of Wang and Louis (2003), a form for the random-effects distribution that allows the conditional and hierarchical mean to be described by the same link function. Our derivations are for the identity link function, the log link, and a collection of links for binary data. We highlight a number of useful connections: (a) it is shown that the bridge distribution for data with a mean on the unit interval is unique; (b) the three approaches are different for unit-interval data with the logit link, but are connected for the probit link; for the latter, there exist closed forms; (c) further results are derived for the bridge distribution in the case of unit-interval data and a Student's t link; (d) in contrast to the unit-interval case, it is shown how large classes of distributions act as bridge distributions when an identity or a logarithmic link is adopted; (e) for these links, the three approaches are either identical or closely connected; (f) it is underscored for a random-intercepts model and logarithmic link, that the data contain no information about the particular distribution for the random intercept, given that the same fit to the data can be ascribed to an entire class of random-intercept distribution; (g) the implications of the difference between the unit-interval case on the one hand and the identity and logarithmic cases on the other, regarding sensitivity to model assumptions, are discussed.

Response to comments on “Marginalized multilevel hurdle and zero‐inflated models for overdispersed and correlated count data with excess zeros”

“…We find this in Kassahun et al, but, for example, also in Molenberghs et al and in various earlier papers.…”

“…Geert Molenberghs 1,2 Alvaro Florez Poveda 1 Wondwosen Kassahun 3 Thomas Neyens 1 Christel Faes 1 Geert Verbeke 1,2 1 I-BioStat, CenStat, Universiteit Hasselt, B-3590 Diepenbeek, Belgium…”

A taxonomy of mixing and outcome distributions based on conjugacy and bridging

“…They made generic SAS code available for all three approaches and for both data types. Wang and Louis (2003) and Molenberghs et al (2012) used, for binary outcomes, the Fourier transform and its inverse to compute bridge distributions. While elegant as a general solution, it is not straightforward in practice even for fairly standard distributions when used as inverse link functions.…”

“…The aim of this paper is to further develop the concepts of bridging, inverse bridging, and self bridging, exploiting properties of the characteristic function and its series expansion. It is not our intention to provide additional applications; for those interested in these we refer to Molenberghs et al (2012), who provide sufficient tools to apply the models discussed here. In the next section, we review existing work on mixing and mixing distributions, as well as the three operations of marginalizing a GLMM, deriving an MMM, and deriving a bridge distribution.…”

“…and operationalized in (9) at the level of the means Molenberghs et al (2012). considered three generic cases.The first one is where the mean function takes values on the entire real line.…”