Semi-parametric marginal models for hierarchical data and their corresponding full models

Semi-parametrically specified models for multivariate, longitudinal, clustered, multi-level, and other hierarchical data, particularly for non-Gaussian outcomes, are ubiquitous because their parameters can most often be conveniently estimated using the important class of generalized estimating equations (GEE). The focus here is on marginal models, to be understood as models that condition neither on random effects nor on other outcomes, but merely on fixed covariates. In spite of their well-deserved popularity, concern could be raised as to whether such models can always be viewed as a partially specified version of a model with full distributional assumptions, or rather whether such a parent simply does not exist. It is shown, through the use of the hybrid marginal–conditional models, that the answer is affirmative. For conventional GEE with a working correlation structure, the Bahadur model is sometimes considered to be the natural parent candidate, but we show that this is a misconception. The result presented here, which is conceptual in nature, is valid whenever the exponential family is used for the semi-parametric specification, or when a straightforward transformation to an exponential family member is possible, implying validity for broad classes of binary, ordinal, nominal, and count data. The result is illustrated in the context of trivariate binary data. Further, as an illustration, many of the models considered are applied to data from a developmental toxicity study.