Generalized estimating equations

It is a privilege to be able to peruse the fine article by Ziegler and Vens [1], as well as the contributions made by nine discussants [2]. Over the last 25 years, generalized estimating equations (GEE) have seen an ever further spreading use. Nonetheless, it is a technique confronted with confusion and, at times, misunderstanding. The user must carefully read the technique’s manual. Let us highlight a few important principles. First, GEE is a method of estimation rather than a model or a modeling family. That said, it is virtually always applied in the context of marginal models, even though Zeger, Liang, and Albert [3] applied it to generalized linear mixed models. Second, comparisons between marginal and hierarchical models (usually termed random-effects models) need to be done with caution. Each family serves its own purpose and precisely which of the two is chosen should depend predominantly on the research question, although pragmatic, computational considerations are perfectly legitimate, too. In this sense, GEE should not be seen as a “downward biased version” of GLMM. Such a view would violate this and the previous principle. Third, and related, there is the delicate relationship between GEE and fully speci fied marginal models. The latter allow for full likelihood or fully Bayesian estimation methodology, a choice one may wish to make should higher-order moments (including variances and correlations) be of interest, perhaps next to marginal mean parameters. When GEE is chosen, this is often because only the first moment is of scientific relevance, upon which then the second moment (correlation) is considered a nuisance. That said, the correlation structure does deserve attention, which is where the strong contribution of Ziegler and Vens lies. While one can misspecify the correlation structure and still reach consistency, important issues remain. Indeed, one can question, along with Chaganty and Sabo, whether asymptotic theory applies when the correlation is misspecified. Results by Molenberghs and Kenward [4] are encouraging in this respect; they show that a full distribution can always be constructed, as soon as marginal mean and correlation are compatible. Such compatibility is not straightforward, as restated by Chaganty and Sabo, and the resulting model may be contrived. But the crucial fact is that there does exist such a model, thence alleviating somewhat the concern, raised by Chaganty and Sabo, about the lack of a solid asymptotic theory. In this respect, the fact that the marginal mean structure has implications for the correlation parameter space, beyond positive definiteness, is a crucial observation pointed out by Chaganty and Sabo. One should not loose sight of the fact that GEE builds on generalized linear models (GLM), with their well-known mean-variance relationship. GLM are, after all, nonlinear models, in spite of the linear predictors they carry, the implications of which are at the same time non-trivial and profound. But there are instances where more than one moment would be of interest. It is then natural to model two or even more moments, such as in GEE2 (second-order generalized estimating equations) or pseudolikelihood. These methods are reviewed in Methods Inf Med 2010; 49: 419–420