Effect of omitted confounders on the analysis of correlated binary data

Marginal analysis using the generalized estimating equation approach is widely applied to correlated observations, as occur in studies with clusters and in longitudinal follow-up of individuals. In this article, we investigate the effect of confounding in such models. We assume that a risk factor x and a confounder z are related by a generalized linear model to the outcome y, which can be binary or ordinal. In order to investigate confounding arising from the omission of z, a joint structure for x and z must be specified. Modeling normally distributed (x,z) as sums of between- and within-individual (or cluster) components allows us to incorporate different degrees of between and within-individual correlation. Such a structure includes, as special cases, cohort and period effects in longitudinal settings and random intercept models. The latter situation corresponds to allowing z to vary only on the between-individual (or cluster) level and to be uncorrelated with x, and leads to attenuation of the coefficient of x in marginal models with the legit and probit links. More complex situations occur when z is allowed to also vary on the within-individual (or cluster) level and when z is correlated with x. We examine the model specification and the expected bias when fitting a marginal model in the presence of the omitted confounder z. We derive general formulas and interpret the parameters and results in an ongoing cohort study. Testing for omitted I covariates is also discussed.