Doubly robust pseudo-likelihood for incomplete hierarchical binary data

Missing data is almost inevitable in correlated-data studies. For non-Gaussian outcomes with moderate to large sequences, direct-likelihood methods can involve complex, hard-to-manipulate likelihoods. Popular alternative approaches, like generalized estimating equations, that are frequently used to circumvent the computational complexity of full likelihood, are less suitable when scientific interest, at least in part, is placed on the association structure; pseudo-likelihood (PL) methods are then a viable alternative. When the missing data are missing at random, Molenberghs et al. (2011, Statistica Sinica, 21,187-206) proposed a suite of corrections to the standard form of PL, taking the form of singly and doubly robust estimators. They provided the basis and exemplified it in insightful yet primarily illustrative examples. We here consider the important case of marginal models for hierarchical binary data, provide an effective implementation and illustrate it using data from an analgesic trial. Our doubly robust estimator is more convenient than the classical doubly robust estimators. The ideas are illustrated using a marginal model for a binary response, more specifically a Bahadur model.