Nested doubly robust estimating equations for causal analysis with an incomplete effect modifier

Abstract

Inference regarding exposure effects within subgroups of individuals and regarding the effect‐modifying role of some covariates plays a central role in research on stratified medicine. Large administrative databases offer an appealing basis for investigating these questions but causal inference can be challenging due to confounding and missing data. We consider the setting where subgroups are defined by the value of an incompletely observed potential effect modifier. We first formulate simple doubly inverse probability‐weighted estimating equations involving one weight to facilitate causal inference with complete data and another weight to adjust for the fact that the effect modifier is only partially observed. We then develop a nested doubly robust (NDR) estimating function which is shown to yield more efficient and robust estimators. In simulation studies, both approaches are shown to yield valid inference in finite samples, but the advantages of the NDR estimators are evident when one or more of the auxiliary models are misspecified. An application to a study of the effect of biological therapy on inflammation in a rheumatology cohort is given for illustration.