Causal inference accounting for unobserved confounding after outcome regression and doubly robust estimation

Causal inference with observational data can be performed under an assumption of no unobserved confounders (unconfoundedness assumption). There is, however, seldom clear subject-matter or empirical evidence for such an assumption. We therefore develop uncertainty intervals for average causal effects based on outcome regression estimators and doubly robust estimators, which provide inference taking into account both sampling variability and uncertainty due to unobserved confounders. In contrast with sampling variation, uncertainty due to unobserved confounding does not decrease with increasing sample size. The intervals introduced are obtained by modeling the treatment assignment mechanism and its correlation with the outcome given the observed confounders, allowing us to derive the bias of the estimators due to unobserved confounders. We are thus also able to contrast the size of the bias due to violation of the unconfoundedness assumption, with bias due to misspecification of the models used to explain potential outcomes. This is illustrated through numerical experiments where bias due to moderate unobserved confounding dominates misspecification bias for typical situations in terms of sample size and modeling assumptions. We also study the empirical coverage of the uncertainty intervals introduced and apply the results to a study of the effect of regular food intake on health. An R-package implementing the inference proposed is available.