Doubly robust estimation under a possibly misspecified marginal structural Cox model

Summary In this article we consider the marginal structural Cox model, which has been widely used to analyse observational studies with survival outcomes. The standard inverse probability weighting method under the model hinges on a propensity score model for the treatment assignment and a censoring model that incorporates both the treatment and the covariates. In such settings model misspecification can often occur, and the Cox regression model’s non-collapsibility has historically posed challenges when striving to guard against model misspecification through augmentation. We introduce a novel joint augmentation to the martingale-based full-data estimating functions and develop rate double robustness, which allows the use of machine learning and nonparametric methods to overcome the challenges of non-collapsibility. We closely examine its theoretical properties to guarantee root-$ n $ inference for the estimand. The estimator extends naturally to estimating a time-average treatment effect when the proportional hazards assumption fails, and we show that it satisfies both the assumption-lean and the well-specification criteria in the context of a causal estimand for censoring survival data; that is, it is a functional of the potential outcome distributions only and does not depend on the treatment assignment mechanism, the covariate distribution or the censoring mechanism. The martingale-based augmentation approach is also applicable to many semiparametric failure time models. Finally, its application to a dataset provides insights into the impact of mid-life alcohol consumption on mortality in later life.