An instrumental variable procedure for estimating Cox models with non-proportional hazards in the presence of unmeasured confounding

Two-stage instrumental variable methods are commonly used for estimating average causal effects in the presence of an unmeasured confounder. In the context of the proportional hazard Cox regression models, this problem has recently received attention with several methods being proposed. Previously, we developed an improved estimator under the incumbent two-stage residual inclusion procedure called '2SRI' by adding a Gaussian frailty in the second stage. We now consider the more complex situation in which the treatment and the unmeasured confounders can have time varying effects, illustrating the method with the case of a step function with one prespecified change point. We prove that, in situations where the effects of the unmeasured confounder or the treatment change during the follow-up, the first stage of the 2SRI algorithm induces a frailty with time varying coefficients in the second stage, which enables incumbent methods and our previously developed procedure to be improved on. A Monte Carlo simulation study demonstrates the superior performance of the proposed extension of 2SRI that we develop. We apply the new procedure to estimate the effect of endarterectomy versus carotid artery stenting on the time to death of patients suffering from carotid artery disease by using linked vascular quality initiative registry-Medicare data.