A Bayesian approach to correct for unmeasured or semi-unmeasured confounding in survival data using multiple validation data sets

Purpose: The existence of unmeasured confounding can clearly undermine the validity of an observational study. Methods of conducting sensitivity analyses to evaluate the impact of unmeasured confounding are well established. However, application of such methods to survival data (“time-to-event” outcomes) have received little attention in the literature. The purpose of this study is to propose a novel Bayesian method to account for unmeasured confounding for survival data. Methods: The Bayesian method is proposed under an assumption that the supplementary information on unmeasured confounding in the form of internal validation data, external validation data or expert elicited prior distributions is available. The method for incorporating such information to Cox proportional hazard model is described.  Simulation studies are performed based on the recently published instrumental variable method to assess the impact of unmeasured confounding and to illustrate the improvement of the proposed method over the naive model which ignores unmeasured confounding. Results: Simulation studies illustrate the impact of ignoring the unmeasured confounding and the effectiveness of our Bayesian approach. The corrected model had significantly less bias and coverage of 95% intervals much closer to nominal. Conclusion: The proposed Bayesian method provides a useful and flexible tool in incorporating different types of supplemental information on unmeasured confounding to adjust the treatment estimates when the outcome is survival data.  It out-performed the naive model in simulation studies based on a real world study.