Two‐stage residual inclusion for survival data and competing risks—An instrumental variable approach with application to SEER‐Medicare linked data

Instrumental variable is an essential tool for addressing unmeasured confounding in observational studies. Two-stage predictor substitution (2SPS) estimator and two-stage residual inclusion (2SRI) are two commonly used approaches in applying instrumental variables. Recently, 2SPS was studied under the additive hazards model in the presence of competing risks of time-to-events data, where linearity was assumed for the relationship between the treatment and the instrument variable. This assumption may not be the most appropriate when we have binary treatments. In this paper, we consider the 2SRI estimator under the additive hazards model for general survival data and in the presence of competing risks, which allows generalized linear models for the relation between the treatment and the instrumental variable. We derive the asymptotic properties including a closed-form asymptotic variance estimate for the 2SRI estimator. We carry out numerical studies in finite samples and apply our methodology to the linked Surveillance, Epidemiology and End Results (SEER)-Medicare database comparing radical prostatectomy versus conservative treatment in early-stage prostate cancer patients.