Statistics Papers

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Statistics in Medicine





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Two-stage instrumental variable methods are commonly used to estimate the causal effects of treatments on survival in the presence of measured and unmeasured confounding. Two-stage residual inclusion (2SRI) has been the method of choice over two-stage predictor substitution (2SPS) in clinical studies. We directly compare the bias in the causal hazard ratio estimated by these two methods. Under a principal stratification framework, we derive a closed-form solution for asymptotic bias of the causal hazard ratio among compliers for both the 2SPS and 2SRI methods when survival time follows the Weibull distribution with random censoring. When there is no unmeasured confounding and no always takers, our analytic results show that 2SRI is generally asymptotically unbiased, but 2SPS is not. However, when there is substantial unmeasured confounding, 2SPS performs better than 2SRI with respect to bias under certain scenarios. We use extensive simulation studies to confirm the analytic results from our closed-form solutions. We apply these two methods to prostate cancer treatment data from Surveillance, Epidemiology and End Results-Medicare and compare these 2SRI and 2SPS estimates with results from two published randomized trials.

Copyright/Permission Statement

This is the peer reviewed version of the following article: [Wan, F., Small, D.S., Bekelman, J.E., & Mitra, N. (2015). Bias in Estimating the Causal Hazard Ratio When Using Two-Stage Instrumental Variable Methods. Statistics in Medicine 34, no. 14: pp. 2235-2265], which has been published in final form at This article may be used for non-commercial purposes in accordance with Wiley Terms and Conditions for Use of Self-Archived Versions.


instrumental variable, two-stage residual inclusion, two-stage predictor substitution, unmeasured confounding, survival, bias



Date Posted: 25 October 2018

This document has been peer reviewed.