  Instrumental variable (IV) methods are widely used for estimating average
treatment effects in the presence of unmeasured confounders. However, the
capability of existing IV procedures, and most notably the two-stage residual
inclusion (2SRI) procedure recommended for use in nonlinear contexts, to
account for unmeasured confounders in the Cox proportional hazard model is
unclear. We show that instrumenting an endogenous treatment induces an
unmeasured covariate, referred to as an individual frailty in survival analysis
parlance, which if not accounted for leads to bias. We propose a new procedure
that augments 2SRI with an individual frailty and prove that it is consistent
under certain conditions. The finite sample-size behavior is studied across a
broad set of conditions via Monte Carlo simulations. Finally, the proposed
methodology is used to estimate the average effect of carotid endarterectomy
versus carotid artery stenting on the mortality of patients suffering from
carotid artery disease. Results suggest that the 2SRI-frailty estimator
generally reduces the bias of both point and interval estimators compared to
traditional 2SRI.
