Instrumental Variable Estimation in a Survival Context

Abstract

Bias due to unobserved confounding can seldom be ruled out with certainty when estimating the causal effect of a nonradomized treatment. The instrumental variable (IV) design offers, under certain assumptions, the opportunity to tame confounding bias, without directly observing all confounders. The IV approach is very well developed in the context of linear regression and also for certain generalized linear models with a non-linear link function. However, IV methods are not as well developed for regression analysis with a censored survival outcome. In this paper, we develop the instrumental variable approach for regression analysis in a survival context, primarily under an additive hazards model, for which we describe two simple methods for estimating causal effects. The first method is a straightforward two-stage regression approach analogous to two-stage least squares commonly used for IV analysis in linear regression. In this approach, the fitted value from a first -stage regression of the exposure on the IV is entered in place of the exposure in the second-stage hazard model to recover a valid estimate of the treatment effect of interest. The second method is a so-called control function approach, which entails adding to the additive hazards outcome model, the residual from a first-stage regression of the exposure on the IV. Formal conditions are given justifying each strategy, and the methods are illustrated in a novel application to a Mendelian randomization study to evaluate the effect of diabetes on mortality using data from the Health and Retirement Study. We also establish that analogous strategies can also be used under a proportional hazards model specification provided the outcome is rare over the entire follow-up.

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

“…We assume that X u is independent of X o and X I . Denote Following the work of Tchetgen Tchetgen et al, we put a key assumption on X u , ie, where ϵ is an error term independent of X e , X I , and X o . Proposition in the Appendix shows that, integrating out X u , we have Note that the same coefficient β e (and β o ) from is remained in .…”

“…There are two commonly used IV approaches, ie, two‐stage predictor substitution (2SPS) and two‐stage residual inclusion (2SRI). For survival outcomes, Tchetgen Tchetgen et al considered these two methods under the additive hazards model and gave conditions under which the causal parameters of interest can be correctly estimated. We note that, while the Cox proportional hazards model has been more widely used in practice for survival data, an important appeal of additive hazards models is that, unlike proportional hazards, a hazards difference is a collapsible effect measure .…”

“…For survival outcomes, Tchetgen Tchetgen et al considered these two methods under the additive hazards model and gave conditions under which the causal parameters of interest can be correctly estimated. We note that, while the Cox proportional hazards model has been more widely used in practice for survival data, an important appeal of additive hazards models is that, unlike proportional hazards, a hazards difference is a collapsible effect measure . This means that, when there is an unobserved (exogenous) covariate for the survival outcome, integrating out this covariate under the additive hazards model still gives a model satisfying the additive hazards assumption.…”

Instrumental Variable Based Estimation Under the Semiparametric Accelerated Failure Time Model

“…The majority of such methods impose strong outcome modeling assumptions by using parametric structural equation models (Tang and Lee, 1998;Muthen and Masyn, 2005;Chen, Hsiao, and Wang, 2011). Recent work has moved away from parametric survival models (Lin et al, 2014;Li et al, 2015;MacKenzie et al, 2014;Tchetgen Tchetgen et al, 2015). However, parametric assumptions on the IV models were still required.…”

Instrumental variable method for time‐to‐event data using a pseudo‐observation approach

“…In the analysis of medical survival data, semiparametric models are widely used. Although some work relating instrumental variable analysis to semiparametric survival models has been conducted, mainly within the frameworks of accelerated failure time models and additive or proportional hazards models (Robins and Tsiatis, ; Brännäs, ; Loeys and Goetghebeur, ; Bijwaard, ; MacKenzie et al, ; Li, Fine, and Brookhart, ; Tchetgen Tchetgen et al, ), theory on instrumental variable analysis for time‐to‐event data is fairly limited. One general approach for handling instrumental variables in case of nonlinear parametric models that has gained some foothold within time‐to‐event analysis of the hazard rate is the two‐stage residual inclusion (Terza, Basu, and Rathouz, ).…”