Causal Proportional Hazards Estimation with a Binary Instrumental Variable

Abstract

Instrumental variables (IV) are a useful tool for estimating causal effects in the presence of unmeasured confounding. IV methods are well developed for uncensored outcomes, particularly for structural linear equation models, where simple two-stage estimation schemes are available. The extension of these methods to survival settings is challenging, partly because of the nonlinearity of the popular survival regression models and partly because of the complications associated with right censoring or other survival features. Motivated by the Prostate, Lung, Colorectal and Ovarian (PLCO) Cancer screening trial, we develop a simple causal hazard ratio estimator in a proportional hazards model with right censored data. The method exploits a special characterization of IV which enables the use of an intuitive inverse weighting scheme that is generally applicable to more complex survival settings with left truncation, competing risks, or recurrent events. We rigorously establish the asymptotic properties of the estimators, and provide plug-in variance estimators. The proposed method can be implemented in standard software, and is evaluated through extensive simulation studies. We apply the proposed IV method to a data set from the Prostate, Lung, Colorectal and Ovarian cancer screening trial to delineate the causal effect of flexible sigmoidoscopy screening on colorectal cancer survival which may be confounded by informative noncompliance with the assigned screening regimen.

Keywords: Causal treatment effect; Cox proportional hazards model; Instrumental variable; Noncompliance.

Figure 1:

Simulation results: convergence rates, mean estimates, and empirical coverage probabilities of 95% confidence intervals: Complier (■); Naive (•); κ (o); κ_v (+); κ_vtr (✳)

Figure 2:

Simulation results: the estimated standard errors and empirical standard deviations of $\widehat{\kappa }$, ${\widehat{\kappa }}_v$, ${\widehat{\kappa }}_{v,tr}$ weighted estimators: Empirical (□); Mean SE (+); Median SE (∇); Mean Bootstrap SE (×); Median Bootstrap SE (✳)