Instrumental variable estimation of a hazard ratio for treatment with a waiting time without specifying its dependence on unmeasured confounders: application to a procedural registry.

The timing of an intervention such as surgery may happen sometime after treatment inception (e.g., diagnosis date) due to being put on a wait-list for treatment or being scheduled for treatment at a later time. If interest lies in the change in risk upon receiving the intervention, a Cox model with a time-dependent exposure may be employed. However, if there exists unmeasured confounding, maximum partial likelihood estimators of the hazard ratio are biased. An instrumental variable is a cause of the exposure of interest but not of the outcome except through the exposure. We propose a new framework involving potential outcomes for each possible waiting time, and derive an estimating equation for the hazard ratio of a treatment with a time-dependent start using an instrumental variable without making any assumptions about the form of involvement of unmeasured confounders. We conducted simulations to evaluate bias of our hazard ratio estimator. We illustrate the approach using two examples: First, we analyzed a procedural registry for which instrumental variable based estimators may be the only unbiased approach. Second, we demonstrate causal hazard ratio estimation for a randomized trial with time-dependent treatment initiation.