Time dependent hazard ratio estimation using instrumental variables without conditioning on an omitted covariate

Todd A Mackenzie,A James O’Malley,Pablo Martinez-Camblor

doi:10.1186/s12874-021-01245-6

Abstract

BackgroundEstimation that employs instrumental variables (IV) can reduce or eliminate bias due to confounding. In observational studies, instruments result from natural experiments such as the effect of clinician preference or geographic distance on treatment selection. In randomized studies the randomization indicator is typically a valid instrument, especially if the study is blinded, e.g. no placebo effect. Estimation via instruments is a highly developed field for linear models but the use of instruments in time-to-event analysis is far from established. Various IV-based estimators of the hazard ratio (HR) from Cox’s regression models have been proposed.MethodsWe extend IV based estimation of Cox’s model beyond proportionality of hazards, and address estimation of a log-linear time dependent hazard ratio and a piecewise constant HR. We estimate the marginal time-dependent hazard ratio unlike other approaches that estimate the hazard ratio conditional on the omitted covariates. We use estimating equations motivated by Martingale representations that resemble the partial likelihood score statistic. We conducted simulations that include the use of copulas to generate potential times-to-event that have a given marginal structural time dependent hazard ratio but are dependent on omitted covariates. We compare our approach to the partial likelihood estimator, and two other IV based approaches. We apply it to estimation of the time dependent hazard ratio for two vascular interventions.ResultsThe method performs well in simulations of a stepwise time-dependent hazard ratio, but illustrates some bias that increases as the hazard ratio moves away from unity (the value that typically underlies the null hypothesis). It compares well to other approaches when the hazard ratio is stepwise constant. It also performs well for estimation of a log-linear hazard ratio where no other instrumental variable approaches exist.ConclusionThe estimating equations we propose for estimating a time-dependent hazard ratio using an IV perform well in simulations. We encourage the use of our procedure for time-dependent hazard ratio estimation when unmeasured confounding is a concern and a suitable instrumental variable exists.

Highlights

Estimation that employs instrumental variables (IV) can reduce or eliminate bias due to confounding
Monte Carlo simulations We evaluated the behavior of the estimating equations we propose in (6) under two scenarios for the marginal timedependent hazard ratio; i) a three piece constant hazard ratio, and ii) a log linear time-dependent hazard ratio
The instrumental variable based estimator we propose is unbiased as an estimator of the marginal hazard ratio in each of the three periods, with a slight tendency toward bias by confounding for large and small hazard ratios; that is, the magnitude of the bias increases as the HR moves away from null and in the direction opposite of the confounding

Summary

Introduction

Estimation that employs instrumental variables (IV) can reduce or eliminate bias due to confounding. Estimation of treatment effects using instrumental variables for outcomes subject to right censoring has received the attention of many applied and several theoretical studies. Stukel et al [1] proposed an ad-hoc estimator of the hazard ratio (HR) based on a linear model. MacKenzie et al [2] proposed a hazard ratio estimator that assumes omitted covariates have an additive effect on the hazard. Tchetgen et al [3] proposed an estimator of an additive hazards model based on two stage residual inclusion. Li et al [4] proposed a consistent estimator of a treatment satisfying an additive hazard model. Martínez-Camblor et al [6] identified the role of frailties in estimation of the hazard ratio via the two-stage residual inclusion algorithm if the treatment and omitted covariate jointly satisfy a Cox model. Wang et al [7] derived an estimator of the marginal hazard ratio using a binary instrument

Methods

Discussion

Conclusion