Recovering the Target Hazard Ratio Under Nonproportional Hazards Induced by an Omitted Covariate: Simulation-based Approach

Abstract

When an omitted covariate whose inclusion would balance proportional hazards is excluded from a proportional hazards model, bias in the estimated treatment effect may arise. The omitted covariate may represent an unobservable biomarker status, an overlooked stratification factor, or a strong continuous prognostic factor. In this paper, we propose a simple simulation-based approach for recovering the target hazard ratio for treatment effect, defined as the hazard ratio from the correctly specified proportional hazards model that includes the omitted covariate. Our approach identifies the target hazard ratio value that generates a band of survival curves enclosing the observed survival curve estimates most, under the minimal assumptions of a Weibull baseline event-time distribution, (a mixture of) uniform censoring, and unit variance of the omitted covariate. Simulation studies indicate that the proposed method recovers the target hazard ratio reasonably well under practical scenarios involving a broad range of true hazard ratios, regardless of the true distribution of the omitted covariate. Consistency of the empirical estimates from the proposed procedure to the true values is proved. We illustrate the proposed method using data from a phase III breast cancer clinical trial.