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

When an omitted covariate whose inclusion would restore proportional hazards is excluded from a proportional hazards model, bias in the estimated treatment effect may arise from two sources: marginalization over the distribution of the omitted covariate and model misspecification caused by fitting a proportional hazards model when omission of that covariate induces nonproportional hazards.

The omitted covariate may represent an unobservable biomarker status, an overlooked stratification factor, or a strong continuous prognostic factor.

Although theoretical frameworks have been proposed to reduce bias due to model misspecification under strong assumptions, these assumptions are often impractical in real-data applications.

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 the tightest band of survival curves enclosing the observed survival curve estimates stratified by treatment group, under the minimal assumptions of a Weibull baseline event-time distribution and unit variance of the omitted covariate.

Simulation studies indicate that (i) modeling the exponentiated omitted covariate through a univariate frailty term can help recover the target hazard ratio to some extent, although not satisfactorily, and (ii) the proposed method recovers the target hazard ratio reasonably well under practical scenarios involving a range of true hazard ratios, Weibull baseline parameters, and random uniform censoring or mixtures thereof, regardless of the true distribution of the omitted covariate.

We illustrate the proposed method using data from a phase III breast cancer clinical trial.