Sensitivity bounds for bias in hazard ratios: A causal hazard perspective

The Cox proportional hazards model has popularized the conventional hazard ratio as a standard measure for assessing the effect of exposure on time-to-event outcomes. However, as noted in Hernán's influential critique, interpreting the hazard ratio as a causal effect can introduce substantial selection bias. While Hernán's work discusses this issue qualitatively, it lacks a method to quantify the extent of bias or the strength an unmeasured confounder would need to account for it. Our study fills this gap by proposing a nonparametric sensitivity analysis framework, inspired by Ding and VanderWeele, which quantifies "how hazardous the hazard ratio is." Using the causal hazard ratio proposed by Aalen as the "true hazard ratio," we derive an upper bound for the bias in the conventional hazard ratio (termed HR bias) in both discrete and continuous time settings, demonstrating that this bias accumulates over time. Additionally, we derive the E-value for HR bias, allowing researchers to evaluate whether unmeasured confounding alone could fully explain the conventional hazard ratio under a certain level of the causal hazard ratio. This approach provides researchers with a robust sensitivity analysis framework, reducing reliance on restrictive assumptions and enhancing the reliability of causal interpretations in survival analysis.