Revisiting the hazards of hazard ratios through simulations and case studies

The hazard has been a central concept in the analysis and interpretation of time-to-event data for over 50 years. At any follow-up time, the hazard is the probability of the event happening in the next unit of time amongst those still at risk. Hazard ratios (HRs) between groups are frequently used to quantify the exposure/treatment's association with the failure time. In a highly cited paper, Hernán criticized HRs, asserting that their decreases over time may reflect simply a built-in selection bias, induced by an unmeasured susceptibility, and should not be interpreted as genuine changes in treatment effect. Hernán supports his arguments mainly by the results of a hormone therapy trial, where the HR for coronary heart events decreased largely during follow-up, with hazards crossing from harmful to protective treatment effect. However, he did not present simulations or algebraic derivations to demonstrate that these changes may reflect just an unmeasured susceptibility. We use simulations and real-world case studies to systematically explore this issue. The first simulation series reveals how the underestimation bias and changes over time in Cox proportional hazards model-based HRs depend on the joint impact of susceptibility on the hazard, its distribution, and the incidence of events; with important bias toward the null occurring only for a very strong susceptibility. Further simulations mimic the hormone therapy trial highlighted by Hernán, to demonstrate that the reported bias and crossing hazards are extremely unlikely to reflect just an unmeasured susceptibility, which suggests some biological reasons for decaying treatment HR, possibly including decreasing treatment adherence. Finally, we present real-world examples of interpretable and clinically plausible time-dependent HRs in cancer research. In conclusion, results of our simulations and real-world case studies suggest that concerns about HR limitations may be overstated, and we encourage researchers to model time-dependent HRs and consider potential biological and clinical causes thereof.