Weighted Hazard Ratio Estimation for Delayed and Diminishing Treatment Effect

Nonproportional hazards (NPH) have been observed in confirmatory clinical trials with time to event outcomes. Under NPH, the hazard ratio does not stay constant over time and the log rank test is no longer the most powerful test. The weighted log rank test (WLRT) has been introduced to deal with the presence of nonproportionality. We focus our attention on the WLRT and the complementary Cox model based on time varying treatment effect proposed by Lin and Leon. We investigate whether the proposed weighted hazard ratio (WHR) approach is unbiased in scenarios where the WLRT statistic is the most powerful test. In the diminishing treatment effect scenario where the WLRT statistic would be most optimal, the time varying treatment effect estimated by the Cox model estimates the treatment effect very close to the true one. However, when the true hazard ratio is large the proposed model overestimates the treatment effect and the treatment profile over time. In the delayed treatment scenario, the estimated treatment effect profile over time is typically close to the true profile. For both scenarios, we have demonstrated analytically that the hazard ratio functions are approximately equal under small treatment effects. When the assumed rate of how quickly the treatment effect profile is diminishing or delaying differs in the analysis from that in the true data generating mechanism, the estimated hazard ratio profile from the WHR approach is biased. Since in practice the true HR time profile may differ from that assumed in the WHR analysis, it may be preferable to use alternative approaches for effect estimation.