Non-proportional hazards have been observed in clinical trials, such as the delayed treatment effect observed in cancer immunotherapies and the diminishing effect caused by the confouding due to follow-up therapies. The log-rank test loses power and the standard Cox model usually produces biased estimates under such conditions. Weighted log-rank tests have been utilized to increase the test power; however, it is not intuitive how to interpret the test result in terms of the clinical effect.
We propose a Cox-model based time-varying treatment effect estimate to complement the weighted logrank test. The score test from the proposed model is equivalent to the weighted log-rank test, and it provides an "effect time-profile" that describes treatment effect as a function of time.
Simulation results show that under non-proportional hazards, the proposed model preserves type-I error, achieves higher power than log-rank tests, and produces unbiased estimates if the weight function is correctly specified. The estimate may still be less biased than the standard Cox model even when the weight function is mis-specified. An example from a real trial also suggests the potential utility of the proposed method.
This approach makes the assumptions of the weighted log-rank test explicit, and the validity of assumptions can be assessed based on prior knowledge or model goodness-of-fit. The effect time-profile provides more elaborate description of the treatment effect, which may be useful for clinical decision making. The proposed method can be routinely conducted to complement weighted log-rank tests, especially in the context where non-proportional hazards are expected.