Estimation of treatment effects in weighted log-rank tests

Non-proportional hazards have been observed in clinical trials. The log-rank test loses power and the standard Cox model generally 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 log-rank test.The score test from the proposed model is equivalent to the weighted log-rank test, and a time-profile of the treatment effect can be obtained by fitting a time-varying covariate Cox model. Simulation results show that the proposed model preserves type-I error and achieve higher power than log-rank tests under non-proportional hazards scenarios. Whereas the standard Cox model produces biased effect estimates, the proposed model produces unbiased estimate if the weight function is correctly specified. It also achieves a better model fit and an enhanced flexibility to accommodate non-proportional hazards compared to the standard Cox model.The proposed 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. It also helps to translate the weighted log-rank test results into quantitative estimates of the treatment effect with intuitive interpretation. The proposed method can be routinely conducted to complement weighted log-rank tests, especially in the setting where non-proportional hazards are expected.