  Non-proportional 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 non-proportionality. We focus our attention on the WLRT and the
complementary Cox model based on the time-varying treatment effect proposed by
Lin and Le\'on (2017) (doi: 10.1016/j.conctc.2017.09.004). We will 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 we note that the proposed model overestimates the treatment
effect and the treatment profile over time. However, 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 certain constraints.
In conclusion, our results demonstrate that in certain scenarios where a given
WLRT would be most powerful, we observe that the WHR from the corresponding Cox
model is estimating the treatment effect close to the true one.
