The pattern of the difference between two survival curves we often observe in randomized clinical trials for evaluating immunotherapy is not proportional hazards; the treatment effect typically appears several months after the initiation of the treatment (i.e., delayed difference pattern). The commonly used logrank test and hazard ratio estimation approach will be suboptimal concerning testing and estimation for those trials. The long-term restricted mean survival time (LT-RMST) approach is a promising alternative for detecting the treatment effect that potentially appears later in the study. A challenge in employing the LT-RMST approach is that it must specify a lower end of the time window in addition to a truncation time point that the RMST requires. There are several investigations and suggestions regarding the choice of the truncation time point for the RMST. However, little has been investigated to address the choice of the lower end of the time window. In this paper, we propose a flexible LT-RMST-based test/estimation approach that does not require users to specify a lower end of the time window. Numerical studies demonstrated that the potential power loss by adopting this flexibility was minimal, compared to the standard LT-RMST approach using a prespecified lower end of the time window. The proposed method is flexible and can offer higher power than the RMST-based approach when the delayed treatment effect is expected. Also, it provides a robust estimate of the magnitude of the treatment effect and its confidence interval that corresponds to the test result.



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