Comparing the survival times among two groups is a common problem in time-to-event analysis, for example if one would like to understand whether one medical treatment is superior to another. In the standard survival analysis setting, there has been a lot of discussion on how to quantify such difference and what can be an intuitive, easily interpretable, summary measure. In the presence of subjects that are immune to the event of interest (`cured'), we illustrate that it is not appropriate to just compare the overall survival functions. Instead, it is more informative to compare the cure fractions and the survival of the uncured sub-populations separately from each other. Our research is mainly driven by the question: if the cure fraction is similar for two available treatments, how else can we determine which is preferable? To this end, we estimate the mean survival times in the uncured fractions of both treatment groups ($MST_u$) and develop permutation tests for inference. In the first out of two connected papers, we focus on nonparametric approaches. The methods are illustrated with medical data of leukemia patients. In Part II we adjust the mean survival time of the uncured for potential confounders, which is crucial in observational settings. For each group, we employ the widely used logistic-Cox mixture cure model and estimate the $MST_u$ conditionally on a given covariate value. An asymptotic and a permutation-based approach have been developed for making inference on the difference of conditional $MST_u$'s between two groups. Contrarily to available results in the literature, in the simulation study we do not observe a clear advantage of the permutation method over the asymptotic one to justify its increased computational cost. The methods are illustrated through a practical application to breast cancer data.

翻译：比较两组间的生存时间是时间-事件分析中的常见问题，例如评估某种治疗方案是否优于另一种。在标准生存分析框架中，如何量化这一差异并选择直观且易于解释的汇总指标已有大量讨论。当存在对目标事件免疫的受试者（即“治愈”者）时，我们证明直接比较总体生存函数并不合适；相反，分别比较治愈率与未治愈亚群的生存信息更具价值。本研究主要源于以下问题：若两种可用治疗方案具有相似的治愈率，如何进一步判断哪种更优？为此，我们估计两组未治愈人群的平均生存时间（$MST_u$）并开发了基于置换检验的推断方法。在这两篇系列论文的第一篇中，我们聚焦于非参数方法，并通过白血病患者医疗数据加以说明。第二部分针对观察性研究中至关重要的混杂因素调整问题，修正未治愈者的平均生存时间：我们采用广泛使用的Logistic-Cox混合治愈模型，在给定协变量条件下估计$MST_u$，并开发了渐近法和置换法以推断两组条件$MST_u$的差异。与现有文献结果不同，本模拟研究未发现置换法相较于渐近法的显著优势足以证明其额外计算成本的合理性。最后通过乳腺癌数据的实际应用演示该方法。