Cure-rates and shape-restricted hazard functions under censoring

在这个项目中，我计划解决生存分析中的两个重要问题。第一个是估计样本中免疫个体（长期存活者）的比例或检测其存在，其中我们观察每个个体的一些时间-事件数据（例如，给药后疼痛缓解的时间，或缓解后复发的时间;未表现出缓解或复发的个体被称为“免疫”）。我建议在不同类型的“截尾”下研究这个问题，而不是确切的时间，我们通常只能观察后者是否大于或小于一些截尾时间（例如，在“情况1区间截尾”下，我们只有检查时间以及是否有问题的事件发生在它之前（“当前状态”））。有趣的是，即使我们没有免疫个体的实际数量，审查上述信息通常也有助于我们得出有效的推论。问题的关键是如何近似计算所需的比例及其抽样变异。

In this project I plan to address two important issues in survival analysis. The first is estimation of the proportion, or testing for presence, of immune individuals (long-term survivors) in a sample where we observe some time-to-event data for each individual (for instance, time to pain-relief following a drug administration, or time to relapse following a remission; the individuals who do not exhibit relief or relapse are termed 'immune'). I propose to study the problem under different types of 'censoring' where instead of the exact time we can often only observe whether the latter is bigger or smaller than some censoring time (for instance, under 'Case-1 interval censoring' we only have the inspection times and whether or not the event in question ocurred before it (the 'current status')). Interestingly, even if we do not have the actual number of immune individuals, censoring information such as above often helps us draw valid inferences. The problem precisely is to figure out how to calculate the required proportion and its sampling variation approximately.