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。第一个是估计或测试免疫个体(长期幸存者)在样本中的存在比例，我们观察每个个体的一些事件发生时间数据(例如，服药后疼痛缓解的时间，或缓解后复发的时间；没有缓解或复发的个体被称为“免疫”)。我建议在不同类型的“审查”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大于还是小于某个审查时间(例如，在“第一类间隔审查”下，我们只有检查时间，以及所讨论的事件是否在它之前发生(“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能帮助我们得出有效的推论。准确地说，问题是如何计算出所需比例及其抽样偏差的近似计算。

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.