The Mixture Cure (MC) models constitute an appropriate and easily interpretable method when studying a time-to-event variable in a population comprised of both susceptible and cured individuals. In literature, those models usually assume that the latter are unobservable. However, there are cases in which a cured individual may be identified. For example, when studying the distant metastasis during the lifetime or the miscarriage during pregnancy, individuals that have died without a metastasis or have given birth are certainly non-susceptible. The same also holds when studying the x-year overall survival or the death during hospital stay. Common MC models ignore this information and consider them all censored, thus yielding in risk of assigning low immune probabilities to cured individuals. In this study, we consider a MC model that incorporates known information on cured individuals, with the time to cure identification being either deterministic or stochastic. We use the expectation-maximization algorithm to derive the maximum likelihood estimators. Furthermore, we compare different strategies that account for cure information such as (1) assigning infinite times to event for known cured cases and adjusting the traditional model and (2) considering only the probability of cure identification but ignoring the time until that happens. Theoretical results and simulations demonstrate the value of the proposed model especially when the time to cure identification is stochastic, increasing precision and decreasing the mean squared error. On the other hand, the traditional models that ignore the known cured information perform well when the curation is achieved after a known cutoff point. Moreover, through simulations the comparisons of the different strategies are examined, as possible alternatives to the complete-information model.

翻译：混合治愈模型为研究包含易感个体与治愈个体的群体中时间至事件变量提供了一种恰当且易于解释的方法。在现有文献中，这些模型通常假设治愈个体是不可观测的。然而，在某些情况下，治愈个体是可以被识别的。例如，在研究终生远处转移风险或妊娠期流产时，未发生转移即死亡或已成功分娩的个体显然属于非易感人群。同样地，在研究x年总生存期或住院期间死亡情况时也存在类似情形。传统的混合治愈模型忽略了此类信息，将其全部视为删失数据，从而导致将低免疫概率错误分配给治愈个体的风险。本研究提出一种融合已知治愈个体信息的混合治愈模型，其中治愈识别时间既可为确定性亦可为随机性。我们采用期望最大化算法推导最大似然估计量。此外，我们比较了处理治愈信息的不同策略：（1）为已知治愈案例分配无限事件时间并调整传统模型；（2）仅考虑治愈识别概率而忽略达到治愈状态的时间。理论结果与仿真实验表明，所提模型在治愈识别时间为随机性时具有显著价值，能有效提高估计精度并降低均方误差。另一方面，当治愈发生在已知截断点之后时，忽略已知治愈信息的传统模型仍表现良好。通过仿真实验，我们进一步检验了不同策略作为完整信息模型替代方案的比较效果。