不同抽樣限制下條件存活函數之無母數平滑估計

Abstract

在釵h追蹤研究中，存活資料常以橫斷面抽樣方法收集，而此種抽樣方法常受制於蒐集條件或抽樣限制，而造成資料上的設限或截切。例如：研究個體因種種原因失去追蹤而造成資料上的設限，或因只能收集已發生起始事件而尚未進展至終止事件的個體，無法觀察到母體中不合如此抽樣條件的個體，因而造成資料上的截切。在此我們考慮存活時間之分佈和起始事件之發生時間相關的右設限和左截切資料。當存活時間和起始事件到設限或截切之時段為獨立時，Kaplan-Meier 估計量為非參數假設下之最大概似估計量。但上述性質在存活時間之分佈和起始事件之發生時間有關時可能不成立。在此我們應用廣義 Kaplan-Meier 估計量的想法來估計給定起始事件時間下之條件存活函數，並且使用 kernel 平滑方法以得到平滑估計量。最後給予其理論及模擬上的探討。

In many follow-up studies, survival data are often collected by a cross-sectional sampling scheme. Such sampling schemes causes censoring or truncation in the data. We consider the case that the survival time distribution depends on the occurrence time of the initiating event in right-censored data and left-truncated data. When the survival time distribution does not depend on the censoring period of time or the truncation period of time, the Kaplan-Meier estimator is the nonparametric MLE for the censored data or truncated data, respectively. But this may not true if the survival time distribution depends on the initiating time. Here we construct a generalized Kaplan-Meier estimator to estimate the conditional survival function of the survival time given the initiating time, and then kernel smooth it with respect to the survival time to obtain a smooth estimator. Theoretical and numerical justifications are given.