傳統上，最常使用Cox比例風險模型來描述長期追蹤（時間相依）共變數與存活時間之間的關係。然而，卻遭遇到長期追蹤測量值並非固定時間測量以及測量誤差存在的問題，另外， 資料缺失(informative missing)也是在參數估計中造成偏誤(bias)重要的原因之一。因此，在本篇文章當中我們使用聯合模型(joint model)來解決此問題。我們使用線性隨機效應模型(linear random effect model)來描述長期追蹤資料，並根據概似比檢定的方法來判斷長期追蹤模型的適合度；另外，使用Cox比例風險模型來描述共變數與存活時間之間的關係，在參數的估計方面，結合這兩個部份建立聯合概似函數利用EM演算法(expectation maximization algorithm)做參數之估計。我們主要利用凝血酶原指標(Prothrombin index)來評估肝硬化的嚴重程度，並進一步探討強體松藥物對於肝硬化病患其治療的療效，並且觀察凝血酶原指標、年齡以及性別對於存活時間之間的相關性。本研究主要是著重在圖形法和聯合模型來對資料做分析。

Typically, the Cox model is the most popular model to describe the relationship between longitudinal covariates(time-dependent) and the survival time. However, to model the survival time with longitudinal covariates may encounter difficulties when the longitudinal measurement are scattered sparsely and contain measurement errors. There is an additional complexity when the longitudinal process can not be observed due to the event time(death or endpoint), which results in informatively missing data. Therefore, in this study, we applied the joint model to overcome these difficulties. We propose a linear random effects model for longitudinal process, and use the likelihood ratio test to choose a proper longitudinal model. The Cox proportion hazards model then used to link the longitudinal biomarkers and event time, and use the EM algorithm to search for the maximum likelihood estimates. We used the prothrombin index to appraise liver cirrhosis progression, and investigated the prothrombin index, age, sex, and their relationship with survival time. Both graphic techniques and joint model approach were used to explore their relationship.

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