在臨床試驗或醫學研究中，普遍探討的問題是：『到底有哪些因素會影響疾病的發展 』或是 『藥物對病情的控制是否具有療效 』。
解決這樣的問題可透過存活分析得到幫助，其中Cox比例風險模型最常被用來描述存活資訊與變數間的關係以便了解上述問題。
這些影響疾病的變數常會隨時間而變動，我們稱此為長期追蹤資料。
在追蹤過程中常會因某些因素導致資料不完整或是測量時有實驗誤差，這些都將影響Cox比例風險函數中參數的估計；
在此，我們將採用隨機效應來描述變數的軌跡以解決上述問題，並以Cox比例風險函數與隨機效應所配適的聯合模型來進行實例分析，
探討D-青黴胺藥物對於原發性膽汁性肝硬化（PBC）病人的存活時間的影響以及其膽紅素值的變化情況。
我們初步地透過多種圖示法觀察，包括：事件歷史圖，等高圖以及3D平滑曲線圖。
接著再進一步地使用聯合模型所得的估計值進行探討。
最後根據這兩種方法我們得到相同的結論：第一，D-青黴胺藥物對於PBC病人的病情並沒有顯著的療效。
第二，PBC病人的膽紅素值會與風險成正比，膽紅素值越高時，風險隨之上升。

The main purpose of this thesis is to investigate the effect of D-penicillamine
and bilirubin to lifetime of Primary Biliary Cirrhosis (PBC) patients simultaneously.
Two methods are presented here, graphic techniques and
joint-modeling approach. Graphic techniques are used to do preliminary
exploring the data, which include Event history plot, Contour plot, and
3D smoothing spline surfaces. The joint modeling approach is conducted
to do statistical inferences on estimated parameters for the data. These
two methods have led to the same conclusions. In particular, the drug
D-penicillamine has no significant effect on survival. Moreover, the time
dependent covariate bilirubin can be well described through a cubic random
coefficient model and has a significant impact on patients’ lifetime.

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