在臨床試驗中，我們常常收集兩種型態的資訊，一種是生物個體的存活資訊，另外一種則是我們感興趣疾病的生物指標資訊。而聯合模型則可以有效地同時處理這兩種型態資料形式的一種統計方法。首先，長期追蹤資料模型裡，在某些疾病中，影響疾病發病風險的生物指標可能不只有一種，因此，我們這裡考慮的是多個生物指標進行模型推倒。而存活模型中，傳統的存活模型上常用的是COX模型，然而，如果其中有部分或全部我們所感興趣疾病的生物指標未服從比例風險的假設時，COX模型則不能使用。因此，我們改採用另外一種存活模型，AFT模型，來解決不服從比例風險原則的問題。在整個統計方法中，我們使用的是蒙地卡羅期望最大值演算法來求得我們要估計的參數，其中，AFT模型中未特定的準線風險函數上，我們使用核心平滑函數來估計，因此我們則可以使用牛頓法來提高我們尋找回歸參數的最大值的效率。在本篇的第三章，我們使用模擬的方式來驗證我們聯立模型的效用。而第四章中，我們採用的是台灣愛滋病的疾病資料來對我們的模型進行實際醫學研究的案例分析。

In many clinic trials, it become very common to collect survival time and time-dependent covariates simultaneously.
In this situation, we are interested not only in the event time but also in the longitudinal covariates.
Joint modeling approach has been successfully handle this kind of data.
In many the literature, the Cox model is mostly widely used survival model.
However, it must follow the proportional hazards assumption, which fails in many medical studies or clinic trials.
In particular, when the data contains several longitudinal biomarkers, it is usually the case that proportionality doesn't hold for part of the biomarkers.
To overcome this case, we propose a joint modeling approach for the accelerated failure time model with multivariate longitudinal covariates.
The estimation is based on a joint likelihood function using Monte Carlo EM algorithm.
The unspeified baseline hazard function is approximated by a kernel smooth function so that Newton-Raphson method can be applied to derive the estimates without closed form in the EM steps.
Simulation studies are conducted to evaluate the performance of the proposed joint model approach.
A case study on Taiwanese AIDS cohort study is used to demonstrate the usefulness of the estimating procedures.

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