聯合模型是個可以同時分析長期追蹤資料與存活資料的方法，此方法使用長期追蹤模型與存活模型來做推論，最常使用的存活模型是Cox比例風險模型。但現實中，並非所有共變量都會滿足比例風險假設。為了解決這問題，本篇研究使用一個較廣義的Cox-Aalen 加乘法模型，再結合長期追蹤模型來描述時間相依共變量與存活時間的關係。為了提升計算效率與解釋能力，我們使用參數化模型來估計基底風險函數。因此，我們建構了聯合概似函數再使用蒙地卡羅EM 演算法來估計未知參數，並且設計了模擬研究來確認本篇研究方法的效果以及使用台灣愛滋病人的資料來做分析。

Joint modeling approaches offer a solution to analyzing both survival and longitudinal processes simultaneously. The existing approaches focus mostly on developing adaptive and flexible longitudinal processes based on a preselected survival model, most commonly the Cox proportional hazards model. When the proportional hazards assumption fails for some of time dependent covariates of interest, an alternative model robust to proportionality assumption may needed to replace the Cox model. By combining the Cox model and Aalen additive hazards model, we propose a joint model of additive-multiplicative hazards model and longitudinal processes to describe the relationship between survival time and time-varying covariates. This general class of hazards regression model does not need proportionality assumption for all longitudinal covariates. Moreover, unspecified baseline hazard is replaced by parametric models in this study to improve computational efficiency and interpretability. A joint likelihood procedure is proposed to estimate the unknown parameters and components through a Monte Carlo EM algorithm. We conduct simulations to check the performance of our method and analyze a real data from a Taiwanese HIV/AIDS cohort study for illustration.

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