傳統非線性混合效應模型其效應與殘差通常都假設為常態分配，但如此可能與實際資料情況不符。因此我們提出一個新的非線性以及非常態假設之混合效應模型，相較於常態假設之模型，其應用可以更為廣泛。在本文中我們將使用在追蹤卵巢癌病情之腫瘤標記 CA125 資料作為範例，使用我們提出的非線性混合效應模型來進行配適。我們發現其效應與殘差並非對稱，對此以非常態假設較能夠貼近真實情況。並且我們期望藉由對 CA125 長期追蹤資料的配適，能夠讓醫生在臨床上對於病患病情的診斷有所助益。我們使用貝氏階層模型的架構逐步建立出我們的模型，當中我們加入病患的癌症期別作為一部分資訊，且考慮病患的復發與否作為模型中的混合因子來對病會之 CA125 進行分析與討論。
　　另外我們針對廣義 Cox 比例風險模型進行推廣。一般來說我們僅會考慮會隨時間變化的風險因子，我們稱之為時間相依因子(time-dependent covariate)，其影響係數並不會隨時間變化；Wang (2015) 將模型推廣，考慮一般風險因子其影響係數會隨時間變化，我們稱之為時變係數(time-varying effect coefficient)，並提供R 軟體函式，利用樣條函數(spline function)對時變係數進行估計。本文將其函式更推廣，考慮時間相依因子其影響亦會隨時間變化，即時間相依因子擁有時變係數，同樣利用樣條函數對時變係數進行估計，本文最會將提供簡單的範例加以說明。

It is common to analyze longitudinal data using nonlinear mixed-effects (NLME) model. And we often use NLME model with normality and homogeneity assumption. However, this assumption may be unrealistic in practice. Our aim is to model the longitu-
dinal profiles of CA125, a tumor marker, in an ovarian cancer study. When fitting these profiles using NLME model, we observed that the distribution of the random effects and
errors are skewed. Hence we propose an NLME model with skewed normal random effects and skewed-t errors. Moreover, we observed that errors and some of the random effects are heterogeneous due to early and late cancer stage. Therefore, we apply the Bayesian hierarchical framework using the heterogeneity and skewness information to construct our new NLME model. Most importantly, we hope that this model can be helpful for doctors during the clinical treatments.
In the second part, we provide a more generalized Cox proportional hazard (Cox PH) model. The traditional Cox PH model has been used to identify the risk factors without considering time-varying effects. A generalized Cox PH model must satisfy the proportional hazard assumption, even though the risk factors are time-dependent. Wang (2015) has provided a more generalized Cox PH model by considering the risk factors which have time-varying effects and shared the R package. Here we extended the model even more. Some of the risk factors which are time-dependent can have time-varying effects simultaneously. We use spline function to approximate the time-varying coefficients and also provide an R function.

論文審定書i
誌謝i
摘要ii
Abstract iii
1 研究動機與目的1
2 資料描述3
2.1 資料處理. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.2 變數介紹. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
3 研究方法5
3.1 模型與分配假設. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2 多維偏斜常態分配. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.3 多維偏斜t 分配. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.4 階層式模型建構. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.5 待估計參數之後驗分配. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 研究結果15
4.1 模型配適結果. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 多種模型結果比較. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5 Cox 比例風險模型推廣21
6 結論與結語26
參考文獻27
A 附錄28
附錄28
A.1 附錄一. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
A.2 附錄二. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

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