在醫學診斷中，通常會記錄病患回診所測量的共變數值，即為時間相依的
共變數值，有了長期追蹤資料的性質，即不適用一般的接受者作業特徵曲線下面積(AUC)來判斷生物指標對於疾病預測能力的程度，因此根據 Heagerty 和 Zheng (2005) 和 van Houwelingen, Putter (2012) 以及 Blanche, Dartigues 和 Jacqmin-Gadda (2013) 所提出的方法，皆可用來估計時間相依的AUC。由於上述的三種方法皆為依據 Heagerty 和 Zheng (2005) 的架構再分別透過不同的估計方法去計算時間相依的AUC，因此本論文主要針對 Heagerty 和 Zheng (2005) 的方法並進一步透過模擬和實例分析來探討隨著時間的不同，AUC 對於生物指標預測疾病能力的程度。由於 Heagerty 和 Zheng (2005) 是使用部分概似法，其需要完整的共變數歷史且不允許有測量誤差，因此本論文預期使用聯合模型可以解決部分概似函數的缺失問題，
使得時間相依的AUC可以有更精確的估計結果。

In the medical diagnosis, it usually recorded the measurements for covariates of patients with returning to clinic which also called time-dependent covariates. With the property of longitudinal data, it is not suitable for using traditional area under the Receiver Operating Characteristic curve (AUC) to distinguish the biomarkers
for predicting ability of diseases. According to the methods in Heagerty & Zheng (2005), van Houwelingen, Putter (2012) and Blanche, Dartigues & Jacqmin-Gadda (2013), all can estimate time-dependent AUC. Since these three kinds of methods are mainly based on the approach in Heagerty & Zheng (2005), each method computes time-dependent AUC by different ways. Hence, we focus on the method in Heagerty & Zheng (2005) and explore AUC for biomarkers via simulation and case study. Due to Heagerty & Zheng (2005) using partial likelihood function to compute AUC that needs complete covariate history and doesn’t allow for measurement error. Consequently, this thesis tries to apply joint model approach to solve the problems of partial likelihood function to obtain a better prediction.

Bamber, D. (1975). The area above ordinal dominance graph and the area below the receiver operating characteristic graph. Journal of Mathematical Psychology, 12, 387-415.

Blanche, P., Dartigues, J. F., & Jacqmin-Gadda, H. (2013). Estimating and comparing timedependent areas under receiver operating characteristic curves for censored event times with competing risks. Statistics in medicine, 32, 5381-5397.

Blanche, P. & packaging by Paul Blanche (2015). timeROC: Time-Dependent ROC Curve and AUC for Censored Survival Data. R package version 0.3. URL: http://CRAN . R-project . org/ package= timeROC.

Carey, J. R., Liedo, P., M ller, H. G., Wang, J. L. & Chiou, J. M. (1998). Relationship of age patterns of fecundity to mortality, longevity, and lifetime reproduction in a large cohort of Mediterranean fruit fly females. J. of Gerontology : Biological Sciences, 53, 245-251.

Cox, D. R. (1972). Regression Models and Life-Tables. Journal of the Royal Statistical Society Series B (Methodological), 34, 187-220.

Dickson, E. R., Grambsch, P. M., Fleming, T. R., Fisher, L. D. and Langworthy, A. (1989). Prognosis in Primary Biliary Cirrhosis: Model for Decision Making. Hepatology, 10, 1-7.

Fleming, T. R. and Harrington, D. P. (1991). Counting Processes and Survival Analysis. John, Wiely and Sons, New York.

Hanley, J. A. and McNeil, B. J. (1982). The meaning and use of the area under a receiver operating characteristic (ROC) curve. Radiology, 143, 29-36.

Harrell, F. E., Lee, K. L. & Mark, D. B. (1996). Multivariable prognostic models: issues in developing models, evaluating assumptions and adequacy, and measuring and reducing errors, Statistics in Medicine 15, 361-387.

Heagerty Patrick J. and Zheng Yingye (2005). Survival Model Predictive Accuracy and ROC Curves. Biometrics, 61, 92–105.

Heagerty, P. J. & packaging by Paramita Saha (2012). risksetROC: Riskset ROC curve estimation from censored survival data. R package version 1.0.4. URL:http://CRAN . R-project . org/ package=risksetROC.

Hosmer, D.W., and Lemeshow, S. (2000). Applied logistic regression (2nd ed.). New York: Wiley.

Hung H, Chiang CT (2010). Estimation methods for time-dependent AUC models with survival data. Canadian Journal of Statistics, 38(1), 8–26.

Putter, H. & packaging by Hein Putter (2015). dynpred: Companion Package to “Dynamic Prediction in Clinical Survival Analysis”. R package version 0.1.2. URL: http://CRAN . R-project . org/ package=dynpred.

Uno H, Cai T, Tian L, Wei LJ (2007). Evaluating prediction rules for t-year survivors with censored regression models. Journal of the American Statistical
Association, 102(478), 527–537.

van Houwelingen HC, Putter H (2012). Dynamic Prediction in Clinical Survival Analysis. Chapman & Hall.

Wulfsohn, M. S. and Tsiatis, A. A. (1997). A Joint Model for Survival and Longitudinal Data Measured with Error. Biometrics, 53, 330-339.

Zheng Y, Cai T, Jin Y, Feng Z (2012). Evaluating prognostic accuracy of biomarkers under competing risk. Biometrics, 68(2), 388–396.

Zhou, X. H., Obuchowski, N. A., McClish, D. K. (2002). Statistical methods in diagnostic medicine, John Wiley and Sons, Inc., New York.

行政院衛生署疾病管制局 出版 (2008)。愛滋病檢驗及治療指引。

吳威霖 (2014)。半母數接受者作業特徵曲線之比較及應用。國立中央大學統計研究所碩士論文。

翁瑄佑(2013)。台灣愛滋病實例研究-以聯合模型探討CD4細胞數以及病毒乘載量對愛滋病患存活時間之關係。國立中央大學統計研究所碩士論文。

羅雅筑 (2012)。利用有序限制性推論之相關方法調整ROC曲線。國立臺北大學統計學系碩士論文。