在存活分析當中, 病患們於不同時間點時重複測量感興趣的長期追蹤共變數是非常普遍的。在這種情況下, 常會因測量誤差或生物體本身的差異, 以及共變數觀測值是否測量得到與存活有關時, 使得推論產生偏差。為了修正偏差,我們使用可以同時配適存活與長期追蹤共變數的聯合模型來解決此問題。聯合模型可以被應用來分析原發性膽汁性肝硬化疾病之病患資料, 主要是探討D-青黴胺治療藥物與年齡層不同以及Mayo 風險評分測量值之變化對存活的影響。最後得到結果為: D-青黴胺對原發性膽汁性肝硬化病情並無顯著的效、
年齡層對存活無顯著差異、以及病患的Mayo 風險評分與風險成正相關, 評分越高其死亡風險越高。而且, 從接受者作業特徵曲線下面積得到, 對於原發性膽汁性肝硬化疾病而言, Mayo 風險評分比生物指標膽紅素有較高的預測準確性能力, 其值越高代表病患有較高的風險會死亡, 其值越低代表病患之病情較輕緩。

In survival analysis, it’s very common that the interesting covariates were measured intermittently at different measurment times for different patients. In this scenario, the repeated measurments could include measurment
errors and measurments can not be observed after the survival time. Those situations could result in biased inferences for study when using Cox partial likelihood. To corret the bias, we use a joint model approach which models survival time and the longitudinal covariates simultaneously. This approach was applied to analyze Primary Biliary Cirrhosis patients data with the main interest of exploring the relationship between longitudinal Mayo risk score and survival. The results
suggested that the drug D-penicillamine and age groups have no significant effect on survival and the longitudinal covariate Mayo risk score can be well described through a cubic random coefficient model and has
a significant impact on patients’ lifetime. Moreover, from AUC (area under the ROC curve) of ROC curve (Receiver Operating Characteristic curve) which suggests that the Mayo risk score has better prediction capacity than the biomarker, bilirubin.

[1] Akritas, M. G. (1994). Nearest neighbor estimation of a bivariate distribution under random censoring. Annals of Statistics 22, 1299-1327.
[2] Cleveland, W. S. (1979). Robust Locally Weighted Regression and Smoothing Scatterplots. Journal of the American Statistical Asso-ciation 74, 829-836.
[3] Cox, D. R. (1972). Regression Models and Life-Tables. Journal of the Royal Statistical Society. Series B (Methodological) 34, 187-220.
[4] Dempster, A. P., Laird, N. M. and Rubin, D. B. (1977). Maximum Likelihood from Imcomplete Data via the EM Algorithm. Journal of the Royal Statistical Society. Series B (Methodological) 39, 1-38.
[5] 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.
[6] Ding, J. and Wang, J. L. (2008). Modeling Longitudinal Data with Nonparametric Multiplicative Random Effects Jointly with Survival Data. Biometrics 64, 546-556.
[7] Dubin, J. A., M¨uller, H. G. and Wang, J. L. (2001). Event history graphs for censored survival data. Statistics in Medicine 20, 2951-2964.
[8] Efron, B. (1979). Bootstrap methods: Another look at the jackknife.Annual Statistician 7, 1-26.
[9] Efron, B. (1994). Missing Data, Imputation and the Bootstrap. Journal of American Statistical Association 89, 463-475.
[10] Fleming, T. R. and Harrington, D. P. (1991). Counting Processes and Survival Analysis. New York: Weiley.
[11] Goldman, A. I. (1992). Eventcharts: Visualizing Survival and Other Timed-Events Data. The American Statistician 46, 13-18.
[12] Gong, Y., Klingenberg, S. L. and Gluud , C. (2004). D-penicillamine for primary biliary cirrhosis. Cochrane Database of Systematic Re-views. New York: Wiley.
[13] Hanley, J. A. (1989). Receiver operating characteristic (ROC) methodology: the state of the art. Critical Reviews in Diagnostic Imaging 29, 307-335.
[14] Heagrty, P. J., Lumley, T. and Pepe, M. S. (2000). Time-dependent ROC curves for censored survival data and a diagnostic marker. Biometrics 56, 337-344.
[15] Henderson, R., Diggle, P. and Dobson, A. (2000). Joint modelling of longitudinal measurements and event time data. Biostatistics 1, 465-480.
[16] Hosmer, D.W., and Lemeshow, S. (2000). Applied logistic regression (2nd ed.). New York: Wiley.
[17] Hsieh, F., Tseng, Y. K. and Wang, J. L. (2006). Joint Modeling of Survival and Longitudinal Data: Likelihood Approach Revisited.Biometrics 62, 1037-1043.
[18] Laird, N. M., Ware, J. H. (1982). Random-Effects Models for Longitudinal Data. Biometrics 38, 963-974.
[19] Lee, J. J., Hess, K. R. and Dubin, J. A. (2000). Extensions and Application of Event Charts. The American Statistician 54, 63-70.
[20] Lehmann, E. L. (1986). Testing Statistical Hypotheses. New York: Wiley.
[21] Louis, T. A. (1982). Finding the observed Fisher information when using the EM algorithm. Journal of the Royal Statistical Society, Series B (Methodological) 44, 226-233.
[22] Markus, B. H., Dickson, E. R., Grambsch, P. M., Fleming, T. R., Mazzaferro, V., Klintmalm, G. B., Wiesner, R. H., Van Thiel, D. H., and Starzl, T. E. (1989). Efficiency of Liver Transplantation in Patients With Primary Biliary Cirrhosis. New England Journal of
Medicine 320, 1709-1713.
[23] Murtaugh, P. A., Dickson, E. R., Van Dam, G. M., Malinchoc, M., Grambsch, P. M., Langworthy, A. L. and Gips, C. H. (1994). Primary biliary cirrhosis: prediction of short-term survival based on repeated patient visits. Hepatology 1, 126-134.
[24] Neuberger, J., Christensen, E., Portmann, B., Caballeria, J., Rodes, J., Ranek, L., Tygstrup, N. and Williams, R. (1995). Double blind controlled trial of D-penicillamine in patients with primary biliary cirrhosis. Gut 26, 114-119.
[25] Pinheiro, J. C. and Bates, D. M. (2000). Mixed-Effects Models in S and S-PLUS. New York: Springer.
[26] Rosner, B. (2006). Fundamentals of Biostatistics(sixth edition). New York: Thomson Learning.
[27] Silverman, B. W. (1984). Spline Smoothing: The Equivalent Variable Kernel Method. The Annals of Statistics 12, 898-916.
[28] Song, X., Davidian, M. and Tsiatis, A. A. (2002). A Semiparametric Likelihood Approach to Joint Modeling of Longitudinal and Timeto-Event Data. Biometrics 58, 742-753.
[29] Taylor, J. M. G., Cumberland, W. G. and Sy, J. P. (1994). A Stochastic Model for Analysis of Longitudinal AIDS Data. Journal of the American Statistical Association 89, 727-736.
[30] Taylor, J. M. G., and Law, N. (1998). Does the Covariance Structure Matter in Longitudinal Modelling for the Prediction of Future CD4 Counts? Statistics in Medicine 17, 2381-2394.
[31] Tseng, Y. K., Hsieh, F. and Wang, J. L. (2005). Joint modeling of accelerated failure time and longitudinal data. Biometrika 92, 587-603.
[32] Tsiatis, A. A. and Davidian, M. (2004). Joint Modeling of Longitudinal and Time-to-Event Data: An Overview. Statistica Sinica 14, 809-834.
[33] Tsiatis, A. A., DeGruttola, V. andWulfsohn, M. S. (1995). Modeling the Relationship of Survival to Longitudinal Data Measured with Error. Applications to Survival and CD4 Counts in Patients with AIDS. Journal of the American Statistical Association 90, 27-37.
[34] Wang, Y. and Taylor, J. M. G. (2001). Jointly Modeling Longitudinal and Event Time Data With Application to Acquired Immunodeficiency Syndrome. Journal of the American Statistical Association 96, 895-905.
[35] Wulfsohn, M. S. and Tsiatis, A. A. (1997). A Joint Model for Survival and Longitudinal Data Measured with Error. Biometrics 53, 330-339.
[36] Yu, M., Law, N. J., Taylor, J. M. G. and Sandler, H. M. (2004). Joint Longitudinal-Survival-Cure Models and Their Application to Prostate Cancer. Statistica Sinina 14, 835-862.
[37] Zeng, D. and Cai, J. (2005). Asymptotic Results for Maximum Likelihood Estimatiors in Joint Analysis of Repeated Measurements and Survival Time. The Annals of Statistics 33, 2132-2163.
[38] Zweig, M. H. and Campbell, G. (1993). Receiver-operator characteristic plots: a fundamental evaluation tool inclinical medicine. Clinical Chemistry 39, 561-577.