在本篇文章中，我們利用免疫球蛋白抗體Igm值以及注射疫苗的不同(IFN+GMK和GMK)來評估術後黑色素細胞瘤病患的復發狀況。主要是利用聯合模型(joint model)的概念來對資料做分析，聯合模型包含了兩種訊息，其一為長期追蹤資料，其二為存活資訊。以聯合模型所求得的參數估計值具有一致性(consistency)、有效性(efficiency)以及漸進常態(asymptotic normality)的性質。在第一部分我們使用線性隨機效應模型(linear random effect model)對長期追蹤資料做配適，並利用概似比檢定診斷長期追蹤模型的適合度。而在第二部分則是使用Cox比例風險模型描述變數與存活時間。結合這兩部分建構出聯合概似模型且利用EM演算法(expectation maximize algorithm)對參數做估計。接著利用Wald type 拔靴法、Percentile拔靴法及BC percentile 法檢視參數的顯著與否。

In this study, we want to investigate the relationship between recurrent time of patients with resected melanoma and their immunoglobulin M serologic. In addition, we are interested in the influence of different types of vaccines. Since the data includes both information of survival and longitudinal processes, joint model approach is applied to analyze the data. The longitudinal data is described by a linear random effects model, and the survival time is fitted by the Cox model. To derive the estimates of all parameters, Monte Carlo EM algorithm is used by taking random effects as missing. The standard error estimates are obtained through bootstrap re-sampling and corrected by both Percentile bootstrap and BC percentile method.

[1]Ciampi, A. and Etezadi-Amoli, J. (1985). A general model for testing the proportional hazards and the accelerated failure time hypothesis in the analysis of censored survival data with covariate. Communications in Statistics, 14, 651-667.
[2]Cox, D. R. (1972). Regression Models and Life-Tables. Journal of the Royal Statistical Society, 34, 187-220.
[3]Cox, D. R. and Oakes, D. (1984), Analysis of Survival Data, Chapman and Hall,London, New York.
[4]Dafni, U. G. and Tsiatis, A. A. (1998). Evaluating surrogate markers of clinical outcome when measured with error. Biometrics, 54, 1445-1462.
[5]Dubin, J. A., Müller, H. G. and Wang, J. L. (2001).Event history graphs for censored survival data. Statistics in Medicine, 20, 2951-2964.
[6]Efron, B., Tibshirani, R. J. (1993). An introduction to the Bootstrap. Chapman & Hall,New York.
[7]Henderson, R., Diggle, P. and Dobson, A. (2000). Joint
modeling of longitudinal measurements and event time data.Biostatistics, 4, 465-480.
[8]Hsieh, F., Tseng, Y. K. and Wang, J. L. (2006). Joint Modeling of Survival and Longitudinal Data: Likelihood Approach Revisited. Biometrics, 62, 1037-1043.
[9]Kaplan, E. L. and Meier, P. (1958). Nonparametric Estimation from Incomplete Observations. Journal of the American Statistical Association, 53, 457-481.
[10]Laird, N. M. and Ware, J. H. (1982). Random-effects models for longitudinal data. Biometrics, 38, 963-974.
[11]Prentice, R. L. (1982). Covariate measurement errors and parameter estimation in a failure time regression model.Biometrika, 69, 331-342.
[12]Schoenfeld, D. (1980). Chi-squared goodness-of-fit tests for the proportional hazards regression model. Biometrika,67, 145-153.
[13]Schoenfeld, D.A. (1982). Partial residuals for the
proportional hazards regression model. Biometrika, 69,
239-241.
[14]Tseng, Y. K., Hsieh ,F. and Wang, J. L. (2005). Joint modeling of accelerated failure time and longitudinal data.Biometrika, 92, 587-603.
[15]Tsiatis, A. A. and Davidian, M. (2001). A semiparametric estimator for the proportional hazards model with longitudinal covariates measured with error. Biometrika,88, 447-458.
[16]Tsiatis, A. A. and Davidian, M. (2004). Joint Modeling of Longitudinal and Time-to-Event Data: An Overview.
Statistica Sinica, 14, 809-834.
[17]Yan, X. Z.(2010) Joint modeling of longitudinal and
survival data-A case study in patients with resected
melanoma. Master thesis, Graduate Institute of Statistics
National Central University
[18]Wulfsohn, M. S. and Tsiatis, A. A. (1997). A Joint Model for Survival and Longitudinal Data Measured with Error. Biometrics, 53, 330-339.
[19]Zeng, D. and Cai, J. (2005). Asymptotic Results for Maximum Likelihood Estimators in Joint Analysis of Repeated
Measurements and Survival Time. The annals of Statistics,
33(5), 2132-2163.
[20]Zeng, D. and Lin, D. Y. (2007a). Maximum Likelihood
Estimation in Semiparametric Regression Models with
Censored Data (with Discussion). Journal of the Royal
Statistical Society, Series B 69, 507-564.
[21]Zeng, D. and Lin, D. Y. (2007b). Efficient Estimation in the Accelerated Failure Time Model. Journal of the American Statistical Association, 102, 1387-1396.