在生物醫學研究的過程中，經常收集到與時間有關的長期追蹤共變數，若將存活資訊與長期追蹤資料分開進行分析，可能會造成不當的推論。聯合模型將長期追蹤資料與存活資訊同時納入分析，使得估計量具有一致性(consistency)、有效性(efficiency)以及漸進常態(asymptotic normality)的良好性質。Cox聯合模型是最為廣泛使用的，當資料不符合Cox比例風險假設時，則以AFT聯合模型(the joint accelerated failure time model)當作替代。在參數的估計方面，結合長期追蹤資料與存活資訊以建立聯合概似函數，使用EM演算法(expectation-maximumalgorithm)做參數之估計。隨著聯合模型發展，已有許多軟體可以直接進行聯合模型分析：軟體R發展了套件：JM(2008)、Tseng & Yang (manuscript)提供了軟體MATLAB的套件：JointModel，此兩個軟體套件在參數估計時，所使用的方法略有差異，我們將以模擬的方式，對兩者進行比較。最後。藉由地中海果蠅資料進行實例估計。

In clinical research studies, the longitudinal data which pays much attention in recent decade collects the observed event time of interest, called failure time or survival time, along with longitudinal covariates. These outcomes are often separately analyzed and lead us to make biased inferences. Thus, a joint modeling approach is necessarily used to analysis these two parts simultaneously. The estimators obtained from joint model have nice properties, such as consistency, efficiency, and asymptotic normality. The joint Cox model is a popular approach for analyzing survival data, and the joint accelerated failure time model is an alternative approach when the Cox proportional hazard assumption fails. In the part of parameter estimation, we link the longitudinal data and event time, and use the expectation-maximum algorithm to search for the maximum likelihood estimates. There are a code, JM, proposed in the R environment (2010), and a program package of MATLAB, JointModel, proposed by Tseng and Yang (2012 manuscript). We will compare JM and JointModel by simulation studies and apply both packages to analyze the Mediterranean fruit fly data.

Brown ER, Ibrahim JG, DeGruttola V (2005). “A Flexible B-spline Model for Multiple Longitudinal Biomarkers and Survival.” Biometrics, 61, 64-73.
Cox DR (1972). “Regression Models and Life Tables (with discussion).” Journal of the Royal Statistical Society B, 34, 187-220.
Cox DR, Oakes D (1984). “Analysis of Survival Data.” London: Chapman and Hall.
Ding J, Wang JL (2008). “Modeling Longitudinal Data with Nonparametric Multiplicative Random Effects Jointly with Survival Data.” Biometrics, 64, 546-556.
Dimitris R (2010). “JM: An R Package for the Joint Modeling of Longitudinal and Time-to-Event Data.” Journal of Statistical of Software.
Follmann D, Wu M (1995). ”An Approximate Generalized Linear Model with Random Effects for Informative Missing Data.” Biometrics, 51, 151-168.
Guo X, Carlin B (2004). “Separate and Joint Modeling of Longitudinal and Event Time Data using Standard Computer Packages.” The American Statistician, 58, 16-24.
Hsieh FS, Tseng YK, Wang JL (2006). “Joint Modeling of Survival and Longitudinal Data: Likelihood Approach Revisited.” Biometrics, 62, 1037-1043.
Henderson R, Diggle P, Dobson A (2000). “Joint Modeling of Longitudinal Measurements and Event Time Data.” Biostatistics, 1, 465-480.
Lange K (2004). Optimization. Springer-Verlag, New York.
R Development Core Team (2010). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria. ISBN 3-900051-07-0, URL http://www.R-project.org/.
Robins J, Tsiatis AA (1992). “Semiparametric Estimation of an Accelerated Failure Time Model with Time Dependent Covariates.” Biometrika, 79, 311-319.
Rizopoulos D, Verbeke G, Lesaffre E (2009), “Fully Exponential Laplace Approximations for the Joint Modeling of Survival and Longitudinal Data.” Journal of the Royal Statistical Society B, 71, 637-654.
Rizopoulos D, Verbeke G, Molenberghs G (2008), “Shared Parameter Models under Random Effects Misspecification.” Biometrika, 95, 63-74.
Song X, Davidian M, Tsiatis A (2002). “A Semiparametric Likelihood Approach to Joint Modeling of Longitudinal and Time-to-Event Data.” Biometrics, 58, 742-753
Tseng YK, Hsieh FS, Wang JL (2005). “Joint Modeling of Accelerated Failure Time and Longitudinal Data.” Biometrika, 92, 587-603.
Tseng YK, Yang YF. “A MATLAB Package for Longitudinal and Survival data with Cox and AFT Models.” Manuscript.
Tsiatis AA, Davidian M (2004). “Joint Modeling of Longitudinal and Time-to-Event Data: an overview.” Statistica Sinica, 14, 809-834.
Wulfsohn MS, Tsiatis AA (1997). “A Joint Model for Survival and Longitudinal Data Measured with Error.” Biometrics, 53, 330-339.
Yao F, Muller HG, Wang JL (2005). “Functional Data Analysis for Sparse Longitudinal Data.” Journal of the American Statistical Association, 100, 577-590.
Yu M, Law NJ, Taylor JMG, Sandler HM (2004). “Joint Longitudinal-Survival Cure Models and their Application to Prostate Cancer.” Statistica Sinica, 14, 835-862.