在生物醫學的研究的過程中，有興趣的是時間相依共變量與存活時間的關聯性，而推估關聯性最常使用的是Cox比例風險迴歸模型。傳統上使用Cox(1972)的部分概似法估計參數，但前提是必須有所有研究對象的完整共變量資訊且不允許誤差。為了減少部分概似法對於有遺失值時其參數估計上的偏差，本研究分別採用二階段方法以及聯合模型(Wulfsohn，1997)來估計參數，目的是比較此三種方法之下，對於不同遺失比例其參數估計值變化情形，以及通過比例風險假設之比例，以利於不同條件之下選擇最有效之方法，模擬結果在不同的共變量軌跡之下，若無遺失比例之發生則可選擇程式效率較高之部分概似法，在有遺失比例發生但測量誤差不大時可選擇二階段方法，若有遺失比例之發生且測量誤差較高時，則需使用聯合模型來估計參數，其結果較二階段模型快速且準確。

The relationship between longitudinal covariates and a failure time process can be assessed using the Cox proportional hazards model. The purpose of the study is to evaluate the performance of three approaches, the partial likelihood, two-stage partial likelihood , and joint model approaches for the Cox model when the covariate is measured in irregular times with measurements error. The results show that partial likelihood is an efficient choice when data has no missing values; two-stage model can be selected for data with small measure error. As a conclusion, joint modeling approach is the best choice in all situations.

[1] Anderson, P. K. and Gill, R. D. (1982). Cox’s regression model forcounting processes, a large sample study. Annals of Statistics 10.1100-1120.
[2] Burden, Richard L. and Faires, J. Douglas (2000). Numerical Analysis (7th ed.). Brooks/Cole
[3] Cox, D. R. (1972). Regression models and life-tables. Journal of the Royal Statistical Society 34(B). 187-220.
[4] Cox, D. R. (1975). Partial likelihood. Biometrika 62. 269-276.
[5] Efron, B. (1994). Missing data, imputation and bootstrap (with Discussion). J. Am. Statist. Assoc. 89. 463-479.
[6] Hsieh, F., Tseng, Y. K., and Wang, J. L. (2006). Joint modelingof survival time and longitudinal data: likelihood approach revisit.Biometrics, 62.1037-1043.
[7] Klein, J. P. and M. L. Moeschberger (1997). Survival Analysis: Techniques for Censored and Truncated Data. Springer.
[8] Lawless, J. F. (1982). Statistical Models and Methods for Lifetime Data. John Wiley & Sons, Inc., New York.
[9] Louis, T. A. (1982). Finding the observed Fisher information when using the EM algorithm. Journal of the Royal Statistical Society, Series B 44. 226-233.
[10] Lin, D. Y. and Ying, Z. (1993). Cox regression with incomplete covariate measurements. Journal of the American Statistical Association 88(424). 1341-1349.
[11] Orchard, T. and Woodbury, M. A. (1972). A missing information principle: Theory and applications. In Proceedings of the 6th Berkeley Symposium on Mathematical Statistics and Probability, Volume 1,697-715. Berkeley: University of California Press.
[12] Prentice, R. L. (1982). Covariate measurement errors and parameter estimation in failure time regression model. Biometrika 69. 331-342
[13] Press, W. H., Teutolsky, S. A., Vetterling, W. T., and Flannery,B.P. (1992). Numerical recipes in FORTRAN: the art of scientific computing. New York, NY, USA: Cambridge University Press, 2nd ed.
[14] Paik, M. C., and Tsai, W. Y. (1997). On using the Cox proportional hazards model with missing covariates. Biometrika 84(3). 579-593.
[15] Schafer, D.W.(1987). Covariate measurement errors in generalized linear models. Biometrika 74. 385-391.
[16] Tierney, L. and Kadane, J. B. (1986). Accurate approximation for posterior moments and marginal densities. Journal of the American Statistical Association 81. 82-86.
[17] Tseng, Y. K., Hsieh F. and Wang J.L. (2005). Joint modeling of accelerated failure time and longitudinal data., Biometrika 92. 587-603.
[18] Tseng, Y. K. and Hsieh, Y. H.. A Joint model approach for evaluating the efficacy of HAART treatment for AIDS patients in Taiwan. Manuscript.
[19] Wulfsohn, M. S. and Tsiatis, A. A. (1997). A joint model for survival and longitudinal data measured with error. Biometrics 53. 330-339.
[20] 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.
[21] Zhou, H. and Pepe, M. S. (1995). Auxiliary covariate data in failure time regression. Biometrika 82. 139-149.
[22] 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). 2132-63.