在存活分析中，半母數模型通常用 Cox 比例風險模型或加速失敗時間模型來描述變數與存活資訊間的關係，而當變數的資料型態為長期追蹤資料時，可以使用聯合模型做配適，以消除因資料遺失所造成的偏誤。在使用聯合模型的文獻中，大多數都是以 Cox 比例風險模型的風險函數作為架構，搭配混合線性模型組合成聯合模型。但是當資料不符合比例性的假設時，使用 Cox 比例風險模型並不合理，因此 Tseng et al.(2005) 提出可以使用加速失敗時間模型來替換 Cox 比例風險模型。本文引用 Tseng et al.(2005) 提出之加速失敗時間模型架構下的聯合模型，介紹如何使用 EM 演算法估計未知參數與未知的基準風險函數，以及使用拔靴法估計參數的標準差。並將模型應用在 1000 隻雌性地中海果蠅的資料上，其中以果蠅的每日產卵數作為繁殖力的指標，存活時間作為老化程度的指標，以模型探討繁殖力與老化間的關係；此外依照總產卵量大小對果蠅進行分組，用以探討總產卵量的多寡是否會影響參數的估計以及變數與存活時間之間的關係。

In survival analysis, Cox proportional hazard model or accelerated failure time model are used to describe the relationship between variate and survival information in semi-parametric model. When the type of the variate data becomes a longitudinal data, the joint model will be suitable. Most literature would use the Cox proportional hazard model as its basis and then combine it with the mixed linear model to form the joint model. However, it will be unreasonable to use Cox proportional hazard model when the data can''t fit in the proportionality assumption. Consequently, Tseng et al. (2005) suggests to use the accelerated failure time model instead. This thesis will use the joint model of linera mixed dffdct model and accelerated failure time model that proposed by Tseng et al. (2005), and to introduce how to estimate the unknown parameter and baseline hazard function by using EM algoritme and estimate the standard error by bootstrap as well. Besides, this model can also be applied to the study of 1000 female medflies.As for the method, we can use the reproductiive egg-laying data as the index of fecundity and the longevity as the index of ageing to observe the relationship between the fecundity and ageing. What''s more, group the medflies according to their total amont of edd-lying data to study if it will affect the estimation of parameter and the relationship between variate and survival time.

1. Carey, J.R., Liedo, P., M¨uller, 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.
2. Chiou, J.M., M¨uller, H.G. & Wang, J.L. (2003). ”Functional quasilikelihood regression models with smooth random effects.” J. Royal Statist. Soc. B 65, 405-423.
3. Chiou, J.M., M¨uller, H.G. & Wang, J.L., Carey, J.R.(2003). ”A functional multiplicative effects model for longitudinal data , with application to reproductive histories of female medflies.” Statistica Sinica 13, 1119-1133.
4. Chiou, J.M., M¨uller, H.G. &Wang, J.L. (2004). ”Functional response models.” Statistica Sinica 14, 675-693.
5. Dubin, J., M¨uller, H.G. & Wang, J.L. (2001). ”Event history graphs of censored survival data.” Statistics in Medicine 20,2951-2964.
6. Efron, B. (1994). ”Missing data, imputation and bootstrap (with Discussion).” J. Am. Statist. Assoc. 89, 463-479.
7. Efron, B. & Tibshirani, R.J. (1993). An Introduction to the Bootstrap. New York: Chapman & Hall.
8. Grizzle, J.E. & Allen, D.M. (1969). ”Analysis of growth and dose response curves.” Biometrics 2, 357-381. 9. Hayflick, L. & Moorhead, P.S.(1961). ”The serial cultivation of human diploid cell strains.” Exp. Cell. Res. 25, 585-621.
10. Henderson, R., Diggle, P. & Dobson, A. (2000). ”Joint modelling of longitudinal measurements and event time data.” Biostatistics 4, 465-480.
11. Hsieh, F., Tseng, Y.K. & Wang, J.L. (2006). ”Joint Modeling of Survival Time and Longitudinal Data: Likelihood Approach Revisit.”Biometrics 62, 1037-1043.
12. Hui, S.L. (1984). ”Curve fitting for repeated measurements made at irregular time points.” Biometrics 40, 691-697.
13. Kirkpatrick, S., Gelatt, C.D.JR. & Vecchi, M.P.(1983). ”Optimization by simulated annealing.” Science 220, 671-680.
14. Laird, N.M. & Ware, J.H. (1982). ”Random-effects models for longitudinal data.” Biometrics 38, 963-974.
15. M¨uller, H.G., Carey, J.R., Wu, D., Liedo, P. & Vaupel, J.W. (2001). ”Reproductive potential predicts longevity of female Mediterranean fruit flies.” roceedings of the Royal Society B 268, 445-450.
16. Nelder, J.A. & Mead, R. (1965). ”A simplex method for function minimization.” Comp. J. 7, 308-313.
17. Prentice, R.L. (1982). ”Covariate measurement errors and parameter estimation in a failure time regression model.” Biometrika 69, 331-342.
18. M.R. Rose, H.K. Passananti & M. Matos. (2004). Methuselah Flies: A Case Study in the Evolution of Aging. Singapore: World Scientific Publishing.
19. Rao, C.R. (1965). ”The theory of least squares when the parameters are stochastic and its application to the analysis of growth curves.” Biometrika 52, 447-458.
20. Taylor, J.M.G., Aumberland, W.G. & Sy, J.P. (1994). ”A stochastic model for analysis of longitudinal AIDS data.” Journal of the American Statistical Association 89, 727-736.
21. Taylor, J.M.G. & Law, N. (1998). ”Does the covariance structure matter in longitudinal modelling for the prediction of future CD4 counts?” Statistics in Medicine 17, 2381-2394.
22. Tsiatis A.A., DeGruttola V. & Wulfsohn 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.
23. Tsiatis A.A. & Davidian M. (2001). ”A semiparametric estimator for the proportional hazards model with longitudinal covariates measured with error.” Biometrika 88, 446-458.
24. Tsiatis A.A. & Davidian M. (2004). ”Joint modeling of longitudinal and time-to-event data: an overview.”Statistica Sinica 14, 809-834.
25. Tseng Y.K., Hsieh F. & Wang J.L. (2005). ”Joint modeling of accelerated failure time and longitudinal data.” Biometrika 92, 587-603.
26. Wang Y. & Taylor J.M.G. (2001). ”Jointly modeling longitudinal and event time data with application to acquired immunodeficiency syndrome.” J. Am. Statist. Assoc. 96, 895-905.
27. Wulfsohn M.S. & Tsiatis A.A. (1997). ”A Joint Model for Survival and Longitudinal Data Measured with Error.” Biometrics 53, 330-339.
28. Yu M., Law N.J., Taylor J.M.G. & Sandler H.M. (2004). ”Joint longitudinal-survival-cure models and their application to prostate cancer.” Statist. Sinica 14, 835-862.
29. Zeng D. & 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.