近十年來，利用雞尾酒療法 (高效抗逆轉錄病毒療法) 治療愛滋病 (後天免疫缺乏症候群)，已經被廣泛提倡且使用。目前也有大量的研究顯示，此種療法對於治療愛滋病確實有正面的效果。而在各種衡量愛滋病情嚴重程度的指標當中，CD4/CD8 比值是其中一種衡量的指標。本篇論文則是希望透過該指標，探討雞尾酒療法對於愛滋病人存活時間的影響，進而瞭解 CD4/CD8 比值和愛滋病人存活時間的關係。由於此類臨床資料通常同時包含長期追蹤資料及存活時間，為了能有效的進行長期追蹤資料的分析，且在不失去存活資訊的情況下，本篇論文使用了 Wulfsohn 和 Tsiatis (1997) 所提出的聯合模型，並且針對此模型做延伸，導入蒙地卡羅的數值方法，使得此聯合模型方更簡便也更有效率。同時，本篇論文也利用了三種圖形對於資料做了初步的描述，如趨勢圖、事件歷史圖 (Dubin, M"uller, and Wang, 2001) 和 3-D 平滑曲面圖。以上三種圖形不僅能同時呈現出長期追蹤資料的資訊和存活資訊，亦有助於瞭解資料的特性。最後，利用以上所述之方法應用於台灣 160 位愛滋病已發病病人的資料，實際分析其療法之成效。

The famous treatment, HAART (Highly Active Antiretroviral Therapy), has been advocated for the recent decade for AIDS (Acquired Immune Deficiency Syndrome) patients. Also, large amount of researches reveal its efficacy for treating AIDS. The biomarker, CD4/CD8 ratio, is one of the important way of detecting the disease progression. The objective of this paper is to probe the impact of HAART on survival time for AIDS patients and identify the association between CD4/CD8 ratio and survival time for AIDS patients. In order to combine effectively both longitudinal and survival information from this kind of data, we propose an extension of joint model in Wulfsohn and Tsiatis (1997). Instead of Gauss-Hermite quadrature, Monte Carlo integration is employed in the joint model to make it easier to implement and more efficient to derive estimates. Meanwhile, graphic skills such as profile, event history graph (Dubin, M"uller, and Wang, 2001), and 3-D surface accompanied contour plots are proffered not only to exhibit simultaneously the longitudinal and survival information but also to detect the features of data. The aforementioned techniques are applied to a clinical data of AIDS patients in Taiwan.

[1] Clementi, M., Forabosco, P., Amadori, A., Zamarchi, R., De Silvestro, G., Di Gianantonio, E., Chieco-Bianchi, L., and Tenconi, R. (1999), “ CD4 and CD8 T lymphocyte inheritance. Evidence for major autosomal recessive genes,” Human Genetics, 105, 337–342.
[2] Cox, D. R. (1972), “Regression models and life tables”. Journal of the Royal Statistical Society, Series B, 34, 187-220.
[3] Dafni, U. (1993), “Evaluating surrogate markers of clinical outcome when measured with error,” Ph.D. dissertation, Harvard School of Public Health.
[4] Dempster, A. P., Laird, N. M., and Rubin, D. B. (1977), “Maximum likelihood from incomplete data via the EM algorithm,” Journal of the Royal Statistical Society, Series B, 39, 1-38.
[5] Dubin, J., M"uller, H. G., Wang, J. L. (2001), “Event history graphs for censored survival data,” Statistics in Medicine, 20, 2951-2964.
[6] Egger, M., Hirschel, B., Francioli, P., Sudre, P., Wirz, M., Flepp, M., Rickenbach, M., Malinverni, R., Vernazza, P., and Battegay, M. (1997), “Impact of new antiretroviral combination therapies in HIV infected patients in Switzerland: prospective multicentre study. Swiss HIV Cohort Study,” British Medical Journal, 315, 1194-1199.
[7] Faucett, C. J. and Thomas, D. C. (1996), “Simultaneously modelling censored survival data and repeatedly measured covariates: a gibbs sampling approach," Statistics in Medicine, 15, 1663-1685.
[8] Haynes, B. F., Pantaleo, G., and Fauci, A. S. (1996), “Toward an understanding of the correlates of protective immunity to HIV infection,” Science, 271, 324–328.
[9] Hsieh, F., Tseng, Y. K., and Wang, J. L. (2006), “Joint modeling of survival time and jongitudinal data: likelihood approach revisit,” Biometrics, 62, 1037-1043.
[10] Jiao, Y., Li, T. S., Xie, J., Han, Y., Qiu, Z. F., Zuo, L. Y., Thomas, M., and Wang, A. X. (2006), “Association between Nef-specific CD8 T-cell responses and disease progression in HIV-1 subtype B infection,” Chinese Medical Journal, 119, 1609-1615.
[11] Kaplan, E. L. and Meier, P. (1958), “Nonparametric estimation from incomplete observations,” Journal of the American Statistical Association, 53, 457-481.
[12] Laird, N. M. and Ware, J. H. (1982), “Random-effects models for longitudinal data,” Biometrics, 38, 963-974.
[13] Palella, F. J., Delaney, K. M., Moorman, A. C., O., L. M., Fuhrer, J., Satten, G. A., Aschman, D. J., and Holmberg, S. D. (1998), “Declining morbidity and mortality among patients with advanced human immunodeficiency virus infection. HIV Outpatient Study Investigators,” New England Journal of Medicine, 338, 853-60.
[14] Prentice, R. L., Williams, B. J., and Peterson, A. V. (1981), “On the regression analysis of multivariate failure time data,” Biometrika, 68, 373-379.
[15] 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.
[16] Taylor, J. M., Fahey, J. L., Detels, R., and Giorgi, J. V. (1989), “CD4 percentage, CD4 number, and CD4:CD8 ratio in HIV infection: which to choose and how to use,” Journal of acquired immune deficiency syndromes and human retrovirology, 2, 114-124.
[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. (2006), “A Joint model approach for evaluating the efficacy of HAART treatment for AIDS patients in Taiwan,” Submitted.
[19] Tsiatis, A. A., DeGruttola, V., and 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.
[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] Wei, L. J., Lin, D. Y., and Weissfeld, L. (1989), “Regression analysis of multivariate incomplete failure time data by modeling marginal distributions,” Journal of the American Statistical Association, 84, 373-379.
[22] Wulfsohn, M. S. and Tsiatis, A. A. (1997), “A joint model for survival and longitudinal data measured with error,” Biometrics, 53, 330-339.