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研究生:徐人華
研究生(外文):Ren-hua Hsu
論文名稱:復發事件存活分析的共享廣義伽瑪脆弱因子之半母數聯合模型
論文名稱(外文):Semi-parametric Joint Model with Generalized Gamma Frailty for Recurrent Event Survival Analysis
指導教授:曾議寬曾議寬引用關係
學位類別:碩士
校院名稱:國立中央大學
系所名稱:統計研究所
學門:數學及統計學門
學類:統計學類
論文種類:學術論文
論文出版年:2015
畢業學年度:103
語文別:中文
論文頁數:57
中文關鍵詞:存活分析
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本研究建立一個可處理長期追蹤資料及多重事件的半母數聯合模型,以Cox比例風險模型或加速失敗模型的無特定分布準線風險配適各事件時間,事件間的相關性以共享脆弱模型描述,其脆弱因子服從廣義伽瑪分布。使用最大概似估計法估計參數時,會將線性混合效應模型及脆弱模型中的隨機效應當作遺失值處理,所以最大期望演算法將被用於尋找最大概似估計量。在E步驟中將使用蒙地卡羅積分法求得複雜積分的近似值,在M步驟中使用Nelder-Mead單純形法尋找最大概似估計量。最後用愛滋病資料驗證此方法的有效性。
In this study, we establish a more general semi-parametric joint model, which can deal with not only the single event but also the multiple events. We use the unspecified baseline hazard with Cox proportional hazards model or accelerated failure time model to fit the multiple event times with correlation between the events described by shared frailty model. We assume that frailty factor is from the generalized gamma distribution. When estimating the parameters, we treat the random effects from linear mixed effect model and shared frailty model as missing values, thus expectation-maximization algorithm can be implemented to find the maximum likelihood estimates. In E-step, Monte Carlo integration method is used to approximate complex integrals. In M-step, we adopt Nelder-Mead simplex method to find the maximum likelihood estimates. AIDS data is used to demonstrate the usefulness of the proposed method.
摘要 i
Abstract ii
致謝 iii
第一章緒論 1
第二章研究方法 7
2.1 Cox 比例風險模型(Cox Proportional Hazards Model) 9
2.2 加速失敗時間模型(Accelerated Failure Time Model) 10
2.3 共享脆弱模型(Shared Frailty Model) 11
2.4 聯合模型(Joint Model) 12
2.5 聯合概似函數(Likelihood Function) 14
2.6 最大期望演算法(Expectation-Maximization algorithm) 18
2.6.1 E 步驟(E-step) 18
2.6.2 M 步驟(M-step) 21
2.7 數值方法(Numerical Methods) 24
2.7.1 蒙地卡羅法(Monte Carlo method) 24
2.7.2 Nelder-Mead 單純形法(Nelder-Mead Simplex method) 25
第三章模擬研究 30
第四章實例分析 35
第五章結論與討論 39
參考文獻 42
Andersen, P. K. & Gill, R. D. (1982). Cox's regression model for counting processes: A large sample study. Annals of Statistics, 10, 1100-1120.
Balakrishnan, N. & Peng, Y. W. (2006). Generalized gamma frailty model. Statistics in Medicine, 25, 2797-2816.
Cox, D. R. (1972). Regression Models and Life Tables (with discussion). Journal of the Royal Statistical Society B, 34, 187-220.
Cox, D. R. & Oakes, D. (1984). Analysis of Survival Data. London: Chapman and Hall.
Dempster, P., Laird, N. M. & Rubin, D. B. (1977). Maximum likelihood from incomplete data via the EM algorithm. Journal of the royal statistical society B, 39, 1-38.
Efron, B. & Tibshirani, R. J. (1994). An introduction to the Bootstrap. New York: Chapman and Hall.
Hsieh, F. S., Tseng, Y. K. & Wang, J. L. (2006). Joint Modeling of Survival and Longitudinal Data: Likelihood Approach Revisited. Biometrics, 62, 1037-1043.
Klein, J. P. (1992). Semiparametric Estimation of Random Effects Using the Cox Model Based on the EM Algorithm. Biometrics, 48, 795-806.
Lagarias, J. C., Reeds, J. A., Wright, M. H. & Wright, P. E. (1998). Convergence Properties of the Nelder--Mead Simplex Method in Low Dimensions. SIAM Journal on Optimization, 9, 112-147.
Lim, H. J., Liu, J. & Melzer-Lange, M. (2007). Comparison of Methods for Analyzing Recurrent Events Data: Application to the Emergency Department Visits of Pediatric Firearm Victims. Acceident Analysis and Prevention, 39, 290-299.
Liu, L. & Huang, X. (2009). Joint analysis of correlated repeated measures and recurrent events processes in the presence of death, with application to a study on acquired immune deficiency syndrome. Journal of the Royal Statistical Society C, 58, 65-81.
Nelder, J. A. & Mead, R. (1965). A simplex method for function minimization. Computer Journal, 7, 308-313.
Prentice, P. L., Williams, B. J. & Peterson, A.V. (1981). On the regression analysis of multivariate failure time data. Biometrika, 68, 373-379.
Tseng, Y. K., Hsieh, F. S. & Wang, J.L. (2005). Joint Modeling of Accelerated Failure Time and Longitudinal Data. Biometrics, 92, 587-603.
Tseng, Y. K., Su, Y. R., Mao, M. & Wang, J. L. (2015). An Extended Hazard Model with Longitudinal Covariates. Biometrika, 102, 135-150.
Tseng, Y. K. & Yang, Y. F. (2014). A kernel smooth approach for joint modeling of accelerated failure time and longitudinal data. Communications in statistics simulation and computation.
Tsiatis, A. A. & Davidian, M. (2004). Joint modelling of longitudinal and time-to-event data: an overview. Statistica Sinica, 14, 809-834.
Vaupel, J. W., Manton, K. G. & Stallard, E. (1979). The impact of heterogeneity in individual frailty on the dynamics of mortality. Demography, 16, 439-454.
Wei, L. J., Lin, D. Y. & Weissfeld, L. (1989). Regression analysis of multivariate incomplete failure time data by modeling marginal distributions. Journal of American Statistical association, 84, 1065-1073.
Wulfsohn, M. S. & Tsiatis, A. A. (1997). A Joint Model for Survival and Longitudinal Data Measured with Error. Biometrics, 53, 330-339.
Yang, Y. F. (2013). Joint Model of Longitudinal and Survival Data-New Approach and Numerical Improvement. Ph.D. Thesis, National Central University.
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, 2132-2163.
Zeng, D. & Lin, D. Y. (2007). Efficient Estimation in the Accelerated Failure Time Model. Journal of the American Statistical Association, 102, 1387-1396.
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