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研究生： 陳怡瑄
I-Hsuan Chen
論文名稱： 使用最大概似估計法探討有母數擴充風險模型
Maximum likelihood estimation for parametric extended hazard model
指導教授： 曾議寬
Yi-Kuan Tseng
口試委員:
學位類別： 碩士
Master
系所名稱： 理學院 - 統計研究所
Graduate Institute of Statistics
論文出版年： 2015
畢業學年度： 103
語文別： 中文
論文頁數： 93
中文關鍵詞： 存活資訊 、擴充風險模型 、概似比檢定
外文關鍵詞： Survival, Extended hazard model, Likelihood ratio test
相關次數： 點閱：14  下載：0 
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    • 半母數存活模型在聯合模型中扮演著很重要的角
    色，有關聯合模型的文獻中，探討存活的部分大多假
    設為半母數模型，已經有許多估計參數的方法被提出，
    但是推導標準差時通常是透過拔靴法(bootstrap method)
    而得到的，使用上相當費時。為補足文獻上的這項缺
    失，因此本篇將探討有母數存活模型，並透過費雪資
    訊(Fisher informatione) 有效率的得到標準差。在參數估
    計上，使用參數模型也比半母數模型更有效率，而且參
    數模型被廣泛應用在工業以及醫學上。而在參數模型
    的部分設定為存活分析中常用的Weibull、Log-logistic、
    Gamma 以及Log-normal 四個分配。本篇使用最大概似
    估計法得到參數估計並計算各分配下擴充風險模型的
    AIC 值與概似比統計量(likelihood ratio statistic)。由於擴
    充風險模型為Cox 模型與AFT 模型之廣義模型，本篇將
    擴充風險模型視為完整模型，將Cox 與AFT 模型視為簡
    約模型，因此概似比檢定可以幫助我們透過巢狀結構去做模型選擇，選擇AFT 模型或是Cox 模型。

    So far, in joint model approaches, semi-parametric survival
    model has been played an important role for modelling
    event time data. Although many approaches have been proposed,
    the estimation encounters difficulties in deriving standard
    error estimates through bootstrap method, which is extremely
    time consuming. Therefore, to complement the literature,
    we employ parametric survival model for the joint
    model with standard error estimates obtained from Fisher information.
    The estimation of parametric joint model is dramatically
    faster than that of semiparametric one and thus is
    feasible for practical application. We assume four common
    parametric distributions in survival analysis, Weibull, Loglogistic,
    Log-normal, and Gamma distribution. We use the
    maximum likelihood approach to estimate parameter and to
    calculate AIC value, and likelihood ratio statistic to do model
    selection. Since the extended hazard model is the generalized model for Cox model and AFT model, we regard the extended
    hazard model as the full model. Also, we consider Cox model
    and AFT model as reduced model. Therefore, LRT can be
    conducted to do model selection through nested structure.

    目錄 摘要...................................................................................................................... i Abstract................................................................................................................ i 致謝辭.................................................................................................................. i 第一章緒論........................................................................................................ 1 1.1 研究動機與背景..................................................................................... 1 1.2 本文架構................................................................................................. 5 第二章統計方法................................................................................................ 6 2.1 存活模型................................................................................................. 7 2.1.1 與時間獨立的共變數存活模型......................................... 7 2.1.2 與時間相依的共變數存活模型......................................... 22 2.2 概似函數................................................................................................. 24 2.3 EM 演算法.............................................................................................. 27 2.3.1 E-step .................................................................................. 28 2.3.2 M-step ................................................................................. 29 2.4 數值方法................................................................................................. 33 2.4.1 蒙地卡羅法(Monte Carlo method).................................... 33 2.4.2 牛頓法(Newton-Raphson method).................................... 34 2.5 單純形法(Nelder-Mead Simplex Method)............................................ 36 2.6 奇異值分解法(singular value decomposition)...................................... 38 2.7 概似比檢定(likelihood ratio test).......................................................... 39 第三章模擬研究................................................................................................ 40 3.1 模擬方法................................................................................................. 40 3.1.1 與時間獨立存活模型......................................................... 40 3.1.2 與時間相依存活模型......................................................... 43 3.2 模擬資料之設定..................................................................................... 45 3.3 模擬結果................................................................................................. 46 第四章實例分析................................................................................................ 57 第五章結論與討論............................................................................................ 62 參考文獻.............................................................................................................. 63 附錄...................................................................................................................... 66 圖目錄 圖3.1 真實模型為Cox 且基準風險函數為Weibull 分配的概似比 圖形，n=100 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 圖3.2 真實模型為AFT 且基準風險函數為Log-logistic 分配的概 似比圖形，n=50 . . . . . . . . . . . . . . . . . . . . . . . . . . . 47 圖3.3 真實模型為AFT 且基準風險函數為Log-logistic 分配的概 似比圖形，n=100 . . . . . . . . . . . . . . . . . . . . . . . . . . 48 圖3.4 真實模型為AFT 且基準風險函數為Log-logistic 分配的概 似比圖形，n=200 . . . . . . . . . . . . . . . . . . . . . . . . . . 48 圖3.5 真實模型為Cox 且基準風險函數為Log-logistic 分配的概 似比圖形，n=50 . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 圖3.6 真實模型為Cox 且基準風險函數為Log-logistic 分配的概 似比圖形，n=100 . . . . . . . . . . . . . . . . . . . . . . . . . . 49 圖3.7 真實模型為Cox 且基準風險函數為Log-logistic 分配的概 似比圖形，n=200 . . . . . . . . . . . . . . . . . . . . . . . . . . 50 表目錄 表3.1 給定四個分配下的基準風險函數h0(t) 與S(tj  Z) . . . . . . . 40 表3.2 給定四個分配下的基準風險函數h0(gi(t)) 與S(gi(t)) . . . . 43 表3.3 各分配下模擬資料的參數設定值. . . . . . . . . . . . . . . . 45 表3.4 真實模型為AFT 模型且基準風險函數為Weibull 分配的參 數估計. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 表3.5 真實模型為Cox 模型且基準風險函數為Weibull 分配的參 數估計. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 表3.6 基準風險函數為Weibull 分配之擴充風險模型p 值. . . . . . 53 表3.7 真實模型為AFT 模型且基準風險函數為Log-logistic 分配 的參數估計. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 表3.8 真實模型為Cox 模型且基準風險函數為Log-logistic 分配 的參數估計. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 表3.9 基準風險函數為Log-logistic 分配之擴充風險模型p 值. . . . 56 表4.1 基準風險函數為Weibull 分配下的參數估計結果. . . . . . . 59 表4.2 基準風險函數為Log-logistic 分配下的參數估計結果. . . . . 59 表4.3 Weibull、Log-logistic 分配下的AIC 之值. . . . . . . . . . . 60 表4.4 Weibull、Log-logistic 分配下的的LRT 值. . . . . . . . . . . 61 表4.5 使用R 統計軟體給定在不同分配下其AIC 值. . . . . . . . . 61

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