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研究生:童宜亮
研究生(外文):Yi-Liang Tung
論文名稱:群聚資料下混合效應模式的半參數化貝氏分析
論文名稱(外文):Semiparametric Bayesian Analysis of Mixed Models for Clustered data
指導教授:蕭朱杏蕭朱杏引用關係
指導教授(外文):Chuhsing Kate Hsiao
學位類別:博士
校院名稱:國立臺灣大學
系所名稱:流行病學研究所
學門:醫藥衛生學門
學類:公共衛生學類
論文種類:學術論文
論文出版年:2004
畢業學年度:92
語文別:中文
論文頁數:121
中文關鍵詞:邊際機率計數資料二元資料馬可夫鏈蒙地卡羅法長期追蹤資料無參數化迴歸懲罰性節點變化係數混合效應模式貝氏因子貝氏模式比較相加性混合效應模式
外文關鍵詞:Longitudinal dataCount dataBinary dataMarkov Chain Monte CarloMarginal likelihoodNonparametric regressionVarying-coefficient mixed modelsPenalized splinesBayesian model comparisonBayes factorAdditive mixed models
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這篇論文的內容是針對群聚資料(包括了長期追蹤資料),提出半參數化混合效應模式(semiparametric mixed models)的完整貝氏分析方法。所討論的半參數化模式包含了相加性混合效應模式(additive mixed models)與變化係數混合效應模式(varying-coefficient mixed models)。我利用懲罰性節點法(penalized spline)來發展無參數迴歸函數的事前分布以進行貝氏分析。所提出的事前分布有別於目前文獻中其他形式的事前分布,主要差異就是它是可積分的(proper);而且不像平滑節點法(smoothing spline)需要利用樣本中不重複的所有解釋變數值來決定節點數目與位置,我所提出的是由分析者自己決定節點數目與位置。這兩個差異使得所提出的方法具有計算上的優勢,特別是處理樣本數很大的資料集,也因為可積分的性質使得模式選取可以利用比較不同模式的邊際概似函數來達成。此外,相較於古典統計學派的方法,所提出的分析法可比較容易的延伸到非連續的資料結構上,並不需要依賴特異的平滑參數估計準則。模式參數的估計過程是利用吉布斯抽樣法(Gibbs sampler)模擬產生事後分布的樣本以進行推論。我仔細的討論不同資料結構下的詳細抽樣步驟,這包括了連續、二元與計數資料。特別是計數資料由於不具有事前與事後分布的共軛特性(conjugate),我提出了更有效率的更新過程以完成吉布斯抽樣法。模擬研究顯示所提出的貝氏分析法能夠正確的估計出無參數迴歸函數的趨勢。除了考慮估計的問題外,更進一步也討論了模式選取的問題。主要是探討半參數化與參數化模式的邊際概似函數估計方法,因而使得所進行的模式比較不再侷限於巢式(nested)的關係。最後,我將所提出的方法應用到AIDS世代研究(Kaslow et al. 1987)的例子上。分析結果顯示會影響HIV感染者體內CD4+細胞數目的解釋變數是感染者感染前的CD4+細胞數目,而HIV感染者體內CD4+細胞數目隨時間變化的趨勢是不斷線性遞減的。在同時考量參數化與半參數化模式後,參數化的線性混合效應模式是資料較支持的可能模式。
In this thesis I consider Bayesian semi-parametric analysis of mixed-effects models for clustered data. Particularly, I consider the additive mixed model and varying-coefficient mixed model, and use nonparametric arbitrary smooth functions to represent the covariate effects. I model the nonparametric functions using the qth-degree polynomial penalized splines with fixed knots, and specify the prior for the corresponding smoothing parameter of each function. A computationally efficient Markov chain Monte Carlo (MCMC) algorithm is proposed to simulate posterior samples for inference. In addition to the continuous response setting, the binary and count data are also considered and discussed in detail. Special attention is necessary due to the non-conjugacy for binary data with logit link and count data with log link. I also develop a modified Metropolis-Hastings algorithms to mix the Markov chain and increase the speed. The simulation studies show that the posterior mean via nonparametric approach captures well the true functional forms. In addition to the estimation, I also address the problem of model choice between the competing parametric and semi-parametric specifications using marginal likelihoods and Bayes factors. Finally, the data of multicenter AIDS cohort study (Kaslow et al. 1987) are considered for illustration.
第一章、緒論……………………………………………………… 1
第一節、迴歸模式的發展背景…………………………………… 1
第二節、引發研究動機的實例…………………………………… 4
第三節、研究目標………………………………………………… 5
第四節、論文大綱………………………………………………… 6
第二章、研究回顧………………………………………………… 8
第一節、無參數化迴歸模式……………………………………… 8
(一)、平滑節點法……………………………………………… 9
(二)、懲罰性節點法…………………………………………… 11
(三)、核估計法………………………………………………… 12
第二節、貝氏模式篩選…………………………………………… 13
(一)、Laplace近似法………………………………………… 14
(二)、MCMC抽樣法……………………………………………… 15
(三)、結合模擬與Laplace近似法…………………………… 17
第三章、相加性混合效應模式之統計分析……………………… 18
第一節、現有的分析方法………………………………………… 18
第二節、連續資料之相加性混合效應模式……………………… 22
(一)、階層式架構與概似函數………………………………… 23
(二)、事前分布的設定………………………………………… 24
(三)、MCMC抽樣法……………………………………………… 26
(四)、模擬研究………………………………………………… 32
第三節、以Probit為連結函數的二元資料相加性混合效應模式 39
(一)、階層式架構與潛在變數法……………………………… 40
(二)、事後分布的模擬………………………………………… 42
(三)、模擬研究………………………………………………… 47
第四節、以Log為連結函數的計數資料相加性混合效應模式… 50
(一)、概似函數與階層式架構………………………………… 51
(二)、MCMC抽樣法……………………………………………… 52
(三)、模擬研究………………………………………………… 56
第四章、變化係數混合效應模式之統計分析…………………… 61
第一節、現有的分析方法………………………………………… 61
第二節、連續資料之變化係數混合效應模式…………………… 64
(一)、事前分布的設定與概似函數…………………………… 64
(二)、MCMC抽樣法……………………………………………… 66
(三)、模擬研究………………………………………………… 69
第三節、二元資料之變化係數混合效應模式…………………… 73
(一)、事後分布的性質………………………………………… 74
(二)、事後分布的模擬………………………………………… 75
(三)、模擬研究………………………………………………… 77
第四節、計數資料之變化係數混合效應模式…………………… 80
(一)、事後分布的性質………………………………………… 80
(二)、事後分布的模擬………………………………………… 81
第五章、模式比較………………………………………………… 83
第一節、連續資料的情況………………………………………… 84
(一)、事後分布機率的估計…………………………………… 86
(二)、概似函數的估計………………………………………… 90
(三)、VCMM之邊際機率估計…………………………………… 91
(四)、模擬研究………………………………………………… 93
第二節、二元資料的情況………………………………………… 96
(一)、Probit連結函數相加性混合效應模式的邊際機率估計 96
(二)、Probit連結函數變化係數混合效應模式邊際機率估計 100
(三)、模擬研究………………………………………………. 100
第三節、計數資料的情況………………………………………… 102
(一)、概似函數的估計………………………………………… 102
(二)、事後分布機率的估計…………………………………… 106
第六章、實例分析………………………………………………… 107
第七章、結論與討論……………………………………………… 115
參考文獻…………………………………………………………… 117
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