請用此 Handle URI 來引用此文件: http://hdl.handle.net/11455/49768
標題: 多變量偏斜t分佈對於長期資料分析之研究
Longitudinal Data Analysis Using the Multivariate Skew t Distribution
作者: 林宗儀
關鍵字: 數學類
基礎研究
Bayesian inference
貝氏推論
最大概式估計
MST分佈
離群值
預測
隨機效應
maximum likelihood estimation
multivariate skew t distribution
outliers
prediction of future values
random effects
摘要: 線性混合效應模型 (LMM; Laird與Ware, 1982) 已被廣泛地應用在分析連續型之長期資料,特別是從富於變化性之生物醫學研究所產生的資料。LMM的重要優勢是它能在模式化實驗對象之間與實驗對象本身因不同觀察時間所產生的變異中提供相當大的彈性。在LMM的架構中,在考量數理推導的易於處理和計算上的方便,隨機效應和誤差項經常被假設為常態分佈。然而,當常態性的隨機項嚴重背離時會使得模型欠缺穩健性,將導致無效的推論和不合理的估計。在過去幾年中,利用更一般化的分佈來找尋具穩健性之線性混合效應模型已受到研究學者的重視。Pinheiro等人(2001) 開發t分佈線性混合效應模型 (TLMM),並由實際例子和模擬演算來顯示此模型能排除離群值影響。而後,Lin 與 Lee (2006; 2007) 分別在古典統計和貝氏統計的觀點下對此方向之研究作更進一步地探討。依據Azzalini 與 Dalla Valle (1996) 和Azzalini 與 Capitaino (1999) 所提出的多變量偏斜常態分佈,Lin 與 Lee (2008) 提出了一個隨機效應為偏斜常態分佈的線性混合效應模型 (SNLMM).在此計畫中,我們致力於偏斜t線性混合效應模型 (STLMM),其隨機效應和誤差項之聯合分佈為Azzalini 與 Capitaino (2003)所定義的多變量偏斜t分配。STLMM有一個的特點是它可以被寫成兩個隨機表示式。此隨機表示式對於建立參數估計之EM類型演算法和其他理論目的是非常有幫助的。此外,LMM、TLMM以及SNLMM皆可視為STLMM的特例。我們首先提出具交換性期望值條件最大化演算法(AECM, Meng 與 van Dyk, 1997)來計算STLMM中參數的最大概似估計值。參數估計值的標準差可透過Louis' 方法 (1982)獲得,此方法執行方便且具有彈性。除此之外,我們對此模型也致力於發展貝氏分析技術。在貝氏處理中,我們所關心的問題是參數先驗分佈的選定、參數的後驗分佈、馬可夫鏈蒙地卡羅 (MCMC) 演算法的執行、其收斂評估、以及貝氏預測分佈等等。提出的方法將藉由實例和模擬研究加以闡述說明。由於受測者提早離開實驗或是不在計劃安排之時間回診等原因,缺失值經常在實際研究中出現 (Laird, 1988; Schluchter, 1988)。每位測試者之測量數量不等或是間歇性遺漏值造成資料缺失的現象會產生不均衡模式。假設缺失值是屬可忽略機制下之隨機遺漏(MAR; Rubin, 1976; Little 與 Rubin, 2002),對僅有部份觀察值之受測者,我們將為沒觀察到部份提供最大概似與貝氏預測估計式。同時,我們也利用這些估計式處理未來反應值之預測。
Linear mixed models (LMM; Laid and Ware, 1982) have been frequently used to analyze continuous longitudinal data arising particularly from a wide variety of biometrical studies. One important strength of LMM is attributed to the fact that it offers a great flexibility in modeling the between- and within-subjects correlations. In the framework of LMM, the random effects and error terms are routinely assumed to have a normal distribution due to their mathematical tractability and computational convenience. However, a serious departure of normality may suffer from lack of robustness and subsequently lead to invalid inference and unreasonable estimates.Over the past few years, seeking a robust extension of LMM with a more general family of distributions has received considerable attention. Pinheiro et al. (2001) developed a t linear mixed model (TLMM) and demonstrated its robustness against outliers via a real example and several simulations. Further developments along this direction have been discussed in Lin and Lee (2006; 2007) from classical and Bayesian perspectives, respectively. Lin and Lee (2008) proposed a skew normal linear mixed model (SNLMM) with random effects following the multivariate skew-normal distribution of Azzalini and Dalla Valle (1996) and Azzalini and Capitaino (1999).In this project, we focus on the use of skew t distribution of Azzalini and Capitaino (2003) for the random effects and Student's t distribution for the within-subject errors, called the skew t linear mixed model (STLMM). An appealing key feature of STLMM is that it can be formulated as two flexible stochastic representations, which are useful for easy setup of EM-type algorithms for estimation and other theoretical purposes. Note that the LMM, TLMM and SNLMM can be viewed as special cases in this family.We shall first present an alternating expectation-conditional maximization (AECM) algorithm introduced by Meng and van Dyk (1997) for the fitting of STLMM. The standard errors of estimates can be obtained by the Louis' method (1982), which has the advantage of ease of implementation and flexibility. In addition, we are also devoted to developing fully Bayesian techniques for this model. In the Bayesian treatment, we are concerned with the issues of chosen priors, posterior distributions of parameters, Markov chain Monte Carlo (MCMC) implementation as well as its convergence assessment, Bayesian predictive distributions, and so forth. The proposed methodologies will be illustrated through some real examples and simulation studies.Missing values arise frequently in practice due partly to early withdrawal or failure to meet scheduled appointments (Laird, 1988; Schluchter, 1988). The resulting missingness yields an unbalanced pattern with unequal number of measurements or intermittent missing values for each subject. Assuming that missing data are missing at random (MAR) with an ignorable mechanism in the sense of Rubin (1976) and Little and Rubin (2002), we shall offer the ML and Bayesian predictors for the unobserved component of a partially observed response vector. The strategy can also be used for the prediction of the unobserved future vectors.
URI: http://hdl.handle.net/11455/49768
其他識別: NSC99-2118-M005-001-MY2
文章連結: http://grbsearch.stpi.narl.org.tw/GRB/result.jsp?id=2124923&plan_no=NSC99-2118-M005-001-MY2&plan_year=99&projkey=PA9907-2438&target=plan&highStr=*&check=0&pnchDesc=%E5%A4%9A%E8%AE%8A%E9%87%8F%E5%81%8F%E6%96%9Ct%E5%88%86%E4%BD%88%E5%B0%8D%E6%96%BC%E9%95%B7%E6%9C%9F%E8%B3%87%E6%96%99%E5%88%86%E6%9E%90%E4%B9%8B%E7%A0%94%E7%A9%B6
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