臺灣博碩士論文加值系統

線性混合模型(LMM) 已成為分析連續型長期資料的一個有效的工具，特別被應用在由生物統計學研究所產生的資料。LMM 的特點是它同時為實驗對象之間和相異時間觀察的變異建立模型。在LMM 的架構下，為了數理推導和統計分析的便利，隨機效應和誤差項經常假定為常態分配。然而，當隨機效應嚴重背離常態性時，將使得模型因穩健性的欠缺而更糟，並導致隨後的無效推論和不合理的估計。基於多變量偏斜t 分佈的架構下，本文提出一個具有穩健性的線性混合效應模型之建模方法來處理上述的問題。

多變量偏斜常態分佈已被普遍地認為是一個有用的統計分析工具，它能處理具有不對稱性的多變量資料。多變量偏斜t 分佈是多變量偏斜常態分佈的一種具穩健性的延伸, 它能同時捕捉資料中厚尾與偏斜的行為。本文會先探討這兩個分佈的一些基本性質並提供可解析的EM 演算法來求得最大概似估計值。

在一個LMM 的延伸模型中，我們對隨機效應及誤差項分別假設服從多變量偏斜t 分佈及多變量t 分佈。所提出的模型具有高度的靈活性，它能為連續型長期資料的隨機效應同時獲取具偏斜及厚尾的資訊。在兩個便利的階層展式中，我們提出一個有效的AECM 演算法來計算參數的最大概似估計值。在這種模式下，我們也討論隨機效應和間斷的遺失值的預測研究。最後，我們透過二組實際的例子來闡述我們所提出的方法。

Linear mixed models (LMM) 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 modelling 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. This dissertation proposes a robust linear mixed modeling framework using the multivariate skew t distribution for dealing with these problems.

The multivariate skew normal distribution has been recognized to be a useful statistical tool for modeling multivariate data with asymmetric behaviors. The multivariate skew t distribution, a robust extension of skew normal, can better capture skewness and fat tails simultaneously. This dissertation reviews some basic properties and provides, respectively, a computational feasible EM-type algorithm to obtain the maximum likelihood estimates for these two skew distributions.

We consider an extension of linear mixed models by assuming a multivariate skew t distribution for the random effects and a multivariate t distribution for the error terms. The proposed model provides flexibility in capturing the effects of skewness and heavy tails simultaneously among continuous longitudinal data. We present an efficient AECM algorithm for the computation of maximum likelihood estimates of parameters on the basis of two convenient hierarchical formulations. The techniques for the prediction of random effects and intermittent missing values under this model are also investigated. The proposed methodologies are illustrated by analyzing two real data sets.

1. Introduction 1
2. Review of the MSN and the MST distributions 4
2.1. Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2. The multivariate skew normal distribution . . . . . . . . . . . . . . . 5
2.3. The multivariate skew t distribution . . . . . . . . . . . . . . . . . . . 8
3. Model and method 16
4. ML estimation via the AECM algorithm 20
5. Prediction of random effects and missing values 25
6. Application 27
6.1. The schizophrenia data . . . . . . . . . . . . . . . . . . . . . . . . . . 27
6.2. The Framingham cholesterol data . . . . . . . . . . . . . . . . . . . . 34
7. Simulation study 39
8. Conclusion and future works 42
Appendix 43
A. Proof of Proposition 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
B. Proof of Proposition 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
C. Proof of Lemma 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
D. Proof of Proposition 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
E. Proof of Proposition 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
F. Parameter estimation for the MSN distribution using an EM-type algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
G. Proof of Lemma 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
H. Proof of Lemma 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
I. Proof of Proposition 6 . . . . . . . . . . . . . . . . . . . . . . . . . . . 51
J. Parameter estimation for the MST distribution using an EM-type approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52
K. Proof of (18) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
L. Proof of Proposition 7 . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
M. Proof of the observed information matrix for the STLMM . . . . . . . 62
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