多元反應變數長期資料之多變量線性混合模型

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姓名

王婉倫(Wan-Lun Wang) 查詢紙本館藏

畢業系所

統計研究所

論文名稱

多元反應變數長期資料之多變量線性混合模型
(Analysis of Multi-outcome Longitudinal Data with Multivariate Linear Mixed Models)

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摘要(中)

在生物醫學或臨床試驗中，每個實驗個體隨著時間被重複測量紀錄多元反應變數和其他解釋變數。涉及分析此類型資料之研究一般稱作「多變量長期追蹤研究」。多變量線性混合效應模型已成為聯合分析多元長期資料的一個普遍使用工具。其主要目的除了描述重複測量值和其他解釋變數之間的關係外，並調查反應變數之間隨時間演變的關係。本文主要目的是基於多變量常態分佈與多變量t分佈去架構具p階自我迴歸(AR(p))之多變量線性混合模型，並以最大概似及貝氏觀點去發展相關的統計理論和演算方法。
在本文的第一部分，我們致力於提供額外的工具來分析多變量線性混合模型，其誤差項假設為根據自我迴歸之序列相關。基於ECM演算法之最大概似估計和透過馬可夫鏈蒙地卡羅(MCMC)程序之貝氏方法將被提出來計算模型之參數估計值。分數檢定統計量被提出來檢定在個體內誤差項之自相關性是否存在。同時我們也討論如何估計隨機效應以及未來值預測等問題。
由於考量常態假設下之模型對於潛在離群值的敏感以及資料的厚尾現象，本文的第二部分提出一個具穩健性一般化的多變量線性混合模型，此模型假設隨機效應和具AR(p)的個體內誤差項之聯合分佈為多變量t分佈。在此模型下，分數檢定統計量也被提出來檢測自相關的存在性。有效的混合ECME-scoring演算法被提出來求得參數之最大概似估計及其標準差。在貝氏結構中，我們利用Gibbs抽樣方法和Metropolis-Hastings演算法來獲得後驗分析推論。
最後，我們將所提出之理論方法應用於後天免疫缺乏症候群(愛滋病；AIDS)臨床試驗研究資料上。除此之外，模擬結果顯示具AR(p)之多變量線性混合模型確實能提供更精準的預測；多變量t線性混合模型確實比傳統的多變量線性混合模型對於離群值在參數估計上更具穩健性。

摘要(英)

In biomedical studies or clinical trials, repeated measures with multiple characteristics and a set of covariates are taken from each subject over a certain period of time. Research involving the analysis of such data is commonly referred to as multivariate longitudinal studies. The multivariate linear mixed model (MLMM) has become a frequently used tool for a joint analysis of more than one series of longitudinal data. The primary objective of the topic is not only to describe the relationship between repeated measures and other possible covariates, but also to investigate the association of the evolutions of characteristics. My dissertation mainly consists of two themes with emphases on maximum likelihood (ML) and Bayesian inferences for MLMMs and its robust t-based extension, called the multivariate t linear mixed model (MtLMM).
In the first theme, we are devoted to providing additional tools for MLMMs in which the errors are assumed to be serially correlated according to an autoregressive process. We present a computationally flexible ECM pocedure for obtaining the ML estimates of model parameters. Moreover, the approximate Bayesian method as well as a fully Bayesian via the Markov chain Monte Carlo (MCMC) sampler in this context are discussed. A score test statistic is derived for testing the existence of autocorrelation among within-subject errors. The techniques for estimation of the random effects and prediction of further responses given past repeated measures are also investigated.
In the second theme, motivated by a concern of sensitivity to potential outliers and data with longer-than-normal tails or possible serial correlation, we aim to develop a robust generalization of the MLMM, which is constructed by using the multivariate t distribution and a parsimonious AR(p) dependence structure for the within-subject errors. A score test for inspection of the existence of autocorrelation is also derived. An efficient hybrid ECME-scoring procedure is provided for computing the ML estimates with standard errors as a by-product. Similarly, a fully Bayesian estimation can then be conducted by utilizing the MCMC algorithm. An approximate Bayesian method is also described for drawing the large-sample Bayesian inference.
The proposed methodologies are demonstrated through an application of the ACTG 175 study. Several simulation studies are undertaken to clarify the applicability of the MLMM and the robust-to-outliers ability of the MtLMM.

關鍵字(中)

★ 經驗貝氏
★ 自我迴歸模型
★ 離群值
★ 預測
★ 分數檢定
★ 隨機效應

關鍵字(英)

★ AR(p)
★ Empirical Bayes estimates
★ Outliers
★ Prediction
★ Random effects
★ Score test

論文目次

1 Introduction 1
1.1 Background and Literature Review......................1
1.2 Motivation and Objective..............................3
1.3 Overview..............................................5
2 Multivariate Linear Mixed Models for Multiple Longitudinal Data 7
2.1 Notation and Setting..................................7
2.1.1 Model formulation...................................8
2.1.2 The score test for autocorrelation..................9
2.2 Maximum Likelihood Estimation........................12
2.2.1 Parameter estimation via the ECM algorithm.........12
2.2.2 Initialization.....................................16
2.3 Bayesian Approach....................................17
2.3.1 Asymptotic methods.................................19
2.3.2 The MCMC method....................................23
2.4 Inference for Random Effects and Prediction..........26
3 Multivariate t Linear Mixed Models with Autoregressive Errors 30
3.1 Model Specification..................................30
3.1.1 The likelihood.....................................30
3.1.2 Large-sample properties............................32
3.2 Maximum Likelihood Approach..........................34
3.2.1 Parameter estimation via the hybrid ECME-scoring procedure................................................34
3.2.2 Notes on implementation............................39
3.3 Bayesian Inference...................................40
3.3.1 Approximate Bayesian method........................41
3.3.2 Implementation of MCMC sampling....................43
3.4 Estimation of Random Effects and Prediction of Future Values...................................................46
4 An Application to the ACTG 175 Study 49
4.1 Description of the Data..............................49
4.2 Preliminary Analysis.................................50
4.3 Analytical Results Based on ML Estimation............52
4.3.1 Model fitting......................................53
4.3.2 Diagnostics........................................60
4.4 Posterior Inference Based on MCMC Outputs............61
4.4.1 Bayesian model selection...........................61
4.4.2 Outlier detection..................................68
5 Simulation Studies 69
5.1 Study I: Utilities of the Model......................69
5.2 Study II: Robustness of the MtLMM....................72
6 Discussion and Conclusion 75
References 78
Appendices 83
A. The Score Vector and the Fisher Information Matrix for Model (2.1)..............................................83
B. Proof of Theorem 2.1..................................83
C. Posterior Distribution of θ in the MLMM via Normal Approximation............................................84
D. Proof of Eqs. (2.24) and (2.25).......................86
E. Proof of (2.27) and (2.28)............................87
F. The Score Vector and the Fisher Information Matrix for Model (3.1)..............................................89
G. Proof of Theorem 3.3..................................91
H. Normal Approximation of the Posterior Distribution of θ in the MtLMM..........................................92

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指導教授

樊采虹(Tsai-Hung Fan)

審核日期

2010-5-24

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