一個分析相關性資料的新方法-複合估計方程式

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姓名

楊棋全(Chi-chuan Yang) 查詢紙本館藏

畢業系所

統計研究所

論文名稱

一個分析相關性資料的新方法-複合估計方程式
(Composite estimating equations for correlated data)

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摘要(中)

在許多研究領域中裡，我們需要適當的統計模型來分析資料。然而，當我們誤用不適合的統計模型時，統計推論的結果可能會有問題。強韌的統計方法放寬了模型的假設條件，允許我們使用某些不適合的統計模型。但是，跟資料真正的統計模型相比，強韌方法的參數估計量會相對比較無效。
在不需知資料真正分配下，本文提出一個新的半母數方法來分析相關或不相關資料，此方法結合兩種估計函數，其中一個估計函數是關於資料的獨立性，另外一個估計函數則是納入資料相關性的部分。此方法我們稱「複合估計方程式」。
我們會探討複合估計函數迴歸參數估計量的大樣本性質，例如漸進常態與有效性。本文目的是在複合估計方程式裡尋求廣義線性模型之最有效性的迴歸參數估計量。在有連結函數與無截距的廣義簡單線性迴歸模型及有截距的簡單線性迴歸模型下，本文已將最佳估計量的解析解推導出來。但在廣義複迴歸模型底下，因為最佳估計量不具有解析解，所以本文另外提出近似方法來尋找最佳估計量。
本文模擬章節關注不同複合估計方程式的比較，同時也將所提的估計方程式與常用於分析相關性資料的廣義估計方程式(Liang and Zeger, 1986)及複合概似函數(Lindsay, 1988)作比較。實例分析則是呈現本文所提方法的效力。

摘要(英)

In many studies we demand the proper statistical distributions for analyzing data. If an improper model is used, the conclusions from statistical inference may be questionable. A robust approach to misspecifications would loosen up the model assumptions and help to overcome the problem originating from the use of improper models. However, compared with the data’s true density, robust methods would be inefficient if the adopted model (or semi-parametric model) is not the true density.
This thesis proposes a semi-parametric means of analyzing correlated or independent data whose underlying distributions need not to be known. The idea is to combine two estimating equations, one for independent data and one to accommodate the nature of within-cluster association existing in data. The proposed method is named the composite estimating equations.
The performance of the composite estimating equations will be investigated in terms of their asymptotic properties, such as the asymptotic normality and the efficiency. The aim of this study is to establish the most efficient estimate of the regression parameter of interest in the composite estimating equations under the generalized linear model. Optimal formulas have been shown in generalized simple linear regression models without intercepts, and in simple linear regression model with intercepts. An analytical expression of the optimal estimate does not exist in generalized multiple regression models. Hence, we adopt the approximation method to deal with problem of optimality.
A simulation is carried out to provide a comparison between various composite estimating equations as well as composite estimating equations with generalized estimating equations (Liang and Zeger, 1986) and composite likelihood (Lindsay, 1988) that are usual used for correlated data. Several examples are used to demonstrate the efficacy of the proposed method.

關鍵字(中)

★ 長期追蹤資料
★ 群組資料
★ 廣義估計方程式
★ 複合估計方程式
★ 複合概似函數
★ 多元負二項
★ 相關性資料

關鍵字(英)

★ Generalized estimating equations
★ Composite estimating equations
★ Composite likelihood
★ Clustered data
★ Longitudinal data
★ Correlated data
★ Multivariate negative binomial

論文目次

Contents
1 Introduction 1
2 Composite estimating equations 5
2.1 Composite estimating equations . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.2 Unbiasedness of CEE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.3 Generalized estimating equations . . . . . . . . . . . . . . . . . . . . . . . . . . 9
2.4 Composite likelihood . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Sandwich variance 13
4 Efficient estimate 15
4.1 Generalized simple linear regression without intercepts for simple versions of CEE 15
4.2 Simple linear regression with intercepts for a simple version of CEE related to
the independent normal score . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
4.3 Administration of optimal problem . . . . . . . . . . . . . . . . . . . . . . . . . 29
5 Simulations 34
5.1 Comparison within the optimal formulas of CEE for p = 1 and p = 2 . . . . . . 35
5.1.1 Independent normal data for p = 1 . . . . . . . . . . . . . . . . . . . . . 35
5.1.2 Dependent data for p = 1 . . . . . . . . . . . . . . . . . . . . . . . . . . 36
5.1.3 Independent normal and dependent data for p = 2 . . . . . . . . . . . . . 37
5.2 Comparison within various versions of CEE for p = 3 . . . . . . . . . . . . . . . 37
5.2.1 Independent normal data for p = 3 . . . . . . . . . . . . . . . . . . . . . 38
5.2.2 Dependent normal data for p = 3 . . . . . . . . . . . . . . . . . . . . . 38
5.3 Compare CEE with GEE and CL . . . . . . . . . . . . . . . . . . . . . . . . . . 40
5.3.1 Independent data for p = 3 . . . . . . . . . . . . . . . . . . . . . . . . . 41
5.3.2 Dependent data for p = 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 42
i
6 Real examples 43
7 Conclusions and future work 47
References 49
Appendix A 51
Appendix B: Tables 53
List of Figures
1 Comparison of B=(mA2)) with it three components . . . . . . . . . . . . . . . . 18
2 Sandwich estimate of (2) under various φ values . . . . . . . . . . . . . . . . . . 31
3 Sandwich estimate of (1) under various φ values . . . . . . . . . . . . . . . . . . 31
4 Sandwich estimate of (8) under various φ values . . . . . . . . . . . . . . . . . . 32
5 Sandwich estimate of (12) under various φ values . . . . . . . . . . . . . . . . . 32
List of Tables
1 Results for the mouth rinse data . . . . . . . . . . . . . . . . . . . . . . . . . . 44
2 Results for the hospital visit data . . . . . . . . . . . . . . . . . . . . . . . . . . 45
3 Results for the small mice data . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

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指導教授

鄒宗山(Tsung-shan Tsou)

審核日期

2011-6-17

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