臺灣博碩士論文加值系統

在許多統計的應用中，羅吉斯回歸（logistic regression）模型常用來分析帶有解釋變數的二元資料形態。當資料中的兩個分類極度不平衡時，模型參數的估計已被證實會有嚴重的偏誤，因而對應的相關風險評估推論將是不精確的。本篇論文中我們應用羅吉斯回歸模型去評估罕見事件的風險變異，我們不是使用某一特定的變數選擇準則去挑選一個最佳的模型，而是透過資料擾動的技巧應用在某些變數選取準則上去提出一個局部的模型平均流程，進而得到不同的風險估計值。然後藉由Kullback-Leibler損失的近似不偏估計量從中選擇。我們所提出的局部模型平均流程考慮了一般配模過程中常忽略的參數估計與模型選取的不確定性，因此我們所提出的方法將具有較佳的表現。我們透過完整的模擬實驗說明所提出方法的有效性，並藉由分析壞死性小腸結腸炎的真實資料說明我們方法的可行性。

In statistical applications, logistic regression has been a popular method for analyzing binary data accompanied by explanatory variables. But when the two classifications are extremely imbalanced, the estimation of model parameters has been shown to be severely biased and hence the inferences of relative risks based on a selected model would be inaccurate. In this paper, we focus on assessing the risk variations of rare events based on logistic regression models. Instead of selecting a best model based on a particular variable selection criterion, we propose a local model averaging procedure based on a data perturbation technique applied to different information criteria to obtain new risk estimates. Then an approximately unbiased estimator of Kullback-Leibler loss is used to choose the best one among them. Our proposed local model averaging procedure accounts for both estimation uncertainty and selection uncertainty, which are generally not considered by other modeling procedures. Therefore the proposed method has superior performance in various situations. We present complete simulations to assess the robustness of our approach and a real data example for necrotizing enterocolitis (NEC) is also applied for illustration.

Contents
1 Background and Motivation 1
2 Logistic Regression Model 4
3 Model Selection and Model Averaging 7
3.1 Model Selection 7
3.2 Model Averaging 8
4 The Proposed Methodology 10
4.1 Local Model Average Estimators 10
4.2 Assessment of LMA Estimators 12
5 Simulation Study 18
5.1 Simulation Scenarios 18
5.2 Simulation Results and Sensitivity Study 19
6 Applications to the Necrotizing Enterocolitis Data 31
7 Conclusion 37

List of Tables
1 True values and MLEs of model parameters for each case of Experiment I, where the values in parentheses are the standard deviations of parameter estimation based on 400 simulation replicates 20
2 True values and MLEs of model parameters for each case of Experiment II, where the values in parentheses are the standard deviations of parameter estimation based on 400 simulation replicates 21
3 Average values of percentage of rare events (Y = 1) for Experiment I and Experiment II based on 400 simulation replicates, where the values in parentheses are the corresponding standard deviations 23
4 KL loss performance of various methods for Experiment I based on 400 simulation replicates, where the values in parentheses are the corresponding standard errors 24
5 KL loss performance of various methods for Experiment II based on 400 simulation replicates, where the values in parentheses are the corresponding standard errors 25
6 Distributions of the selected variables for AIC and BIC criteria under various cases of Experiment I and Experiment II based on 400 simulation replicates, where the symbol “ ∗ ” represents that the underlying true model is exactly selected 28
7 Times of our proposed adaptive-LMA method averages around AIC or BIC criterion for various cases of Experiment I and Experiment II based on 400 simulation replicates 29
8 KL loss performance of our proposed adaptive-LMA method with respect to various perturbation sizes τ and T = 400 for Case 3 of Experiment I and Experiment II based on 400 simulation replicates, where the values in parentheses are the corresponding standard errors 29
9 KL loss performance of our proposed adaptive-LMA method with respect to various MC sample sizes T and τ = 0.5 for Case 3 of Experiment I and Experiment II based on 400 simulation replicates, where the values in parentheses are the corresponding standard errors 29

List of Figures
1 Plots of averaged KL loss values with respect to five cases for Experiment I based on 400 simulation replicates 26
2 Plots of averaged KL loss values with respect to five cases for Experiment II based on 400 simulation replicates 27
3 Risk estimates of severe NEC for 188 individuals based on various methods 34
4 Frequency of the selected models (variables) for the BIC-LMA method based on 400 perturbed data..........36

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