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研究生:陳瓊梅
研究生(外文):Chyong-Mei Chen
論文名稱:具有不同變異性的半參數轉換治癒模型
論文名稱(外文):Semiparametric Transformation CureModels with Heteroscedasticity
指導教授:陳珍信陳珍信引用關係
學位類別:博士
校院名稱:國立臺灣大學
系所名稱:流行病學研究所
學門:醫藥衛生學門
學類:公共衛生學類
論文種類:學術論文
論文出版年:2004
畢業學年度:92
語文別:英文
論文頁數:79
中文關鍵詞:治癒模型科克斯模型不等變異性之風險迴歸模型轉換模型鞅論估計方程式
外文關鍵詞:transformation modelsmartingale processescure modelsheteroscedastic hazards regression modelestimating equations
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存活分析(survival analysis)是生物醫學研究中評估治療效果的一個很有用的統計方法。典型的存活模型假設所有的人(個體)都會經歷研究中所感興趣的事件而至終每個人都會有有限長的事件時間。近二十年來, 能夠考慮到群體中一些人最後會完全治癒而不會有事件時間的治癒模型(cure model)已逐漸受到重視。這種治癒模型也可以處理不具易感受性(nonsusceptible)的問題, 例如, 群體中有些人注定會得某一疾病而有發病年齡的問題, 但是又有些人卻不會得此疾病而沒有何時發病的問題。以往的治癒模型假設未能治癒之人的發病時間模型為典型的存活模型, 例如, 科克斯模型(Cox model)或加速壽命模型(accelerated failure time model)。這些的存活模型都可表示成轉換模型(transformation model),但是卻未能考慮到變異性的存在。

在這篇論文裡, 我們要提出一個更廣義的治癒模型,具有不同變異性的半參數轉換治癒模型。在這個治癒模型裡,我們考慮治癒與否的二元變量隨機變數(binary random variable)服從邏吉斯迴歸模型(logistic regression model),而那些未治癒之人的事件時間則服從不等變異性轉換模型(heteroscedastic transformation model)。這種不等變異性轉換模型可以描述存活曲線的相交現象。

我們的治癒模型的建構動機來自Hsieh (2001)與Lu和Ying (2004)。我們在Hsieh的不等變異性之風險迴歸模型(heteroscedastic hazards regression mode)下考慮治癒的機率, 亦可以說,在Lu和Ying的半參數轉換治癒模型下考慮不等變異性的問題。只要令不等變異性轉換模型的誤差項服從極值分配(extreme value distribution)或羅吉斯分配(logistic distribution),則其對應的就是, Hsieh的不等變異性之風險迴歸模型與不等變異性之比例勝算比模型(heteroscedastic proportional odds model)。

此模型的參數估計方法動機來自無母數的最大概似估計量(nonparametric maximum likelihood estimates)與鞅論 (martingale theory)。所得的估計量具有一致性(consistency), 近似常態分配(asymptotically normal distribution)且其近似變異--共變異矩陣(asymptotical variance-covariance matrix)具有確切的形式。若是轉換函數是給定的, 則本論文所提得的迴歸參數的估計方程式(estimating equatrions)即等於由Godambe的估計函數理輪(estimating function theory)所導的估計方程式; 在此種情形下, 此估計量達到了Godambe定義的最佳化。由模擬可以驗證這些大樣本性質。當具易感受性之人的事件時間為等變異性時, 此論文所提的估計量之有效性可以藉模擬與Lu和Ying所提的估計量做個比較。本論文亦為所提的模型與估計方法舉一實例。
Survival analysis is a very useful statistical methodology to evaluate the efficiency of a treatment in the biomedical study with censored data. Typical survival models assume that the events will occur eventually for all subjects such that everyone has a finite failure time. In the recent two decades, the cure models have received more attentions to consider the problem that some people are cured permanently without failure times. The cure models can also be used to deal with nonsusceptible problems, such as age onset problems in which some people are susceptible to a disease but some are not. A cure model considers the distribution of event times of the susceptible with a typical survival model, for example, Cox''s regression model and the accelerated failure time model. These models share the same representations of transformation models but do not consider the heteroscedasticity.

This dissertation proposes a more general class of failure models, semiparametric heteroscedastic transformation cure models, to fit the binary random variable for the occurrence of cure by a logistic regression and the event time of the non-cured by a heteroscedastic transformation model. The heteroscedastic transformation model can describe the phenomenon of crossing in survival curves.

Being motivated from Hsieh (2001) and Lu and Ying (2004), our proposed model adds the cure probability to Hsieh''s heteroscedastic hazards regression model, or, equivalently, the heteroscedasticity to the semiparametric transformation cure model of Lu and Ying. The family of heteroscedastic transformation models allows different variances in a transformation model, and therefore includes Hsieh''s heteroscedastic hazards regression model and the heteroscedastic proportional odds model by, respectively, specifying the extreme value distribution and the logistic distribution for error terms.

The principle of constructing estimating equations for the regression parameters is motivated from nonparametric maximum likelihood estimates and the martingale theory. Given a specific form of the transformation function, these estimating equations of regression parameters can be identically constructed based on Godambe''s estimating function theory (1985) for martingale processes. In this situation, our approach is optimal in Godambe''s criterion, while the optimality is not attainable in the reduced case of Lu and Ying. The relevant statistical properties of the estimators from the estimating equations include consistency, asymptotically normal distribution and a closed form of the asymptotical variance-covariance matrix. Simulation studies are performed to validate the large sample properties. We also compare the efficiency with estimators of Lu and Ying under the homoscedasticity through simulation studies. A real data analysis is conducted as an illustration.
1 Introduction 1
1.1 Background 1
1.2 Cure Models 2
1.3 Motivation 3
1.4 Organization of the Dissertation 4

2 Literature Review 5
2.1 The Transformation Model 5
2.2 The Heteroscedastic Hazards Regression Model 6
2.3 The Semiparametric Transformation Cure Model 8
3 Semiparametric Heteroscedastic Transformation Cure Models 10
3.1 The Proposed Model and Its Assumptions 10
3.2 Model Discrimination 12

4 Estimating Procedures for the Model 18
4.1 Nonparametric Maximum Likelihood Estimation 18
4.2 Estimating Equations 20
4.3 Computational Algorithm 22
4.4 Comparisons of Models 23
4.4.1 The Homoscedastic Transformation Cure Model 23
4.4.2 The Heteroscedastic Hazards Regression Model 24

5 Large Sample Properties and Finite Sample Properties 25
5.1 Consistency and Asymptotic Normality 25
5.1.1 Sketch of the Proof 25
5.1.2 Regularity Conditions 26
5.1.3 Asymptotic Results 27
5.2 Godambe''s Theory of Estimating Equations 28

6 Simulation Results and A Real Data Analysis 31
6.1 Simulation Studies 31
6.1.1 Simulations on the Heteroscedastic Transformation Cure Model 31
6.1.2 Simulations on the Homoscedastic Transformation Cure Model 33
6.2 Analysis of Breast Cancer Data 36
6.2.1 Analysis under Homoscedasticity 38
6.2.2 Analysis under Heteroscedasticity 40

7 Discussion 48
7.1 Heterogeneity versus Heteroscedasticity 48
7.2 Multiple Crossing Points 49
7.3 Efficiency in Model Estimation 49
7.4 Time-dependent Covariates 50
7.5 Comparison with the Bounded Cumulative Hazards Model 50
7.6 Computational Problems 51
7.7 Future Research Topics 52
7.7.1 Testing Problems 52
7.7.2 Interval Censored Data 53

Appendices 54
A Score Functions of the Transformation Function 54
B Proof of Lemma 1 56
C Proof of Large Sample Properties 58
D Proof of Lemma 2 72
E The Optimal Estimating Function in Godambe''s Theory 75

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