本論文針對存活資料，考慮“不受感染體質”(nonsusceptibility)者之存在，在混合模式架構下提出半母數迴歸分析方法。我們採用邏輯斯模式分析解釋變數與“發病與否”的關係。針對受感染體質者之“潛在發病時間”，我們探討兩類迴歸模式之推論問題。第一類模式包含常見的加速失敗模式和位移模式，我們利用計數程序之機率性質以建構估計函數，並進一步提出模式選取方法。第二類為線性轉換模式，包含等比風險模式與等比勝負比模式。我們採用概似函數法做為參數估計的原則，除了分析獨立設限的情況外，並進一步提出當存在競爭風險時，如何修正模式假設與推論方法。兩個研究方向都利用 EM 的技巧，以補插法處理感染體質不確定之觀測值。我們透過模擬實驗評估所提出方法在有限樣本下之表現。

In this thesis, we consider semiparametric regression analysis for survival data in presence of non-susceptibility or cure. The mixture framework is adopted in analysis of such data. The incidence rate is assumed to follow the logistic regression model and the latency distribution is studied under two types of semiparametric regression models. One class refers to the semi-parametric linear regression model which includes the AFT and location-shift models as special cases. We propose estimating functions and also a model checking procedure based on properties of counting processes. The other class is known as transformation models which contain the proportional hazards model and proportional odds model. The likelihood principle is adopted for parameter estimation. We examine two situations of independent and dependent censoring respectively. In both research directions, the principle of EM is applied to handle uncertain susceptibility status. Simulation results are provided to examine the finite-sample properties of the proposed methods.

1 Introduction 1
1.1 Literature background 1
1.2 Outline of the thesis 2
2 Literature Review for Semi-parametric Linear Models with Cure 4
2.1 Overview 4
2.2 M-Estimation by Li & Taylor (2002) 7
2.3 Log-rank type Estimation by Zhang & Peng (2007)8
2.4 Sketch of Numerical Algorithm for EM-type Estimation 9
3 Proposed Approach for Semiparametric Linear Models 11
3.1 Martingale estimating function based on complete data 11
3.2 The proposed estimating functions 12
3.3 Large sample analysis 13
3.4 Numerical algorithm and variance estimation 17
3.4.1 Re-sampling based on bootstrap approach 17
3.4.2 Re-sampling based on pivotal estimating function 18
3.5 Model Checking 20
3.6 Simulation analysis 22
3.6.1 Data generation 22
3.6.2 Simulation results 22
4 Literature Review for Transformation Models with Cure 25
4.1 Background 26
4.2 Different model expressions 26
4.3 Likelihood approach under the PH model 26
4.4 Moment approach for the class of transformation models 29
5 Proposed Approach for Transformation Cure Model under Independent Censoring 32
5.1 Model assumptions 32
5.2 Likelihood analysis for complete data 33
5.3 Imputation for handling missing data 36
5.4 Numerical algorithms 37
5.5 Simulation analysis 37
5.5.1 Data generation 37
5.5.2 Simulation results 38
6 Proposed Approach for Transformation Cure Model Under Dependent Censoring 40
6.1 Model assumptions 40
6.2 Imputation under the new models 41
6.3 Likelihood analysis under dependent censorship 42
6.4 Score equations under dependent censorship 45
6.5 Numerical algorithm 49
6.6 Simulation analysis 50
6.6.1 Data generation 50
6.6.2 Simulation results 52
Conclusion 53
References 54
Appendix 56
Tables and figures 69

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