倖存分析中，可以透過比較因果倖存函數來評估不同治療方法間的好壞。在資料收集的過程中可能會發生兩種偏誤，一種為選擇偏誤，來自於母體與研究對象之間解釋變數分布的不同; 另一種則為不同治療下，實驗組及對照組之間自變數分布不均產生的偏誤，若未適當處理這些偏誤可能導致估計上出現誤差，結果也不具有因果解釋。本篇論文的目的為: 當資料同時存在這兩種偏誤時，利用選擇機率倒數加權法修正後的的治療機率倒數加權法以及具雙重穩健性質的治療機率倒數加權法正確估計因果倖存函數，最後藉由數值模擬驗證所提出的方法並實際應用在肝細胞癌的資料上。

In survival analysis, causal survival functions can be uesd to compare the effectiveness among treatments. However, there may exist two kinds of biases during the process of data collection. One is selection bias due to some patients with certain characteristics are more sutible for some specific treatments and the other is caused by uneven explanatory variable distributions among treatments. It would lead to some estimation errors and could not make causal inferences without correcting these biases. This thesis proposed to use the inverse selection probability weighting to correct selection bias problem and inverse propensity scores weighting to correct uneven explanatory variable distributions among treatments when the data exists both kinds of biases. Two types of causal survival function are considered. One is inverse propensity scores weighted Kaplan-Meier estimator and the other is augmented inverse propensity scores weighted Kaplan-Meier estimator. Both are adjusted by inverse selection probability. The proposed methods are examined through simulation studies and applied to a real data.

1 Introduction 1
2 Literature Review 3
2.1 Survival Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.1 Notation and Definition . . . . . . . . . . . . . . . . . . . . 3
2.1.2 Kaplan-Meier Estimator . . . . . . . . . . . . . . . . . . . . 4
2.2 Selection Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.2.1 Cox Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Causal Inferences . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3.1 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.3.2 Average Treatment Effects . . . . . . . . . . . . . . . . . . . 7
2.3.3 Causal Survival Function . . . . . . . . . . . . . . . . . . . 7
2.4 Restricted Mean Survival Time . . . . . . . . . . . . . . . . . . . . 8
3 Methodology 9
3.1 Notations and Model Assumptions . . . . . . . . . . . . . . . . . . 9
3.2 Propensity Score . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.3 IPSWKME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3.4 AIPWSKME . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4 Asymptotic Properties 16
4.1 Parameter θ in Propensity Score Model . . . . . . . . . . . . . . . 16
4.2 Parameter β in Survival Model . . . . . . . . . . . . . . . . . . . . 16
4.3 Restricted Mean Survival Time for IPSWKME . . . . . . . . . . . 17
4.4 Restricted Mean Survival Time for AIPSWKME . . . . . . . . . . . 19
5 Simulation Studies 24
5.1 Data Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
5.2.1 Compare Different Scenarios . . . . . . . . . . . . . . . . . . 25
5.2.2 Compare Different Estimators . . . . . . . . . . . . . . . . . 27
6 Real Data Analysis 28
6.1 Data Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28
6.2 Analysis Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
7 Discussion 31
Reference 32
Appendix 34
A. Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
B. Real Data Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48
C. Asymptotic properties . . . . . . . . . . . . . . . . . . . . . . . . . . 59
C1. Parameter θ in Propensity Score Model . . . . . . . . . . . . . 59
C2. Parameter β in Survival Model . . . . . . . . . . . . . . . . . . 60
C3. Restricted Mean Survival Time for IPSWKME . . . . . . . . . 60
C4. Restricted Mean Survival Time for AIPSWKME . . . . . . . . 63

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