決定治療的準則(a treatment regime) 是根據每個病患的身體狀況去指派個別的治療方法。在倖存資料中，常會存在著選擇偏誤以及自變數在接受不同治療的兩群分佈不平均的問題。因此我們會使用治療機率倒數加權的方法以及Rubin因果模型來修正，否則結果不具一致性(consistent)也沒有因果推論(causal inference)的解釋。本論文主要研究利用修正後治療機率倒數加權(IPSW)以及具雙穩健性的方法(AIPSW)去估計最佳治療的準則，使得整體病人或是接受特定治療的病人在此準則下接受的治療可以達到限制平均生存時間(RMST)最大。我們提出的方法會以模擬驗證並應用在真實資料中。

A treatment regime is the way of assigning treatment to each patients according to their own characteristics. In our survival data, there might exist both selection bias and imbalanced covariates between different treatments. The inverse probability weighting method and Rubin causal model are used to adjust those biases. Besides, these methods not only provide the consistency to estimators but also make the causal inference available. In this thesis, we propose both Inverse propensity score weighted (IPSW) and augmented Inverse propensity score weighted (AIPSW) to estimate the optimal treatment regime that maximizes the restricted mean survival time (RMST) for all patients and those who received the specified treatment. Finally, we evaluate the proposed methods by the simulation studies and also applied those methods to the real data.

1 Introduction 1
2 Literature Review 3
2.1 Survival Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.1 Kaplan-Meier Estimator . . . . . . . . . . . . . . . . . . . . 3
2.1.2 Cox Propotional Hazard Regression model . . . . . . . . . . 4
2.2 Selection Bias . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 Causal Inference . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
2.3.1 Propensity score . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3.2 Assumptions . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3.3 Causal survival function . . . . . . . . . . . . . . . . . . . . 6
2.3.4 Average Treatment Effect (ATE) . . . . . . . . . . . . . . . 6
2.3.5 Average Treatment Effect on the Treated (ATT) . . . . . . . 7
3 Model and Methodology 7
3.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 Estimation Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2.1 Propensity score . . . . . . . . . . . . . . . . . . . . . . . . 8
3.2.2 Balancing weights . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2.3 Inverse propensity score weighted estimators . . . . . . . . . 10
3.2.4 Augmented inverse propensity score weighted estimators . . 11
3.2.5 Computational Aspects . . . . . . . . . . . . . . . . . . . . . 14
3.2.6 Restricted mean survival . . . . . . . . . . . . . . . . . . . . 14
4 Asymptotic Properties 15
4.1 Inverse propensity score weighted estimators . . . . . . . . . . . . . 15
4.2 Augmented inverse propensity score weighted estimators . . . . . . 16
5 Simulation Studies 17
5.1 Data Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.2 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
5.2.1 Compare Different Scenarios . . . . . . . . . . . . . . . . . . 18
5.2.2 Compare Different Estimators . . . . . . . . . . . . . . . . . 20
6 Real Data Analysis 22
6.1 Data Description . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.2 Analysis Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
7 Conclusion 26
Reference 27
Appendix 28
A Proof of Theorems 28
A.1 Proof of Theorem 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
A.2 Proof of Theorem 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
A.3 Proof of Theorem 3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
A.4 Proof of Theorem 4 . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
B Simulation and Real data results 53
B.1 Simulation results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
B.2 Real data results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

[1] Cheng, Y.J. and Wang M.C. Estimating propensity scores and causal survival
functions using prevalent survival data. 2012. Biometrics, 68(3), 707-716.
[2] Horvitz, D.F. and Thompson, D.J. A generalization of sampling without
eeplacement from a finite universe. Journal of the American Statistical Association. 1952; 47(260), 663-685.
[3] Jiang R, Lu W, Song R, Davidian M. On estimation of optimal treatment
regimes for maximizing t-year survival probability. J R Stat Soc Ser B Stat
Methodol. 2016 To appear.
[4] Jiang, R., Lu, W., Song R., Hudgens, M.G. Double robust estimation of optimal treatment regimes for survival data – with application to an HIV/AIDS
study. 2017. Annals of Applied Statistics, 11(3), 1763-1786.
[5] Lunceford, J.K. and Davidian, M. Stratification and weighting via the propensity score in estimation of causal treatment effects: a comparative study. 2004.
Statistics in Medicine, 23(19), 2937-60.
[6] Rosenbaum, PAUL & Rubin, Donald. (1983). The Central Role of the Propensity Score in Observational Studies For Causal Effects. Biometrika. 70. 41-55.
10.1093/biomet/70.1.41.
[7] Zhao YQ, Zeng D, Laber EB, Song R, Yuan M, Kosorok MR. Doubly robust learning for estimating individualized treatment with censored data.
Biometrika. 2015; 102:151–168. MR3335102. [PubMed: 25937641]