倖存資料中，要比較兩種治療方式的好壞，通常會觀察其風險函數或是倖存函數，來探討不同治療方式的優劣性。在本文章中，我們考慮的倖存資料存在選擇偏差，在選擇偏差的情況下會產生兩個問題，第一個是當倖存資料中存在兩組治療病人其自變數分布的不平衡，導致估計出來的倖存函數就會沒有因果關係的解釋，我們使用傾向分數來解決此問題。第二個問題是資料在有選擇偏誤的情況下，則會導致傾向分數以及倖存函數的估計上出現偏差，可能導致嚴重的錯誤結果，因此需要進行修正。文章中進行了模擬實驗的驗證以及實際案例的應用。

In this paper, we compare causal estimation between two treatments when survival data are subject to selection bias. Based on this biased data for causal estimation, there exist two statistical challenges. First, when the distributions of covariates are imbalanced between the treatment groups, the effect of treatment has no causal interpretation. In this paper, we use the estimated propensity score to adjust for the imbalance between the treatment groups. Second, selection bias will cause the estimated deviation even when we estimate the propensity score and survival function. Different types of weighting approaches are proposed. The proposed
methods are examined through the simulation studies and applied to real data.

目錄
1 緒論. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 文獻回顧. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1 倖存分析. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2.1.1 Kaplan-Meier 估計量. . . . . . . . . . . . . . . . . . . . . . 4
2.2 選擇偏誤(Selection bias) . . . . . . . . . . . . . . . . . . . . . . . 4
2.3 因果推論(Causal inference) . . . . . . . . . . . . . . . . . . . . . . 4
2.3.1 傾向分數(Propensity score) . . . . . . . . . . . . . . . . . . 5
2.3.2 潜在结果(Potential outcomes) . . . . . . . . . . . . . . . . 5
2.3.3 兩個假設. . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3.4 因果倖存函數(Causal survival function) . . . . . . . . . . . 6
3 模型方法7
3.1 模型假設. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
3.2 選擇偏誤下之傾向分數. . . . . . . . . . . . . . . . . . . . . . . . . 7
3.3 選擇偏誤下之因果倖存函數. . . . . . . . . . . . . . . . . . . . . . 8
3.3.1 治療機率倒數加權(Inverse Probability Treatment Weighting)
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3.3.2 分層估計法(Stratification) . . . . . . . . . . . . . . . . . . 9
3.3.3 無母數回歸(Nonparametric Regression) . . . . . . . . . . . 10
3.4 重抽法(Resampling method) . . . . . . . . . . . . . . . . . . . . . 10
4 數值模擬12
4.1 資料以及參數設定. . . . . . . . . . . . . . . . . . . . . . . . . . . 12
4.2 模擬結果分析. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2.1 比較不同模式. . . . . . . . . . . . . . . . . . . . . . . . . . 13
4.2.2 比較不同估計方法. . . . . . . . . . . . . . . . . . . . . . . 15
5 實例分析16
5.1 資料說明. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5.2 模型分析結果. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
6 結論18
參考文獻19
附錄21
A 附表與附圖21
A.1 數值模擬結果. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
A.2 實例分析. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

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