我們研究了在Cox比例風險模式(Cox proportional hazards model)分析盛行倖存資料(Prevalent survival data)時模式選擇議題，然而在此研究中我們遇到了一些挑戰。 第一，在盛行倖存資料中我們蒐集到許多項共變數(Covariates)，研究的目標是希望從這些共變數挑僅選出些許重要的共變數即可，包含連續型與離散型的共變數；第二，盛行倖存資料事實上是有偏誤之抽樣方法(Biased sampling)。我們提供了一套方法，不僅可以同時進行模式選擇與參數估計，還可以矯正因抽樣方法造成的偏誤。更進一步地，我們的方法可以允許使用不同的懲戒函數(Penalty function)，適用於連續型與離散型的共變數。模擬結果顯示我們提供的機制是穩定的，並且可以選出正確的模式。我最後，們也將此方法應用在一筆有關女性乳癌的真實資料上進行分析。

We study the variable selection problem in Cox proportional hazards model for prevalent survival data. In this study, we face some challenges. Firstly, from many potential predictors, we would like to select a small number of key risk factors, including continuous or discrete variables. Secondly, data were collected from a prevalent sampling which is exactly a biased sampling scheme. The proposed method not only can select and estimate variables simultaneously but also can correct the sampling bias. Further, the proposed method can allow for different penalty functions, including continuous or discrete variables. The results of simulation study show that the proposed procedure is stable and more accurate to select the true model. We also apply the proposed method to a real data.

1 緒論1
1.1 倖存資料. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 模式挑選. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
2 文獻回顧5
2.1 符號定義與資料型態. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.2 部分概似函數. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 模式挑選與懲戒部分概似函數. . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 群懲戒函數. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.5 正規解路徑. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3 研究方法14
3.1 群懲戒log-部分概似函數. . . . . . . . . . . . . . . . . . . . . . . . . . . 14
3.2 演算法. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
3.3 變異數估計. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 調整參數之選取. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4 數值模擬23
5 實例分析26
5.1 女性乳癌資料. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
6 結論30
7 附錄31
7.1 附錄一. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
7.2 附錄二. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
7.3 附錄三. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

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