本篇研究中，我們的目的是在Cox 比例風險函數中同時進行估計、變數選取及變數分群。Tibshirani (1996) 在目標函數中加入L1-norm 懲戒函數進行估計讓估計參數具有稀疏性，以此有效的同時達到估計以及變數選取的效果。在傳統的變數分群方法中，變數通常會根據從前的知識來進行分群，而這種分群方法通常被認定太過主觀。本篇研究中，我們應用Tibshirani et al. (2005) 的手法於Cox 比例風險函數的偏概似函數上，Fused LASSO 懲戒函數著重在參數和參數差的L1-norm，其中參數的L1懲戒函數使得參數估計值受到壓縮而達到稀疏性的性質，而參數差的L1 懲戒函數將鄰近的參數差進行壓縮，鄰近的參數得到相同估計值藉此進行變數分群。這種以數據自我統計的方法是較為客觀的，並且我們可以同時估計、變數選取及變數分群。在模擬方面，我們考慮四種模型比較:LASSO、Generalized LASSO、Fused LASSO、和正常的Cox model，以這些模型來分別比較這些懲戒函數所帶來的效果，並且實際應用在一筆肺癌經過輔助化療後基因位點資料分析。

In this work, our aims are to do model selection, coefficient estimation and variable grouping imultaneously in Cox’s proportional hazards model.Tibshirani (1996) added L1 norm penalty function to objective function to obtain the sparsity of coefficient estimation, which is an efficient way to domodel election and coefficient estimation at one time. In traditional variable grouping methods, variables are grouped based on the prior knowledge, which is often be judged too subjective. In this work, we apply Tibshirani et al. (2005) to the partial likelihood of Cox model. The Fused LASSO penalty focuses on the combination of L1 norm and the difference of L1 norm: L1 penalty shrinkages coefficients to ensure the sparseness of coefficient
estimates, while the difference of L1 penalty shrinkages the difference between the neighboring coefficients, which makes variables be grouped in the sense of nvolving same coefficient estimates. This data adaptive approach is more objective and we can estimate, select and group variables simultaneously. In our simulation, we consider three different cases: LASSO, generalized LASSO and Fused LASSO to compare the effects of the L1 and the difference of L1 penalty and apply to analysis Gene Signature for Adjuvant Chemotherapy in Resected Non–Small-Cell Lung cancer data.

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