在研究基因表現量資料時，通常因為做為解釋變數的基因數量遠大於樣本數，造成模型之迴歸係數無法估計。有鑑於此，變數篩選就是一個在建模前必要的步驟。學者們陸續提出懲罰線性迴歸（penalized linear regression）來進行變數篩選。常見的變數篩選方法有Tibshirani（1996）的最小絕對緊縮與選擇算子（Least Absolute Shrinkage and Selection Operator, LASSO）、Fan 和Li（2001）的平滑修剪絕對離差（Smoothly Clipped Absolute Deviation, SCAD）、Zou 和Hastie （2005） 的elastic net，及Zou（2006）的適應性最小絕對緊縮與選擇算子（Adaptive LASSO）。後來多數學者已經將上述方法，從連續型反應變數推廣至二元反應變數，而形成懲罰邏輯式迴歸（penalized logistic regression）。
在稀少事件下，學者King and Zeng（2001）提出方法修正模型參數的最大概似估計量（Maximum Likelihood Estimator, MLE）之偏誤會增大的問題。他們也提供修正此偏誤的方法。近年，多數學者採用Firth（1993）所提出的懲罰最大概似估計法（penalized maximum likelihood method）降低MLE 的偏誤。學者Leitgöb（2013）的模擬比較了兩者，結果顯示，建議使用後者的懲罰最大概似估計法降低MLE 的偏誤。
本論文將於高維度資料且事件發生機率很小時，用模擬方式探討 LASSO、SCAD以及Adaptive LASSO 三種懲罰項（penalty）建立的邏輯式迴歸，在篩選解釋變數上的表現。 經由模擬發現，LASSO 的預測結果較其它兩者差。建議研究者，在稀少事件下，應用懲罰邏輯式迴歸進行變數篩選及建模預測，使用Adaptive LASSO。

It's well known that the accuracy of MLE of the regression coefficient in logistic regression model is seriously affected by rare events. Less attention is given to the performance of variable selection in logistic regression with rare events. Therefore, this thesis studies the performance of three variable selection methods, LASSO (Least Absolute Shrinkage and Selection Operator), SCAD (Smoothly Clipper Absolute Deviation), and Adaptive LASSO, when event rate is low and the number of explanatory variables is much larger than sample sizes.
A simulation study is conducted to compare the accuracy in selecting important explanatory variables of logistic regression model. Based on limited simulation scenarios, when event rate is as low as 0.05, the simulation results recommended using Adaptive LASSO to select important explanatory variables. Consequently, Adaptive LASSO is recommended for variable selection and prediction with rare events data.

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