利用生物標記進行疾病診斷是目前醫學上一個重要的課題，根據生物標記正確地將病人判斷為患者的機率為真陽率 (True-positive rate；TPR)，反之，將健康者誤診為患者的機率為偽陽率 (False-positive rate; FPR)。在可能的生物標記臨界值之下，真陽率相對於偽陽率的曲線即為該生物標記的受試者操作曲線(Receiver Operating Characteristic curve; ROC curve)。此外，真陽率與偽陽率的最大差異，為Youden指數，記做J。傳統上利用此一曲線下的面積(Area under the ROC curve; AUC)評估生物標記之診斷效率。本文為比較兩個服從二維常態分布生物標記在診斷疾病的效率，建立AUC 差異及 J 差異的聯合信賴域。當生物標記的分布未知時，本文也利用自助法(Bootstrapping method)建立AUC差異及J差異的無母數聯合信賴域。除了藉模擬方法研究上述聯合信賴域的覆蓋機率，也利用CA19-9及CA125兩種生物標記在診斷胰臟腫瘤的資料說明本文所提聯合信賴域的應用。

To do diagnosis based on biomarkers is an important issue in medicine study. The true-positive rate (TPR) is the probability of a patient diagnosed to be ill and the false-positive rate (FPR) is the probability of a healthy person diagnosed to be ill. Under all possible cut off points of a biomarker, the curve of TPR vs. FPR is called the Receiver Operating Characteristic curve (ROC curve) of the biomarkers. Moreover, the maximum difference between TPR and FPR, called the Youden index, denote as J. The area under the ROC curve (AUC) and the Youden index are conventionally used to evaluate the efficiency of the biomarkers. To compare the efficiency of two biomarkers that are distributed according to a bivariate normal distribution, we construct a joint confidence region for the difference of AUC and that of Youden index. When the bivariate distribution of the two biomarkers is unknown, we suggest construct a nonparametric joint confidence region for AUC-difference and J-difference based on the bootstrapping method. In spite of studying the coverage probability of the confidence regions mentioned above based on a simulation method, we also use two biomarkers, CA19-9 and CA125, to diagnose the pancreatic cancer based on the proposed confidence regions.

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