Joint Statistical Inference for the Area under the ROC Curve and Youden Index under a Density Ratio Model

The receiver operating characteristic (ROC) curve is a valuable statistical tool in medical research. It assesses a biomarker’s ability to distinguish between diseased and healthy individuals. The area under the ROC curve ($AUC$) and the Youden index (J) are common summary indices used to evaluate a biomarker’s diagnostic accuracy. Simultaneously examining $AUC$ and J offers a more comprehensive understanding of the ROC curve’s characteristics. In this paper, we utilize a semiparametric density ratio model to link the distributions of a biomarker for healthy and diseased individuals. Under this model, we establish the joint asymptotic normality of the maximum empirical likelihood estimator of $(AUC,J)$ and construct an asymptotically valid confidence region for $(AUC,J)$. Furthermore, we propose a new test to determine whether a biomarker simultaneously exceeds prespecified target values of $AU{C}_0$ and $J_0$ with the null hypothesis $H_0:AUC\le AU{C}_0$ or $J\le J_0$ against the alternative hypothesis $H_a:AUC>AU{C}_0$ and $J>J_0$. Simulation studies and a real data example on Duchenne Muscular Dystrophy are used to demonstrate the effectiveness of our proposed method and highlight its advantages over existing methods.