Joint hypothesis testing of the area under the receiver operating characteristic curve and the Youden index

In the receiver operating characteristic (ROC) analysis, the area under the ROC curve (AUC) serves as an overall measure of diagnostic accuracy. Another popular ROC index is the Youden index (J), which corresponds to the maximum sum of sensitivity and specificity minus one. Since the AUC and J describe different aspects of diagnostic performance, we propose to test if a biomarker beats the pre-specified targeting values of AUC₀ and J₀ simultaneously with H₀ : AUC ≤ AUC₀ or J ≤ J₀ against H_a : AUC > AUC₀ and J > J₀. This is a multivariate order restrictive hypothesis with a non-convex space in H_a, and traditional likelihood ratio-based tests cannot apply. The intersection–union test (IUT) and the joint test are proposed for such test. While the IUT test independently tests for the AUC and the Youden index, the joint test is constructed based on the joint confidence region. Findings from the simulation suggest both tests yield similar power estimates. We also illustrated the tests using a real data example and the results of both tests are consistent. In conclusion, testing jointly on AUC and J gives more reliable results than using a single index, and the IUT is easy to apply and have similar power as the joint test.