Figure 1.

Bi-normal ROC curves when Σ_D and ${\mathrm{\Sigma }}_{\overline{D}}$ are not proportional. Assume that the clinically relevant FPR region is [0, 0.2], indicated by the vertical line drawn at 80% specificity. Two biomarkers are combined linearly to form a composite test. The solid and dashed ROC curves in gray correspond to composite tests combined using Su and Liu’s coefficients and Liu et al.’s coefficients, respectively; the solid ROC curve in black corresponds to the composite tests combined by the proposed parametric method (Section 2). Although Su and Liu’s ROC curve may have the largest AUC and Liu et al’s ROC curve has the largest pAUC over a certain specificity region that is not necessarily the clinically relevant region, the ROC curve corresponding to the proposed method has the largest pAUC over the clinically relevant FPR region. The parameters used: Example 1, μ_D = (3,3), ${\mathit{\mu }}_{\overline{D}}$ = (0,0), σ_D = (3,2), ${\mathit{\sigma }}_{\overline{D}}$ = (1,2), and correlation 0.5; Example 2, μ_D = (1,3), ${\mathit{\mu }}_{\overline{D}}$ = (0,0), σ_D = (3,2), ${\mathit{\sigma }}_{\overline{D}}$ = (1,2), and correlation 0.5. σ_D and ${\mathit{\sigma }}_{\overline{D}}$ are the marginal standard deviations of the disease and non-disease groups, respectively.