Table 2 Formulas for the different confidence intervals from section Methods

Logit-transformation-based (LT)

$\text{expit}\left(\text{logit}(\widehat{\mathit{\text{AUC}}})\pm z·s/\left(\widehat{\mathit{\text{AUC}}}(1-\widehat{\mathit{\text{AUC}}})\sqrt{n}\right)\right)$

Mann-Whitney (MW)

$\widehat{\mathit{\text{AUC}}}\pm z·s/\sqrt{n}$

Bamber (Bamber)

$\widehat{\mathit{\text{AUC}}}\pm z·\mathit{\text{se}}$

Binormal (Binormal)

$\mathit{\text{AU}}{C}^{\ast }\pm z·s^{\ast }/\sqrt{n}$

Wilson (Wilson)

$\left(\widehat{\mathit{\text{AUC}}}+0.5t\right)/\left(1+t\right)\pm \sqrt{\widehat{\mathit{\text{AUC}}}(1-\widehat{\mathit{\text{AUC}}})t+0.25t^2}/\left(1+t\right)$

... with continuity correction (Wilson-cc)

lower: $\left(2n\widehat{\mathit{\text{AUC}}}+z^2-1-z\sqrt{z^2-2-1/n+4\widehat{\mathit{\text{AUC}}}(n(1-\widehat{\mathit{\text{AUC}}})+1)}\right)/$ (2(n + z ²))

upper: $\left(2n\widehat{\mathit{\text{AUC}}}+z^2+1+z\sqrt{z^2+2-1/n+4\widehat{\mathit{\text{AUC}}}(n(1-\widehat{\mathit{\text{AUC}}})+1)}\right)/$ (2(n + z ²))

Agresti-Coull (A-C)

$\widetilde{\mathit{\text{AUC}}}\pm z\sqrt{\frac{\widetilde{\mathit{\text{AUC}}}(1-\widetilde{\mathit{\text{AUC}}})}{n+4}}$

Clopper-Pearson (C-P)

lower: (k · f(α/2,2k,2(n - k + 1)))/ (n - k + 1 + k · f(α/2,2k,2(n - k + 1)))

upper: ((k + 1)f(1 - α/2,2(k + 1),2(n - k)))/ (n - k + (k + 1)f(1 - α/2,2(k + 1),2(n - k)))

Modified Wald (Wald)

$\widehat{\mathit{\text{AUC}}}\pm z\sqrt{\frac{\widehat{\mathit{\text{AUC}}}(1-\widehat{\mathit{\text{AUC}}})}{0.75n-1}}$

... with continuity correction(Wald-cc)

$\widehat{\mathit{\text{AUC}}}\pm z\sqrt{\frac{\widehat{\mathit{\text{AUC}}}(1-\widehat{\mathit{\text{AUC}}})}{0.75n-1}}+1/(2n)$