Table 1 An overview of definitions for proposed measures and concepts in sections that follow with the same name

1. The horizontal partial area under the curve (a section that follows)

This partial area denoted pAUC_x, was suggested by Walter [2] and is defined for part or an ROC curve r(·) defined by TPR = [y1, y2] with inverse function r⁻¹(·):

\( {pAUC}_x:={\int}_{y_1}^{y_2}1-{r}^{-1}(y) dy \)

2. The concordant partial area under the curve (a section that follows)

This partial area denoted pAUC_c (Fig 1b) is defined for part of an ROC curve r(·) defined by FPR = [× 1, × 2] and TPR = [y1, y2], with inverse function r⁻¹(·):

\( {\displaystyle \begin{array}{c} pAU{C}_c\triangleq \frac{1}{2} pAU C+\frac{1}{2} pAU{C}_x\\ {}=\frac{1}{2}{\int}_{x_1}^{x_2}r(x) dx+\frac{1}{2}{\int}_{y_1}^{y_2}1-{r}^{-1}(y) dy\end{array}} \)

3. The concordance matrix for ROC data (a section that follows)

A matrix that depicts the exact relationship between the unique scores of positives and negatives in data and their corresponding points along a matrix border that exactly matches the (empirical) ROC curve. It geometrically and procedurally equates area measures AUC and pAUC_c to the statistics c and c_∆.

4. The partial c statistic for ROC data (a section that follows)

This statistic denoted c_∆ is defined for ROC data with P actual positives {p_1…P} and N actual negatives {n_1…N} and a partial curve specified by a subset of J positives and K negatives, i.e., \( \left\{{p}_{1\dots J}^{\prime}\right\} \) and \( \left\{{n}_{1\dots K}^{\prime}\right\} \); with Heaviside function H(·) and classification scores g(·). We present simple c_∆ (the non-interpolated version) here:

\( {\displaystyle \begin{array}{c}\mathrm{simple}\ {\mathrm{c}}_{\varDelta}\triangleq \frac{1}{2 JN}\sum \limits_{j=1}^J\sum \limits_{k=1}^NH\left(g\left({p}_j^{\prime}\right)-g\left({n}_k\right)\right)\\ {}+\frac{1}{2 PK}{\sum}_{j=1}^P{\sum}_{k=1}^KH\left(g\left({p}_j\right)-g\left({n}_k^{\prime}\right)\right)\end{array}} \)