## 3 Results
### Quantitative Evaluation of the Explanation Framework  
From a visual inspection of a number of examples, including those shown in Figure 6, it seems that to predict the breast density category the CNN is actually \"looking\" at the appropriate image information, namely the higher-intensity regions of the mammograms. A possibility could be to quantitatively compare the area of the saliency map over a predefined threshold and a hand-crafted pixel-wise ground truth for dense areas generated by a radiologist. That would be an extremely time-consuming task; thus, we discarded this option and proposed a straightforward method to evaluate whether the maps and the higher-density breast areas are spatially correlated. Moreover, for this classification task it is not fair to use pixel wise ground truth since the class assessment by physicians is made by observing the entire image and not pixel wise. It is apparent from a visual inspection of a number of examples, including those shown in Figure 6, that to predict the breast density category the CNN actually relies on an intelligible and appropriate image information, namely the higher-intensity regions of the mammograms. To quantify the extent of this hypothesized direct relationship, we computed the [PERSON]'s rank correlation coefficient r between the grey-scale image and the grad-CAM map converted into a grey-scale image. The box plots obtained for the correctly classified mammograms in the test set of the four density categories are shown in Figure 7 separately for the CC and MLO projections. We performed the Kruskal-Wallis test [32], which is a non-parametric ANOVA test, to measure whether there is a significant difference in the [PERSON]'s rank correlations  
Figure 6: Example of comparison between grad-CAM maps obtained with the original image (on the **left**) and with the segmented image (on the **right**) for various density classes. Figure 7: Box plots of the [PERSON]’s rank correlation coefficients r obtained for the correctly classified mammograms across the four density categories. The correlation values obtained on the cranio-caudal (CC) and medio-lateral-oblique (MLO) projections are separately shown. The box plots are centered on the median and the boxes represent the interquartile range. among the four classes separately for CC and MLO projections. We obtained a p-value less than 0.05 for both tests and, hence, we can affirm that there is a significant difference among the classes. However, the Kruskal-Wallis test does not state if all the groups are significantly different. For this reason we computed the Dunn test [33] with correction for multiple comparisons, which is the post hoc analysis for Kruskal-Wallis test. The Dunn test showed a significant difference among A, B and C classes, while this is not true for the D class (Table 4). From the boxplots, it can be noticed that high median values of r are generally obtained on mammograms belonging to higher density classes. For the C class the correlation values show the highest median value and the most compact distribution, thus indicating that the CNN classifier is actually considering the higher density areas as the ones to take into account to assign the mammogram to a breast density category. For mammograms of the D class (i.e., those with higher density) this is not always true. The grad-CAM is not in general activated consistently with the higher-density areas of the breast, as depicted in the mammograms. However, the large spread of the r values for this category hampers drawing generalized conclusions. For mammograms of the B categories a positive median r values still indicates a systematic overlap between the higher-intensity areas of the grad-CAM maps and the mammograms. By contrast, the situation is controversial for the mammograms belonging to the A category.