The relationship between Gini terminology and the ROC curve

The objectives of this note are to correct a common error and to clarify the connection between the Gini terminology as used in the economic literature and the one used in the diagnostic and classification literature. More specifically, the connection between the area under the receiver operating characteristic (ROC) curve, which is frequently used in the diagnosis and classification literature, and the Gini terminology, which is mainly used in the economic literature, is clarified. It is shown that the area under the ROC curve is related to the covariance between the two vectors $$Y=\{y_i\}_{i=1}^{n_0}$$Y={yi}i=1n0 and $$\{i/{n_0}\}_{i=1}^{n_0}$${i/n0}i=1n0. Here $$y_i$$yi is the number of items classified to group 1 lying between the $$(i-1)\mathrm{th}$$(i-1)th and the $$i\mathrm{th}$$ith items classified to group 0, and $$n_0$$n0 is the number of items in group 0.