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I understand what a ROC curve is. However, I do not understand the Gini coefficient in the context of binary classification.

All the resources I have checked state that $Gini = 1 - (2 \times AUC_{ROC})$.

How is this equality derived? As an economist, it troubles me to think of a Gini coefficient in this context.

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  • $\begingroup$ Aside: Gini, Corrado. (1912) "On the measure of concentration with especial reference to income and wealth" Reprinted in Memorie di metodologica statistica (Ed. Pizetti E, Salvemini, T). Rome: Libreria Eredi Virgilio Veschi (1955). $\endgroup$
    – Alexis
    Commented Mar 2, 2022 at 20:54
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    $\begingroup$ Watch out: the name Gini coefficient has been applied to at least three different measures. Trust equations used, not names. $\endgroup$
    – Nick Cox
    Commented Mar 2, 2022 at 21:25
  • $\begingroup$ @NickCox Is that right? Can you recommend a good "Hey look at these three different formal measures!"-type article or blog? $\endgroup$
    – Alexis
    Commented Mar 2, 2022 at 21:47
  • $\begingroup$ Not readily. The main point is that there are literatures that don’t really touch, notably income inequality and CART. $\endgroup$
    – Nick Cox
    Commented Mar 2, 2022 at 22:17

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It's complicated. It appears that the Gini coefficient based on the ROC curve was invented by analogy with the economic Gini coefficient based on the Lorenz curve. It's not a perfect analogy – e.g., the Lorenz curve is necessarily convex and the ROC curve is only necessarily monotone. They do happen to be the same in some situations

Also, if you have fitted probabilities $\hat p_i$ for each individual, and you have a well-calibrated model (ie, $\hat p_i$ really does estimate $P[Y=1|X=x_i]$), there is some relationship between the two in concept. If you have a well-calibrated model, you can summarise how well it predicts by considering the variability in the predictions – since, ex hypothesi, different predictions for different people reflects genuine discriminatory power. So, you can look at the $\hat p$ and ask how unequal they are, with more inequality implying better discrimination.

However, the simple relationship between AUC and the economic Gini coefficient doesn't hold in general.

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