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I noticed that software packages sometimes use the function:

.5 + abs(AUC-.5)

when the AUC is less than $0.5$. In other words, it has:

$$ 0.5+|AUC-0.5| $$

as a way to report AUC's under 0.5. Is this just a result of an AUC under $0.5$ being a worthless predictor and is the function above just reversing the labels to whatever prediction scheme is used to calculate the AUC? Thanks.

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If the AUC is below 0.5, then the equation you state gives the AUC when the predictor flips its predictions. When this occurs the flipped predictions are better than the original, at least by the AUC metric. AUCs substantially lower than 0.5 usually indicate some sort of programming error though, if the predictor is not working at all you should get AUC scores close to 0.5, but they can be slightly above or below 0.5 (how close depends on the size of your dataset). I would guess that the software you are referring to does this flip mostly to avoid confusing users.

I find the best interpretation of AUC is not as area-under-the-curve, but as the probability that the predictor correctly ranks a randomly chosen pair of positive and negative instances. Obviously, just ranking pairs randomly gets you an expected 50% success rate, hence the 0.5 baseline for AUC.

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