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Standard ROC curves look at how setting various thresholds on a continuous measure can be used to predict a two-level ordinal outcome (example: antibody level -> (not sick, sick) ). This can then be integrated over the whole range of thresholds yielding the classical Area Under the Curve (AUC).

The AUC has an interesting alternative definition: it is also the probability that, for a given pair of subjects, the continuous measure will rank them correctly with respect to the ordinal outcome.

In the case where this ordinal outcome has more than two levels (example: not sick, a little sick, very sick) , this alternative definition can be used to get a natural generalization of the concept of AUC.

My question is: does this generalization have a name? Is it a standard or at least known measure? Are there any more widely used alternatives?

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I'm no expert, but I have recently read a paper exploring some of the AUC extensions to multiclass. They called their extension Volume Under the ROC Surface and they also explored some other extension options. Here's a direct link to the work of Ferri et al:

Volume Under the ROC Surface for Multi-class Problems

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    $\begingroup$ If I understand correctly, the problem they tackle is a bit different (and more complicated), since they seem to assume any classifiers working on any inputs, whereas I use the original ROC curve with a single scalar input and the trivial threshold classifiers. Good finding anyways, thanks! $\endgroup$ – static_rtti Jun 4 '13 at 14:48

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