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Modelling the distribution of rare and invasive species often occurs in situations where reliable absences for evaluating model performance are unavailable. However, predictions at randomly located sites, or ‘background’ sites, can stand in for true absences. The maximum value of the area under the receiver operator characteristic curve, AUC, calculated with background sites is believed to be 1 − a/2, where a is the typically unknown prevalence of the species on the landscape. Using a simple example of a species' range, I show how AUC can achieve values > 1 − a/2 when test presences do not represent each inhabited region of a species__ range in proportion to its area. Values of AUC that surpass 1 − a/2 are associated with higher model predictions in areas overrepresented in the test data set, even if they are less environmentally suitable than other regions the species occupies. Pursuit of high AUC values …