I have a situation where I am fitting two simple logistic regression models - one model with the variable of interest included as the only predictor, and the other model with the log of the variable of interest included as the only predictor. Both models have the same Area Under the Curve, and I would like to know how to explain why this occurs. I am sure this is not due to chance, but rather has something to do with how AUC is calculated, and it's interpretation.
It is because the AUC is invariant to monotonic changes of variable, of which the log-transform is a special case. The AUC is the probability that a randomly selected case has a higher risk than a control. While the raw difference in risk may not be the same for those two models, the case will still have a higher risk when calculated using either the log-transformed predictor or the untransformed predictor.
It should give us some pause and doubt about AUC that it makes no use whatsoever of the actual risk predicted by the model, but rather the ordering of groups according to a predicted risk (be it arbitrary or otherwise). The axes on a ROC are just sensitivity and 1-specificity.