# ROC/AUC: compare the discrimination ability of a single predictor and a model

I would like to compare a risk prediction test (model) with a single predictor (continuous variable):

Let's say I have a risk prediction testA (e.g. a logistic regression model) that gives a risk probability for cancer (yes/no) between 0-1 when it is applied on a patient with the characteristics B, C, D, E (test A uses all of these characteristics B-E).

Now there has been a new characteristic X found (a continuous variable such as systolic blood pressure) and we think that it could be a good predictor of the cancer state (yes/no) of a patient.

If I want to compare the testA and the characteristic X now to decide which is the better discriminator for the cancer state (yes/no), how could I do that?

1. Steyerberg EW, Vickers AJ, Cook NR, et al. Assessing the Performance of Prediction Models: A Framework for Traditional and Novel Measures. Epidemiology. 2010;21(1):128–138.

and thought about performing a ROC analysis and compare the AUC of the testA with the AUC of the characteristic X using a DeLong test.

However, since characteristic X ist not really model, I am not sure if my approach makes any sense.

The $$c$$-index (concordance probability; AUROC; see also Somers' $$D_{xy}$$ rank correlation coefficient) is not sensitive enough for comparing two models. See http://fharrell.com/post/addvalue for the most sensitive measures.