Statistics for Area under the ROC curve

I have a question regarding statistical evaluation of the AUC. In their paper (http://www.jstor.org/stable/2531595), DeLong et al. describe a method to evaluate AUC curves. (Another good explanation can be found in the book "Statistics with Confidence: Confidence Intervals and Statistical Guidelines" by Altman et al.).

As far as I understood, we compute the $\text{AUC}$ and the standard deviation $\sigma$ of the Kernel matrix. Assuming the normal distribution $\mathcal{N}(\text{AUC},\sigma)$ it is possible to compute confidence intervals.

My question is about the normality assumption:

1. The $\text{AUC}$ usually lies in the interval $[0,1]$ but the interval for the normal distribtion is $(-Inf, Inf)$. Is this problem really negligible? (This problem e.g. is solved in pROC package by just restricting the CI to $[0,1]$)

2. The $Beta$ distribution is defined on the interval $[0,1]$ and has the shape parameters $\alpha$ and $\beta$. Can we estimate them given the data like we are able to do it for the AUC?

To give an example: Given a vector c(T,F,F,F,T,F,F,T,F,F) the $\text{AUC} = 0.619$ and $\sigma = 0.237$ which results in 95% CI $(0.156, 1.083)$.

library(pROC)
temp.in <- c(T,F,F,F,T,F,F,T,F,F)
pROC::auc(pROC::roc(controls=which(temp.in), cases=which(!temp.in)))
pROC::ci.auc(pROC::roc(controls=which(temp.in), cases=which(!temp.in)))

Intead of using the normal distribution I would like to use the $Beta$ distribution. But how we can estimate $\alpha$ and $\beta$ for $Beta$ distribution given c(T,F,F,F,T,F,F,T,F,F)?

An alternative given by  is to compute the interval for the logit AUC:

$log \left( \frac{AUC}{1-AUC} \right) \pm \phi ^{-1} \left( 1 - \frac{\alpha}{2} \right) \frac{\sqrt{AUC}}{AUC(1 - AUC)}$

so that you get an asymmetric interval. In your case, you would get a 95% CI $(0.38, 0.81)$.

If you are frequently dealing with high AUCs and small sample sizes, you may want to have a look at  that shows there is no single method that can optimally compute confidence interval for all ROC curves.

 Pepe MS, The Statistical Evaluation of Medical Tests for Classification and Prediction, OUP 2003, p. 107

 Obuchowski NA, Lieber ML, Confidence bounds when the estimated ROC area is 1.0, Acad Radiol. 2002, 9 (5) p. 526-30

• Thank you very much for the alternative way and the hints. Although assymetric intervals are probably better suited to model the CI of the AUC, I think that they are still able to be geater than 1 in some cases. How I would comute $\alpha$ and $\beta$ estimates for $Beta$-distribution is still open. – Drey Feb 27 '14 at 11:40
• @Drey no they can't possibly be outside [0,1] – Calimo Feb 27 '14 at 13:05
• Okay, I think I don't understand what the $\phi^{-1}$ stands for. – Drey Feb 28 '14 at 13:58
• @Drey it is the inverse (or quantile) of the normal CDF... – Calimo Feb 28 '14 at 14:54
• @Drey $\phi$ being the CDF of $\mathcal{N}$ – Calimo Feb 28 '14 at 16:07