The Binormal ROC Model and the U-Statistic 

Recall that the CDF of a normal distribution is 

(1.1)

 

(1.2)

 

Assuming that and that we can compute the ROC curve for two normal distributions as follows: 

(1.3)

 

The area under the ROC curve (AUC) can be computed as follows: 

(1.4)

 

The U-statistic for the binormal model with sample sizes and can be computed from the AUC as follows:  

(1.5)

 

Computing P-Values from the U-Statistic 

At large samples, the U-statistic is normally distributed with mean 

(2.1)

 

and the standard deviation is  

(2.2)

 

We can therefore compute a Z-score: 

(2.3)

 

A two-tailed p-value is thus 

(2.4)