**Parametric Exploration of the Mann-Whitney U Test**

Abraham Nunes MD PhD MBA

*Dalhousie University, Halifax, Nova Scotia, Canada*

**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) |