# 37 Somers’ D

Somers’ $$D$$ has an asymptotically $$Normal$$ Distribution . It may take any value between -1 and 1. It is used to measure classification agreement between a predictor and outcome variable.

Somers’ $$D$$ is related to a form of a concordance index. Concrodance in measured between 0 and 1 and can effectively be calculated by rescaling Somers’ $$D$$. The rescaling can be accomplished by:

$C = \frac{D+1}{2}$

## 37.1 Theorems for Somers’ $$D$$

### 37.1.1 Theorem: Distribution of Somers’ Derived Concordance

Let $$D \sim$$ Normal$$(\mu, \sigma^2)$$. Then $$C \sim$$ Normal $$(\frac{\mu+1}{2},\frac{\sigma^2}{4})$$.

Proof:

$D \sim Normal(\mu,\sigma^2) \Rightarrow (D+1) \sim Normal(\mu+1,\sigma^2) \\ (D+1) \sim Normal(\mu+1,\sigma^2) \Rightarrow \frac{D+1}{2} \sim Normal(\frac{\mu+1}{2}, \frac{\sigma^2}{4})$

By definition, $$C=\frac{D+1}{2}$$, so $$C \sim Normal(\frac{\mu+1}{2},\frac{\sigma^2}{4})$$.

Note: when the dependent variable is a binary response, the Concordance Index is equal to the area under the Receiver Operator Characteristic (ROC) curve, or AUC.