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.