Discrimination (pseudo) $R^2$ vs. C-index

Question

In the context of binary logistic regression. Both pseudo $R^2$ and C-index measures the discrimination of the model. But why do you need both ? can you gain something from one but not from the other?

Answer 1

The $c$-index, i.e., the concordance probability, which has an equivalence with Somers' $D_{xy}$ rank correlation between predicted and observed, and with the area under the ROC curve, is a good way to describe pure predictive discrimination of a binary regression model. But it doesn't sufficiently reward extreme predictions that are "right". So it is not sensitive enough for comparing two models. The gold standard statistical information measure is the log likelihood (log likelihood + log prior if you are Bayesian). Pseudo $R^2$ is a function of the log likelihood so it is also a gold standard.

For binary $Y$ there are other measures that are sensitive, and are a little more interpretable than $R^2$. These are detailed here.