The $R^2$ of a model measures how well a model fits the data and is a measure of the shared variation between two (or more) variables. Its equivalent measure for logistic regression is the pseudo-$R^2$. A pseudo-$R^2$ is sometimes presented alongside the area under the receiver operator characteristic (ROC) as a measure of a model's predictive accuracy.
I'm curious as to whether there is any straightforward relationship between these two metrics. Does a model with a higher pseudo-$R^2$ necessarily have a larger AUC ROC? Are there any situations where a model can have a low pseudo-$R^2$ but a high AUC ROC? It seems intuitive that the two measures are necessarily correlated, but I've been wrong many times in the past.