Impossibility results for equating the Youden Index with average scoring rules and Tjur $R^2$-like metrics

We consider the Youden index fas well as measures evaluating predicted probabilities for the maximum-likelihood estimate of a logistic regression model with predictor the classifier. We give impossibility results showing that the Youden index can not equal any average of a real scoring rule nor any metric averaging over binary outcomes (0s and 1s) for any continuous real-valued scoring rule. This shows the obstructions of such potential equivalences and highlights the distinct roles these metrics play in diagnostic assessment.