On the Impossibility of Equating the Youden Index with Tjur's $R^2$-like metrics in $2\times2$ Tables

In 2017, Hughes claimed an equivalence between Tjurs $R^2$ coefficient of discrimination and Youden index for assessing diagnostic test performance on $2\times 2$ contingency tables. We prove an impossibility result when averaging over binary outcomes (0s and 1s) under any continuous real-valued scoring rule. Our findings clarify the limitations of such a possible equivalence and highlights the distinct roles these metrics play in diagnostic test assessment.