In 2017, Hughes claimed an equivalence between Tjurs $R^2$ coefficient of discrimination and the 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 finding clarifies the limitations of such a possible equivalence and highlights the distinct roles these metrics play in diagnostic test assessment.

翻译：2017年，Hughes声称Tjur的$R^2$判别系数与用于评估2×2列联表诊断测试性能的约登指数之间存在等价性。我们证明了在任何连续实值评分规则下对二元结果（0和1）进行平均时的不可能性结果。我们的发现澄清了这种可能等价性的局限性，并强调了这些度量在诊断测试评估中所扮演的不同角色。