Department of Radiology, The University of Chicago Medical Center, IL 60637-1470, USA.
Acad Radiol. 2010 Aug;17(8):960-968.e4. doi: 10.1016/j.acra.2010.04.001.
Although an ideal observer's receiver operating characteristic (ROC) curve must be convex-ie, its slope must decrease monotonically-published fits to empirical data often display "hooks." Such fits sometimes are accepted on the basis of an argument that experiments are done with real, rather than ideal, observers. However, the fact that ideal observers must produce convex curves does not imply that convex curves describe only ideal observers. This article aims to identify the practical implications of nonconvex ROC curves and the conditions that can lead to empirical or fitted ROC curves that are not convex.
This article views nonconvex ROC curves from historical, theoretical, and statistical perspectives, which we describe briefly. We then consider population ROC curves with various shapes and analyze the types of medical decisions that they imply. Finally, we describe how sampling variability and curve-fitting algorithms can produce ROC curve estimates that include hooks.
We show that hooks in population ROC curves imply the use of an irrational decision strategy, even when the curve does not cross the chance line, and therefore usually are untenable in medical settings. Moreover, we sketch a simple approach to improve any nonconvex ROC curve by adding statistical variation to the decision process. Finally, we sketch how to test whether hooks present in ROC data are likely to have been caused by chance alone and how some hooked ROCs found in the literature can be easily explained as fitting artifacts or modeling issues.
In general, ROC curve fits that show hooks should be looked on with suspicion unless other arguments justify their presence.
尽管理想观察者的接收机操作特性（ROC）曲线必须是凸的，即其斜率必须单调递减，但发表的对经验数据的拟合常常显示出“钩子”。这种拟合有时基于这样一种论点被接受，即实验是用真实的而不是理想的观察者进行的。然而，理想观察者必须产生凸曲线的事实并不意味着凸曲线仅描述理想观察者。本文旨在确定非凸 ROC 曲线的实际影响，以及可能导致非凸经验或拟合 ROC 曲线的条件。
本文从历史、理论和统计角度来看待非凸 ROC 曲线，我们简要描述了这些角度。然后，我们考虑了具有各种形状的群体 ROC 曲线，并分析了它们所隐含的医学决策类型。最后，我们描述了抽样变异性和曲线拟合算法如何产生包含钩子的 ROC 曲线估计值。
我们表明，群体 ROC 曲线中的钩子意味着使用了不合理的决策策略，即使曲线不穿过机会线，因此在医学环境中通常是不可行的。此外，我们概述了一种简单的方法，通过在决策过程中添加统计变化来改进任何非凸 ROC 曲线。最后，我们概述了如何测试 ROC 数据中存在的钩子是否可能仅仅是由于偶然原因造成的，以及文献中发现的一些带有钩子的 ROC 曲线如何可以很容易地解释为拟合伪影或建模问题。
一般来说，除非有其他论据证明其存在的合理性，否则应怀疑显示钩子的 ROC 曲线拟合。
