论文标题
筛选曲线的流行阈值和几何形状
Prevalence Threshold and the Geometry of Screening Curves
论文作者
论文摘要
筛选测试的积极预测价值（$ρ$）与其目标流行率$ ϕ $之间的关系是比例的 - 尽管完全不是线性的，而是特殊情况。因此，存在一个局部曲率极端的点，仅定义为敏感性$ a $和特异性$ b $的函数，超出了测试的$ρ$相对于$ ϕ $的变化速率。在此，我们显示了探索这种现象的数学模型并定义了$ pervister $ $ $ $ $ $ $ $ $ $ $（$ ϕ_e $）点，其中此更改发生为： $ ϕ_e = \ frac {\ sqrt {a \ left（-b+1 \ right）}+b-1} {（\ varepsilon-1）} $其中$ \ varepsilon $ = $ a $+$ a+$ $ b $。 使用其自由基共轭物，我们获得了方程式的简化版本： $ \ frac {\ sqrt {1-b}} {\ sqrt {a}+\ sqrt {1-b}} $。 从流行率阈值中，我们推断出患病率和积极预测值之间的更广泛的关系，这是$ \ varepsilon $的函数，该函数代表筛选的基本定理，在此定义为： $ \ displayStyle \ lim _ {\ varepsilon \ to 2} {\ displaystyle \ int_ {0}^{1}}}}} {ρ（ϕ）dx} = 1 $ $ 了解这项工作中描述的概念可以有助于实时筛选测试的有效性，并有助于指导对进行筛查的不同临床场景的解释。
The relationship between a screening tests' positive predictive value, $ρ$, and its target prevalence, $ϕ$, is proportional - though not linear in all but a special case. In consequence, there is a point of local extrema of curvature defined only as a function of the sensitivity $a$ and specificity $b$ beyond which the rate of change of a test's $ρ$ drops precipitously relative to $ϕ$. Herein, we show the mathematical model exploring this phenomenon and define the $prevalence$ $threshold$ ($ϕ_e$) point where this change occurs as: $ϕ_e=\frac{\sqrt{a\left(-b+1\right)}+b-1}{(\varepsilon-1)}$ where $\varepsilon$ = $a$+$b$. Using its radical conjugate, we obtain a simplified version of the equation: $\frac{\sqrt{1-b}}{\sqrt{a}+\sqrt{1-b}}$. From the prevalence threshold we deduce a more generalized relationship between prevalence and positive predictive value as a function of $\varepsilon$, which represents a fundamental theorem of screening, herein defined as: $\displaystyle\lim_{\varepsilon \to 2}{\displaystyle \int_{0}^{1}}{ρ(ϕ)dϕ} = 1$ Understanding the concepts described in this work can help contextualize the validity of screening tests in real time, and help guide the interpretation of different clinical scenarios in which screening is undertaken.
