论文标题
关于阳性似然比（LR+）的几何定义定义的定理
Theorems on the Geometric Definition of the Positive Likelihood Ratio (LR+)
论文作者
论文摘要
从筛选的基本定理（FTS）中，我们获得了以下数学关系，将疾病的预测试概率$ ϕ $传达给筛选测试的正预测值$ρ（ϕ）$： $ \ displayStyle \ lim _ {\ varepsilon \ to 2} {\ displaystyle \ int_ {0}^{1}}}}} {ρ（ϕ）dx} = 1 $ $ 其中$ \ varepsilon $是筛选系数 - 敏感性（$ a $）和特定性（$ b $）的总和。但是，鉴于筛选平面上不变点，$ \ varepsilon $的相同值可能会产生筛选曲线的不同形状，因为$ \ varepsilon $不尊重传统的交换性能。为了比较两条筛选曲线之间的性能与$ \ varepsilon $值，我们得出了两个阳性似然比（LR+）的几何定义（LR+），定义为进行阳性测试的可能性导致疾病患者的可能性通过疾病的阳性测试结果而导致疾病的阳性测试的可能性分裂，这有助于筛查测试，从而区分筛查的表现，这两种筛查的表现都在筛查。第一个定义使用在垂直轴上创建的角度$β$，由原点不变和流行阈值$ ϕ_e $之间的界线使$ lr+ = \ frac {a} {1-b} = cot^2 {（β）} $。第二个定义将两行$（y_1，y_2）$从曲线上的任何点投射到平面上不变点，并将LR+定义为其衍生物$ \ frac {dy_1} {dx} {dx} $和$ \ frac {dy_2}} {dx} {dx} $的比率。使用筛选平面上的流行阈值和不变点的概念，本文提出的工作在整个流行频谱中提供了一个新的几何定义（LR+）的新几何定义，并描述了一个正式的措施，以比较两个筛选测试的性能，其筛选系数$ \ varepsilon $相等。
From the fundamental theorem of screening (FTS) we obtain the following mathematical relationship relaying the pre-test probability of disease $ϕ$ to the positive predictive value $ρ(ϕ)$ of a screening test: $\displaystyle\lim_{\varepsilon \to 2}{\displaystyle \int_{0}^{1}}{ρ(ϕ)dϕ} = 1$ where $\varepsilon$ is the screening coefficient - the sum of the sensitivity ($a$) and specificity ($b$) parameters of the test in question. However, given the invariant points on the screening plane, identical values of $\varepsilon$ may yield different shapes of the screening curve since $\varepsilon$ does not respect traditional commutative properties. In order to compare the performance between two screening curves with identical $\varepsilon$ values, we derive two geometric definitions of the positive likelihood ratio (LR+), defined as the likelihood of a positive test result in patients with the disease divided by the likelihood of a positive test result in patients without the disease, which helps distinguish the performance of both screening tests. The first definition uses the angle $β$ created on the vertical axis by the line between the origin invariant and the prevalence threshold $ϕ_e$ such that $LR+ = \frac{a}{1-b} = cot^2{(β)}$. The second definition projects two lines $(y_1,y_2)$ from any point on the curve to the invariant points on the plane and defines the LR+ as the ratio of its derivatives $\frac{dy_1}{dx}$ and $\frac{dy_2}{dx}$. Using the concepts of the prevalence threshold and the invariant points on the screening plane, the work herein presented provides a new geometric definition of the positive likelihood ratio (LR+) throughout the prevalence spectrum and describes a formal measure to compare the performance of two screening tests whose screening coefficients $\varepsilon$ are equal.
