Theorems on the Geometric Definition of the Positive Likelihood Ratio (LR+)

From the fundamental theorem of screening (FTS) we obtain the following mathematical relationship relaying the pre-test probability of disease $\phi$ to the positive predictive value $\rho(\phi)$ of a screening test: $\displaystyle\lim_{\varepsilon \to 2}{\displaystyle \int_{0}^{1}}{\rho(\phi)d\phi} = 1$ 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 $\beta$ created on the vertical axis by the line between the origin invariant and the prevalence threshold $\phi_e$ such that $LR+ = \frac{a}{1-b} = cot^2{(\beta)}$. 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.