Prevalence Threshold and the Geometry of Screening Curves

TL;DR

This paper mathematically explores the relationship between screening test predictive value and prevalence, introducing a prevalence threshold that impacts test interpretation and proposing a fundamental theorem of screening.

Contribution

It defines the prevalence threshold mathematically and derives a generalized relationship between prevalence and predictive value, advancing theoretical understanding of screening test performance.

Findings

Derived the prevalence threshold formula $rac{ ext{sqrt}(1-b)}{ ext{sqrt}(a)+ ext{sqrt}(1-b)}$.

Established a fundamental theorem linking the integral of predictive value over prevalence to 1 as sensitivity and specificity approach optimal values.

Provided insights to interpret screening test validity in clinical scenarios.

Abstract

The relationship between a screening tests' positive predictive value, $\rho$, and its target prevalence, $\phi$, 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 $\rho$ drops precipitously relative to $\phi$. Herein, we show the mathematical model exploring this phenomenon and define the $prevalence$ $threshold$ ($\phi_e$) point where this change occurs as: $\phi_e=\frac{\sqrt{a\left(-b+1\right)}+b-1}{(\varepsilon-1)}$ 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: $\displaystyle\lim_{\varepsilon \to 2}{\displaystyle \int_{0}^{1}}{\rho(\phi)d\phi} = 1$ 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.