Prevalence Threshold and the Geometry of Screening Curves

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.