# Prevalence Threshold and the Geometry of Screening Curves

arXiv: https://arxiv.org/abs/2006.00398 · HTML: https://mlsanity.com/paper/2006.00398/

## Key facts

- Cited by 7 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 31 May 2020.

## Built on by (most cited first, 7 of 7)

- [The Dutch Draw: Constructing a Universal Baseline for Binary Prediction Models](https://mlsanity.com/paper/2203.13084/) (2022)
- [Prevalence Threshold and bounds in the Accuracy of Binary Classification Systems](https://mlsanity.com/paper/2112.13289/) (2021)
- [Information Threshold, Bayesian Inference and Decision-Making](https://mlsanity.com/paper/2206.02266/) (2022)
- [Testing the Consistency of Performance Scores Reported for Binary Classification Problems](https://mlsanity.com/paper/2310.12527/) (2023)
- [Never mind the metrics -- what about the uncertainty? Visualising confusion matrix metric distributions](https://mlsanity.com/paper/2206.02157/) (2022)
- [Transfer Learning-based Real-time Handgun Detection](https://mlsanity.com/paper/2311.13559/) (2023)
- [A Priori Determination of the Pretest Probability](https://mlsanity.com/paper/2401.04086/) (2024)

## Abstract (arXiv)

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 (φe) point where this change occurs as: φe=frac√aleft(-b+1right)+b-1(varepsilon-1) where varepsilon = a+b. Using its radical conjugate, we obtain a simplified version of the equation: frac√1-b√a+√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: displaystylelimvarepsilon → 2displaystyle int01ρ(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.