Prevalence Threshold and the Geometry of Screening Curves
share
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 and maximum 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 at a differential pace relative to ϕ. Herein, we show the mathematical model exploring this phenomenon and define the prevalence threshold ϕ_e point where this change occurs as: ϕ_e=√(a(-b+1))+b-1/(ε-1), where ε = a+b. From the prevalence threshold we deduce a more generalized relationship between prevalence and positive predictive value as a function of ε, which represents a fundamental theorem of screening, herein defined as: lim_ε→ 2∫_0^1ρ(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.
READ FULL TEXT
