Generalizing the Prevalence Threshold Under Prevalence-Dependent Diagnostic Accuracy

When a diagnostic test returns a positive result, the probability that the patient truly has the condition depends on how common the condition is in the population being tested. The prevalence threshold marks the point on that relationship below which a positive result stops being trustworthy. Its familiar closed form assumes that sensitivity and specificity stay fixed as the population changes, yet large empirical surveys of the diagnostic literature show that measured accuracy shifts systematically with prevalence, because prevalence acts as a marker for differences in disease spectrum, case mix, referral and verification. This study asked how the prevalence-threshold framework should be generalised when accuracy varies in this way, which of its classical properties survive, and whether the difference matters in practice.

Sensitivity and specificity were allowed to vary smoothly across populations indexed by prevalence, producing a prevalence-indexed positive likelihood ratio and a corresponding adaptive Bayesian screening curve. We derived the resulting threshold, introduced a dimensionless evidence-drift function that measures how quickly the strength of diagnostic evidence changes as the population changes, and established conditions for the threshold to exist, to be unique, and to be computable by simple iteration. Full derivations and proofs are provided in the Supplementary Appendix. We then applied the framework to two sources of published data: summary accuracy estimates for SARS-CoV-2 rapid antigen tests from a Cochrane review, and pooled prevalence-accuracy regression coefficients from a meta-epidemiological analysis of 6909 diagnostic accuracy studies.

The generalised threshold is defined implicitly rather than by a closed formula, because the likelihood ratio must be evaluated at the threshold itself. Three properties that identify the same point under fixed accuracy-balance on the anti-diagonal of the probability square, unit slope, and maximal curvature-separate once accuracy varies with prevalence, and the size of that separation is itself informative. Evidence drift governs the slope at the threshold and supplies a sufficient condition for uniqueness. In the rapid antigen application, sensitivity fell and specificity rose between the higher-prevalence symptomatic setting and the lower-prevalence asymptomatic setting, so the positive likelihood ratio more than doubled as prevalence fell; carrying the symptomatic likelihood ratio into the asymptomatic setting understated post-test probability by roughly 19 percentage points. In the meta-epidemiological calibration, drift was negative across every clinical subgroup examined and large enough in several to abolish the classical unit-slope property.

The classical prevalence threshold is the constant-accuracy, zero-drift case of a broader adaptive theory. The framework supplies explicit, computable criteria for judging when a likelihood ratio estimated in one population can be carried to another, and the worked applications indicate that the correction is not merely theoretical.