2020/12/13 by Jacques Balayla, Balayla, Jacques
Computer Science · Decision Sciences · Mathematics · #Bayesian Methods and Mixture Models #FOS: Computer and information sciences #Methodology (stat.ME) #Probability and Risk Models #Statistical Methods in Clinical Trials #stat.ME
paper · pdf · doi:10.48550/arxiv.2012.07066
openalex publication_date 2020/12/13 · openalex created_date 2021/09/27 · arxiv created 2021/10/10 · arxiv updated 2021/10/12 · openalex updated_date 2026/07/28
From the fundamental theorem of screening (FTS) we obtain the following mathematical relationship relaying the pre-test probability of disease ϕ to the positive predictive value ρ(ϕ) of a screening test: limε → 2 ∫01ρ(ϕ)dϕ = 1 where ε is the screening coefficient - the sum of the sensitivity (a) and specificity (b) parameters of the test in question. However, given the invariant points on the screening plane, identical values of ε may yield different shapes of the screening curve since ε does not respect traditional commutative properties. In order to compare the performance between two screening curves with identical ε values, we derive two geometric definitions of the positive likelihood ratio (LR+), defined as the likelihood of a positive test result in patients with the disease divided by the likelihood of a positive test result in patients without the disease, which helps distinguish the performance of both screening tests. The first definition uses the angle β created on the vertical axis by the line between the origin invariant and the prevalence threshold ϕe such that LR+ = (a)/(1-b) = cot2(β). The second definition projects two lines (y1,y2) from any point on the curve to the invariant points on the plane and defines the LR+ as the ratio of its derivatives (dy1)/(dx) and (dy2)/(dx). Using the concepts of the prevalence threshold and the invariant points on the screening plane, the work herein presented provides a new geometric definition of the positive likelihood ratio (LR+) throughout the prevalence spectrum and describes a formal measure to compare the performance of two screening tests whose screening coefficients ε are equal.