Invariant Points on the Screening Plane: a Geometric Definition of the Likelihood Ratio (LR+)
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From the fundamental theorem of screening we obtain the following mathematical relationship relaying the positive predictive value (ρ(ϕ)) of a screening test to the prevalence (ϕ) of disease: lim_ε→ 2∫_0^1ρ(ϕ)dϕ = 1 where ε is equal to the sum of the sensitivity (a) and specificity (b) parameters of the test in question. However, identical values of ε may yield different shapes of the screening curve since ε does not respect traditional commutative properties given the invariant points of the screening plane. In order to distinguish between two screening curves with identical ε values, we make use of the angle β created by the line between the origin invariant and the prevalence threshold ϕ_e to make a right-angle triangle. In this work, we provide derivation of this angle β and show its value to be: β = arctan(Ψ) = arctan(√(1-b/a)) From the above relationship, we derive the positive likelihood ratio (LR+), defined as the likelihood of a test result in patients with the disease divided by the likelihood of the test result in patients without the disease, as follows: LR+ = a/1-b = cot^2(β) 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 ε are equal.
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