Derivation of Generalized Equations for the Predictive Value of Sequential Screening Tests
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Using Bayes' Theorem, we derive generalized equations to determine the positive and negative predictive value of screening tests undertaken sequentially. Where a is the sensitivity, b is the specificity, ϕ is the pre-test probability, the combined positive predictive value, ρ(ϕ), of n serial positive tests, is described by: ρ(ϕ) = ϕ∏_i=1^na_n/ϕ∏_i=1^na_n+(1-ϕ)∏_i=1^n(1-b_n) If the positive serial iteration is interrupted at term position n_i-k by a conflicting negative result, then the resulting negative predictive value is given by: ψ(ϕ) = [(1-ϕ)b_n-]∏_i=b_1+^b_(n-1)+(1-b_n+)/[ϕ(1-a_n-)]∏_i=a_1+^a_(n-1)+a_n++[(1-ϕ)b_n-]∏_i=b_1+^b_(n-1)+(1-b_n+) Finally, if the negative serial iteration is interrupted at term position n_i-k by a conflicting positive result, then the resulting positive predictive value is given by: λ(ϕ)= ϕ a_n+∏_i=a_1-^a_(n-1)-(1-a_n-)/ϕ a_n+∏_i=a_1-^a_(n-1)-(1-a_n-)+[(1-ϕ)(1-b_n+)]∏_i=b_1-^b_(n-1)-b_n- The aforementioned equations provide a measure of the predictive value in different possible scenarios in which serial testing is undertaken. Their clinical utility is best observed in conditions with low pre-test probability where single tests are insufficient to achieve clinically significant predictive values and likewise, in clinical scenarios with a high pre-test probability where confirmation of disease status is critical.
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