[Paper Review] Derivation of Generalized Equations for the Predictive Value of Sequential Screening Tests

Jacques Balayla|arXiv (Cornell University)|Jul 26, 2020

Advanced Statistical Methods and Models4 references4 citations

TL;DR

This paper derives generalized Bayesian equations to compute the positive and negative predictive values of sequential and orthogonal screening tests, accounting for varying sensitivity, specificity, and pre-test probability. It demonstrates that repeated or combined testing significantly improves predictive accuracy—especially in low-prevalence settings—by mathematically modeling the impact of test sequences interrupted by conflicting results.

ABSTRACT

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: $ρ(ϕ) = \frac{ϕ\displaystyle\prod_{i=1}^{n}a_n}{ϕ\displaystyle\prod_{i=1}^{n}a_n+(1-ϕ)\displaystyle\prod_{i=1}^{n}(1-b_n)}$ 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: $ψ(ϕ) = \frac{[(1-ϕ)b_{n-}]\displaystyle\prod_{i=b_{1+}}^{b_{(n-1)+}}(1-b_{n+})}{[ϕ(1-a_{n-})]\displaystyle\prod_{i=a_{1+}}^{a_{(n-1)+}}a_{n+}+[(1-ϕ)b_{n-}]\displaystyle\prod_{i=b_{1+}}^{b_{(n-1)+}}(1-b_{n+})}$ 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: $λ(ϕ)= \frac{ϕa_{n+}\displaystyle\prod_{i=a_{1-}}^{a_{(n-1)-}}(1-a_{n-})}{ϕa_{n+}\displaystyle\prod_{i=a_{1-}}^{a_{(n-1)-}}(1-a_{n-})+[(1-ϕ)(1-b_{n+})]\displaystyle\prod_{i=b_{1-}}^{b_{(n-1)-}}b_{n-}}$ 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.

Motivation & Objective

• To develop generalized mathematical equations that compute the predictive value of sequential screening tests under varying sensitivity, specificity, and pre-test probability.
• To address the limitations of single-test screening in low-prevalence conditions where positive predictive value (PPV) drops sharply.
• To model scenarios where sequential testing is interrupted by conflicting results (e.g., a positive after negative tests), and derive the resulting predictive values.
• To provide a formal framework for orthogonal testing—using multiple independent tests in sequence—to improve diagnostic certainty beyond single-test performance.
• To quantify the geometric partitioning of predictive curves and identify the prevalence threshold at which PPV and NPV curves intersect, enabling optimal test strategy selection.

Proposed method

• Uses Bayes’ Theorem to derive the combined positive predictive value (PPV) for n serial positive tests: $\rho(\phi) = \frac{\phi \prod_{i=1}^{n} a_i}{\phi \prod_{i=1}^{n} a_i + (1-\phi) \prod_{i=1}^{n} (1-b_i)}$.
• Derives the negative predictive value (NPV) when a positive test interrupts a sequence of negative results: $\psi(\phi) = \frac{[(1-\phi)b_{n-}]\prod_{i=1}^{n-1}(1-b_i)}{[\phi(1-a_{n-})]\prod_{i=1}^{n-1}a_i + [(1-\phi)b_{n-}]\prod_{i=1}^{n-1}(1-b_i)}$.
• Derives the PPV when a negative test interrupts a sequence of positive results: $\lambda(\phi) = \frac{\phi a_{n+} \prod_{i=1}^{n-1}(1-a_i)}{\phi a_{n+} \prod_{i=1}^{n-1}(1-a_i) + (1-\phi)(1-b_{n+}) \prod_{i=1}^{n-1}b_i}$.
• Introduces the concept of geometric partitioning of predictive curves using definite integrals to quantify the Net Diagnostic Potential (NDP) and Positive Diagnostic Potential (PDP).
• Derives the intersection point $\phi_i$ of PPV and NPV curves by solving $\frac{a\phi}{a\phi + (1-b)(1-\phi)} = \frac{b(1-\phi)}{\phi(1-a) + b(1-\phi)}$, yielding a quadratic solution: $\phi_i = \frac{-b^2 + b - \sqrt{ab(ab - a + 1 - b)}}{a^2 - b^2 - a + b}$.
• Proposes a method to handle degenerate cases where sensitivity equals specificity by perturbing specificity slightly ($b \Rightarrow b + db$) to avoid division by zero.

Experimental results

Research questions

• RQ1 How can the positive predictive value be mathematically modeled when multiple independent screening tests are applied in sequence, especially when results are inconsistent?
• RQ2 What is the predictive value following a sequence of negative tests interrupted by a single positive test, and how does it depend on test characteristics?
• RQ3 At what pre-test probability does the positive predictive value curve intersect the negative predictive value curve, and what is the significance of this intersection?
• RQ4 How does sequential or orthogonal testing improve diagnostic certainty in conditions with low pre-test probability where single tests are unreliable?
• RQ5 What is the geometric partitioning of predictive value curves, and how can it be quantified using integrals to assess diagnostic potential?

Key findings

• The combined positive predictive value for n serial positive tests is given by $\rho(\phi) = \frac{\phi \prod_{i=1}^{n} a_i}{\phi \prod_{i=1}^{n} a_i + (1-\phi) \prod_{i=1}^{n} (1-b_i)}$, which increases with the number of positive tests.
• When a sequence of negative tests is interrupted by a positive result, the resulting positive predictive value is $\lambda(\phi) = \frac{\phi a_{n+} \prod_{i=1}^{n-1}(1-a_i)}{\phi a_{n+} \prod_{i=1}^{n-1}(1-a_i) + (1-\phi)(1-b_{n+}) \prod_{i=1}^{n-1}b_i}$, which depends on the number and performance of prior negative tests.
• The intersection point $\phi_i$ of the PPV and NPV curves is determined by solving a quadratic equation derived from equating the standard PPV and NPV formulas, yielding $\phi_i = \frac{-b^2 + b - \sqrt{ab(ab - a + 1 - b)}}{a^2 - b^2 - a + b}$.
• In low-prevalence settings, sequential testing can overcome the limitations of single tests by significantly increasing the PPV, even when individual tests have moderate sensitivity and specificity.
• The Net Diagnostic Potential (NDP) and Positive Diagnostic Potential (PDP) are quantified using definite integrals: $NDP = \int_0^{\phi_i} \sigma(\phi) d\phi - \int_0^{\phi_i} \rho(\phi) d\phi$ and $PDP = \int_{\phi_i}^1 \rho(\phi) d\phi - \int_{\phi_i}^1 \sigma(\phi) d\phi$, providing a geometric measure of diagnostic utility.
• When sensitivity and specificity are equal, the denominator in the intersection formula becomes zero; this is resolved by perturbing specificity slightly ($b \Rightarrow b + db$) to obtain a stable estimate of $\phi_i$.