Derivation of Generalized Equations for the Predictive Value of Sequential Screening Tests

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, $\phi$ is the pre-test probability, the combined positive predictive value, $\rho(\phi)$, of $n$ serial positive tests, is described by: $\rho(\phi) = \frac{\phi\displaystyle\prod_{i=1}^{n}a_n}{\phi\displaystyle\prod_{i=1}^{n}a_n+(1-\phi)\displaystyle\prod_{i=1}^{n}(1-b_n)}$ 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: $\psi(\phi) = \frac{[(1-\phi)b_{n-}]\displaystyle\prod_{i=b_{1+}}^{b_{(n-1)+}}(1-b_{n+})}{[\phi(1-a_{n-})]\displaystyle\prod_{i=a_{1+}}^{a_{(n-1)+}}a_{n+}+[(1-\phi)b_{n-}]\displaystyle\prod_{i=b_{1+}}^{b_{(n-1)+}}(1-b_{n+})}$ 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: $\lambda(\phi)= \frac{\phi a_{n+}\displaystyle\prod_{i=a_{1-}}^{a_{(n-1)-}}(1-a_{n-})}{\phi a_{n+}\displaystyle\prod_{i=a_{1-}}^{a_{(n-1)-}}(1-a_{n-})+[(1-\phi)(1-b_{n+})]\displaystyle\prod_{i=b_{1-}}^{b_{(n-1)-}}b_{n-}}$ 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.