On the Formalism of The Screening Paradox

Bayes' Theorem imposes inevitable limitations on the accuracy of screening tests by tying the test's predictive value to the disease prevalence. The aforementioned limitation is independent of the adequacy and make-up of the test and thus implies inherent Bayesian limitations to the screening process itself. As per the WHO's $Wilson-Jungner$ criteria, one of the prerequisite steps before undertaking screening is to ensure that a treatment for the condition screened exists. However, in so doing, a paradox, henceforth termed the screening paradox, ensues. If a disease process is screened for and subsequently treated, its prevalence would drop in the population, which as per Bayes' theorem, would make the tests' predictive value drop in return. Put another way, a very powerful screening test would, by performing and succeeding at the very task it was developed to do, paradoxically reduce its ability to correctly identify individuals with the disease it screens for in the future. Where $J$ is Youden's statistic (sensitivity [$a$] + specificity [$b$] - 1), and $\phi$ is the prevalence, the ratio of positive predictive values at subsequent time $k$, $\rho(\phi_{k})$, over the original $\rho(\phi_{0})$ at $t_0$ is given by: $\zeta(\phi_{0},k) = \frac{\rho(\phi_{k})}{\rho(\phi_{0})} =\frac{\phi_k(1-b)+J\phi_0\phi_k}{\phi_0(1-b)+J\phi_0\phi_k}$ In this manuscript, we explore the mathematical model which formalizes said screening paradox and explore its implications for population level screening programs. In particular, we define the number of positive test iterations (PTI) needed to reverse the effects of the paradox as follows: $n_{i\phi_e}=\left\lceil\frac{ln\left[\frac{\omega\phi_e\phi_k-\omega\phi_e}{\omega\phi_e\phi_k-\phi_k}\right]}{2ln\omega}\right\rceil$ where $\omega$ is the square root of the positive likelihood ratio (LR+).