Abstract
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 for exists. However, when applying screening programs in closed systems, 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—over some time t. 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. Given their theoretical nature, clinical application of the concepts herein reported need validation prior to implementation. Meanwhile, an understanding of how the dynamics of prevalence can affect the PPV over time can help inform clinicians as to the reliability of a screening test’s results.
Highlights
Data Availability Statement: All relevant data are within the paper and its Supporting information files
As the prevalence in a population drops with successful population-level screening and treatment, the positive predictive value of the screening test drops, and the false discovery rate, which is equivalent to the complement of the positive predictive value, increases
As we described above, the critical factor is where lie the initial prevalence level φ0, the subsequent prevalence level φk, their difference k, and how they relate to the prevalence threshold, φe, below which the screening paradox becomes more pronounced
Summary
Bayes’ Theorem describes the probability of an event occurring based on prior knowledge of conditions related to that specific event [1]. Bayes theorem’ has applications in innumerable fields. Not surprisingly, it has significant implications in epidemiological modelling as well. From Bayes’ theorem, we can derive the positive predictive value ρ(φ) (PPV) of a screening test, defined as the percentage of patients with a positive screening test that do have the disease screened for, as follows [1]: rð Þ 1⁄4. A ð1Þ a þ ð1 À bÞð1 À Þ where ρ(φ) = PPV, a = sensitivity, b = specificity and φ = prevalence. The PPV ρ(φ) is a function of the disease prevalence, φ. Ρ(φ) increases and vice-versa [3]
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