Bayesian Updating and Sequential Testing: Overcoming Inferential Limitations of Screening Tests

TL;DR

This paper develops a mathematical model to determine how sequential testing can overcome Bayesian limitations in screening tests, improving positive predictive value across different prevalence levels.

Contribution

It introduces a formula for the number of test iterations needed to achieve specific predictive values, addressing Bayesian limitations in screening accuracy.

Findings

Derived a formula for test iterations to reach desired predictive values.

Provided reference tables for practical application across various sensitivities, specificities, and prevalences.

Demonstrated how sequential testing can mitigate Bayesian limitations in screening tests.

Abstract

Bayes' Theorem confers inherent limitations on the accuracy of screening tests as a function of disease prevalence. We have shown in previous work that a testing system can tolerate significant drops in prevalence, up until a certain well-defined point known as the $prevalence$ $threshold$, below which the reliability of a positive screening test drops precipitously. Herein, we establish a mathematical model to determine whether sequential testing overcomes the aforementioned Bayesian limitations and thus improves the reliability of screening tests. We show that for a desired positive predictive value of $\rho$ that approaches $k$, the number of positive test iterations $n_i$ needed is: $ n_i =\lim_{\rho \to k}\left\lceil\frac{ln\left[\frac{\rho(\phi-1)}{\phi(\rho-1)}\right]}{ln\left[\frac{a}{1-b}\right]}\right\rceil$ where $n_i$ = number of testing iterations necessary to achieve $\rho$, the desired positive predictive value, a = sensitivity, b = specificity, $\phi$ = disease prevalence and $k$ = constant. Based on the aforementioned derivation, we provide reference tables for the number of test iterations needed to obtain a $\rho(\phi)$ of 50, 75, 95 and 99$\%$ as a function of various levels of sensitivity, specificity and disease prevalence.