Setting aside the administrative and feasibility concerns, while such an interpretation is theoretically correct, the reality ought to be more nuanced, as there are confounding factors that might make the same result recur upon serial testing on the same patient. Independence of serial testingįrom the concepts described in this work, one might easily suggest that simply repeating the same screening test multiple times increases confidence that a positive result is a true positive. Doing 3 would by definition guarantee that one is above the desired threshold, but doing 2 tests would yield a lower PPV than that desired. In other words, the ceiling function in this context serves to suggest that when say, 2.8 tests are needed to achieve a desired PPV, one is better off doing 3 tests given the discrete nature of tests. For the case of screening tests, it implies that a whole rather than a decimal number of tests (rounded to the nearest, higher, positive integer) ought to be performed. In practical terms, the ceiling function assigns the nearest higher positive integer to a number. In other words, the intersection between the NPV and PPV as per the following equation hovers around 40–60% prevalence for values of sensitivity and specificity greater than 50% (clinically useful ones). ( 8) and ( 11), and the fact that most conditions have a prevalence well below 20% then it follows that if prior to reaching the desired positive predictive value, a negative test result is obtained, the individual is more likely to be disease-free, since \(\sigma (\phi ) \gg \rho (\phi )\) at a low prevalence level of disease (Fig. Given the extremes of the domains of each predictive function as per Eqs. Conversely, a condition whose consequences are less severe but whose treatment may lead to significant morbidity might benefit from a higher degree of diagnostic certainty prior to initiating therapy or proceeding to an invasive diagnostic test. For severe conditions whose treatment is rather innocuous but whose potential consequences are severe, a lower threshold to initiate treatment might be acceptable. The aforementioned relationship holds for a number of identical sequential tests that are positive until the \(n_i\) iteration reaches the desired positive predictive value. Figure 2 provides a graphic representation of the \(n_i\), which given its geometric shape we define as the tablecloth function. Tables 1, 2, 3 and 4 provide different reference values of n as a function of the prevalence \(\phi\) and the sensitivity and specificity for a \(\rho\) of 99, 95, 75 and 50%, respectively. \(n_i =\lim _\right]\) increases, the number of test iterations n needed to achieve a desired positive predictive value decreases as per Eq. Resultsįor a given PPV ( \(\rho\)) approaching k, the number of positive test iterations needed given a prevalence of disease ( \(\phi\)) is: We likewise derive the equation which determines the number of iterations of a positive test needed to obtain a desired positive predictive value, represented graphically by the tablecloth function. We use Bayes’ theorem to derive the positive predictive value equation, and apply the Bayesian updating method to obtain the equation for the positive predictive value (PPV) following repeated testing. Herein, we establish a mathematical model to determine whether sequential testing with a single test overcomes the aforementioned Bayesian limitations and thus improves the reliability of screening tests. Bayes’ theorem confers inherent limitations on the accuracy of screening tests as a function of disease prevalence.
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