Abstract
Background
Bayes’ theorem confers inherent limitations on the accuracy of screening tests as a function of disease prevalence. 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.
Methods
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. 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.
Results
For a given PPV ($$\rho$$
ρ
) approaching k, the number of positive test iterations needed given a prevalence of disease ($$\phi$$
ϕ
) is:
$$n_i =\lim _{\rho \rightarrow k}\left\lceil \frac{ln\left[ \frac{\rho (\phi -1)}{\phi (\rho -1)}\right] }{ln\left[ \frac{a}{1-b}\right] }\right\rceil \qquad \qquad (1)$$
n
i
=
lim
ρ
→
k
l
n
ρ
(
ϕ
-
1
)
ϕ
(
ρ
-
1
)
l
n
a
1
-
b
(
1
)
where $$n_i$$
n
i
= number of testing iterations necessary to achieve $$\rho$$
ρ
, the desired positive predictive value, ln = the natural logarithm, a = sensitivity, b = specificity, $$\phi$$
ϕ
= disease prevalence/pre-test probability and k = constant.
Conclusions
Based on the aforementioned derivation, we provide reference tables for the number of test iterations needed to obtain a $$\rho (\phi )$$
ρ
(
ϕ
)
of 50, 75, 95 and 99% as a function of various levels of sensitivity, specificity and disease prevalence/pre-test probability. Clinical validation of these concepts needs to be obtained prior to its widespread application.
Multi-dimensional sequential testing and detection
