  From the fundamental theorem of screening we obtain the following
mathematical relationship relaying the positive predictive value ($\rho(\phi)$)
of a screening test to the prevalence ($\phi$) of disease:
  $\displaystyle\lim_{\varepsilon \to 2}{\displaystyle
\int_{0}^{1}}{\rho(\phi)d\phi} = 1$
  where $\varepsilon$ is equal to the sum of the sensitivity ($a$) and
specificity ($b$) parameters of the test in question. However, identical values
of $\varepsilon$ may yield different shapes of the screening curve since
$\varepsilon$ does not respect traditional commutative properties given the
invariant points of the screening plane. In order to distinguish between two
screening curves with identical $\varepsilon$ values, we make use of the angle
$\beta$ created by the line between the origin invariant and the prevalence
threshold $\phi_e$ to make a right-angle triangle. In this work, we provide
derivation of this angle $\beta$ and show its value to be:
  $\beta = arctan(\Psi) = arctan\left(\sqrt{\frac{1-b}{a}}\right)$
  From the above relationship, we derive the positive likelihood ratio (LR+),
defined as the likelihood of a test result in patients with the disease divided
by the likelihood of the test result in patients without the disease, as
follows:
  $LR+ = \frac{a}{1-b} = cot^2{(\beta)}$
  Using the concepts of the prevalence threshold and the invariant points on
the screening plane, the work herein presented provides a new geometric
definition of the positive likelihood ratio (LR+) throughout the prevalence
spectrum and describes a formal measure to compare the performance of two
screening tests whose $\varepsilon$ are equal.
