Why are there only two parameters that characterize a binary test? The facile, but incorrect, response is that there are only two error conditions, false positive and false negative, so once we know those two numbers we are done. But those two percentages in isolation do not tell the whole story.
Our 2x2 matrix contains four cells, and the numbers in them are percentages. Once we have any three cells filled, the number in the fourth cell is determined, because the sum of the numbers in the four cells must be 100.
From the three independent cells we can form six ratios. Three of them are inverses of others, and contain no new information. We are left with three seemingly independent ratios, but in fact there are only two. For example, choosing from A, B, and C the ratios A/B and B/C, the ratio A/C contains no new information, as A/C = (A/B) * (B/C).
Any two ratios that come from the 2x2 matrix characterize the binary test. We know that the pairs (e, f), (LR+, LR-), and (sensitivity, specificity) must all contain the same information, as they are just different combinations of the same four numbers. It follows that there must be a transformation that takes any pair into any other pair, and the nomogram encapsulates those transformations.
The author would like to thank his friend and colleague Ron Doerfler for reviewing the manuscript and massaging the nomogram.