Fagan nomogram

Why a straight line gives the right answer

Bayes' theorem in odds form says that the odds after the test are the odds before it, multiplied by the likelihood ratio of the result:

Take logarithms of both sides and the multiplication becomes an addition:

An addition is a thing a ruler can do. Lay the pre-test axis on the left carrying $\log_{10}$ of the odds, and the post-test axis on the right carrying the same scale, and join two points with a straight line. At the midpoint between them the line is at the average of its two endpoints:

So the middle axis reads the likelihood ratio — but at half the decade spacing of the outer two. That halving is not a drawing convention: it falls out of the algebra, and it is why the LR scale on every printed nomogram looks compressed next to its neighbours. The straight line is not an approximation or a lookup trick. It is Bayes' theorem, drawn in log-odds.

How to read it, and what it is for

The nomogram makes visible the one thing a formula hides: how much the test actually moved you. A likelihood ratio of 1 sits at the centre of the middle axis, and the line through it comes out flat — the result taught you nothing. The further the LR is from 1, the steeper the line, and the further the answer travels.

It also makes the failure mode obvious. Start from a very low pre-test probability and even a spectacular likelihood ratio barely lifts the line off the floor. This is the same fact as a low PPV in screening, seen sideways: no test rescues a prior that was never there.

What the numbers mean

LR+ above 10, or LR− below 0.1: decisive in most settings. 5 to 10, or 0.1 to 0.2: moderate. 2 to 5, or 0.2 to 0.5: weak, and rarely enough to change a decision on its own. An LR of 1 is worthless — the result is equally common in the ill and the healthy, and the line comes out flat.

A worked case: D-dimer, and why a bad PPV can be a good test

A patient has a Wells-based pre-test probability of pulmonary embolism of 15%. The D-dimer assay has a sensitivity of 97% and a specificity of only 40%, so its negative likelihood ratio is $(1 - 0.97) / 0.40 = 0.075$. The test comes back negative. Put 15% on the left axis and 0.075 in the middle, and the line lands on 1.3% — below the threshold at which imaging and anticoagulation are justified.

A specificity of 40% is indefensible in a test meant to confirm a diagnosis, and the PPV of a positive D-dimer is about 22%. None of that matters here, because ruling out is not what a PPV measures. Judge a test against the decision it is meant to support, and the instrument for a rule-out decision is the negative likelihood ratio.