Negative Likelihood Ratio (LR−)

The negative likelihood ratio is $(1-\mathrm{Se})/\mathrm{Sp}$: the false-negative rate over the true-negative rate. It is the multiplier for the result the test gives most of the time, and because it is normally below 1 it pulls the odds down. The smaller it is, the harder a negative result rules out.

The calibration runs the other way from LR+ and the reciprocals are worth memorising: below 0.1 is a large shift, 0.1 to 0.2 moderate, 0.2 to 0.5 small, and above 0.5 close to useless. An LR− of 0.1 divides the odds by ten. The pair of a 90 % sensitive and 95 % specific test gives $0.10/0.95 = 0.105$, so a negative result cuts the odds by roughly a factor of ten — meaningful, and nothing like proof.

Notice the asymmetry the algebra forces. LR− has sensitivity in its numerator and specificity in its denominator, so it is driven overwhelmingly by sensitivity: pushing specificity from 90 % to 99 % barely moves it, while pushing sensitivity from 90 % to 99 % moves it by a factor of ten. This is the arithmetic behind SnNout, and it is why a rule-out test is chosen for sensitivity and a rule-in test for specificity. Neither aid is a heuristic; both are readings of these two fractions.

What the ratio cannot do is rescue a bad starting point. A negative result with an LR− of 0.1 on someone whose pre-test odds were 9 to 1 leaves post-test odds of 0.9 to 1 — still nearly a coin toss, from a strongly negative result on a good test. A negative result is not an all-clear; it is a division. Where the answer lands depends on where the odds were when the division was applied, and that is the whole content of the next page.