Likelihood ratios and the Fagan nomogram

Written and reviewed by a practicing emergency physician · Updated July 2026

A positive result carries more weight coming from some tests than from others, and the likelihood ratio is the number that says exactly how much. It asks one blunt question about a result: how many times more often does it show up in people who have the condition than in people who don't? A reactive 4th-generation HIV test is roughly 200 times more common in someone who has HIV than in someone who doesn't, so it should move your estimate a long way. A positive low-dose CT for lung cancer is only about four times more common in disease than in health, so it should move your estimate far less. Same kind of tool, very different pull — and the likelihood ratio is what lets you put both on one scale.

Every test yields two of these ratios. The positive likelihood ratio, LR+, grades a positive result; the negative likelihood ratio, LR−, grades a negative one. Both are built straight out of sensitivity and specificity:

• LR+ = sensitivity ÷ (1 − specificity) — the true-positive rate divided by the false-positive rate.
• LR− = (1 − sensitivity) ÷ specificity — the false-negative rate divided by the true-negative rate.

Read the numerator as "how often this result happens in the sick" and the denominator as "how often the same result happens in the healthy." A ratio of 1 means the result is equally common in both groups, so it tells you nothing. The further above 1 an LR+ climbs, the harder a positive pushes toward disease; the closer to 0 an LR− falls, the harder a negative pushes away from it.

Reading two real tests off their accuracy

Take the two tests above and turn the crank. The 4th-generation HIV assay runs about 99.8% sensitive and 99.5% specific:

• LR+ = 0.998 ÷ (1 − 0.995) = 0.998 ÷ 0.005 = 199.6 ≈ 200
• LR− = (1 − 0.998) ÷ 0.995 = 0.002 ÷ 0.995 ≈ 0.002

A reactive result is about 200 times more likely in someone with HIV; a non-reactive result is about 500 times more likely in someone without it (1 ÷ 0.002 ≈ 500). Either result, on its own, is close to decisive. (The confirmatory algorithm that follows a reactive screen is worked through in HIV testing, by the numbers.)

Low-dose CT is a different animal — about 93.1% sensitive but only 76.5% specific:

• LR+ = 0.931 ÷ (1 − 0.765) = 0.931 ÷ 0.235 = 3.96 ≈ 4
• LR− = (1 − 0.931) ÷ 0.765 = 0.069 ÷ 0.765 ≈ 0.09

A positive scan multiplies the odds of lung cancer by only about four; a negative scan divides them by about eleven (1 ÷ 0.09 ≈ 11). Notice the asymmetry — this test rules out better than it rules in. What separates it from the HIV assay is almost entirely specificity: 76.5% versus 99.5%. A 23.5% false-positive rate is a large denominator under LR+, and it drags the ratio down toward the unhelpful end.

Why prevalence barely moves the ratio

Nothing in LR+ or LR− mentions how common the disease is — the ratios are assembled entirely from sensitivity and specificity. That is what makes them portable: the same LR+ ≈ 200 applies whether HIV turns up in 1 person per 1,000 or 1 per 20. Predictive value behaves the opposite way. PPV and NPV fold prevalence in through Bayes' theorem, so the identical test reads very differently in a low-risk population than a high-risk one — the engine behind the base-rate fallacy. A likelihood ratio deliberately strips prevalence out and describes the test's discriminating power on its own.

"To first order" is the honest hedge. Sensitivity and specificity are not carved in stone; they shift with the mix of patients actually tested, an effect called spectrum bias. A test measured on advanced, unmistakable disease looks more sensitive than the same test used to catch early disease, and referral patterns can push specificity around too. So a likelihood ratio measured in one population is a well-travelled estimate, not a universal constant. Use published LRs freely, but hold them a little loosely.

Odds, and one turn of the crank

Likelihood ratios multiply odds, not probabilities, so the arithmetic has to run in odds. Converting is quick in both directions: odds = p ÷ (1 − p), and back again p = odds ÷ (1 + odds). A pre-test probability of 0.4% becomes pre-test odds of 0.004 ÷ 0.996 ≈ 0.00402 — roughly 1 in 249.

From there, Bayes' theorem collapses to a single multiplication:

pre-test odds × LR = post-test odds

Run a positive HIV screen from that 0.4% starting point: 0.00402 × 200 ≈ 0.80 post-test odds, which converts back to 0.80 ÷ 1.80 ≈ 0.44 — about a 44% chance. Even an LR+ near 200 lands only at a coin flip, because the starting point was so low; that is precisely why a first reactive HIV screen is confirmed with a second, different assay rather than acted on. Reassuringly, the same 44% drops out of a plain 2×2 count: among 100,000 people at 0.4% prevalence, the assay flags about 399 of the 400 true cases plus about 498 healthy people, so only 399 of roughly 897 positives — again about 44% — are real. The odds path and the natural-frequency path are one theorem in two costumes. The full step-by-step lives on pre-test to post-test probability, and the calculator animates the same update under Bayesian updating.

How strong is strong?

A rough convention, set out in the JAMA Users' Guides to the Medical Literature and later distilled by McGee, sorts the ratios into bands. An LR+ above 10 drives a large, often decisive rise in probability; 5 to 10 is moderate; 2 to 5 is small; anything between 1 and 2 barely registers. The negatives mirror it: an LR− below 0.1 is a strong rule-out, 0.1 to 0.2 is moderate, 0.2 to 0.5 is small, and near 1 is useless. By that ruler the HIV assay is off the scale at both ends (≈ 200 and ≈ 0.002), while low-dose CT sits in the "small" band for a positive (≈ 4) and only just crosses into strong rule-out territory for a negative (≈ 0.09).

McGee also offered a bedside shortcut that skips the odds algebra entirely: across the middle range of pre-test probability, likelihood ratios of 2, 5, and 10 add roughly 15, 30, and 45 percentage points, and ratios of 0.5, 0.2, and 0.1 subtract about the same. It drifts near the extremes — the HIV coin-flip is exactly where the shortcut breaks down — but for everyday numbers it is close enough to reason with out loud.

The Fagan nomogram

Thomas Fagan compressed that one multiplication into a picture in a one-paragraph 1975 letter to the New England Journal of Medicine. It is three parallel scales: pre-test probability on the left, likelihood ratio down the middle, post-test probability on the right. Lay a straightedge on your pre-test probability, pivot it through the test's likelihood ratio, and read the answer where the line meets the right-hand scale. The scales are logarithmic — that is the trick that quietly turns "multiply the odds" into "draw a straight line." One chart handles any test at any starting probability, no calculator required.

This tool draws that nomogram live: move the prevalence, sensitivity, or specificity sliders and the line and its endpoints follow. Open either scenario above and watch it in the test panel — the huge-LR HIV line swings nearly to the top on a positive, while the modest low-dose-CT line lifts only partway up.