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AKT Statistics Guide · Part 3 · Chapter 10

Likelihood Ratios & the Fagan Nomogram

One number that turns a test result into a change in probability — the “shove” that moves your patient from a pre-test hunch to a post-test answer.

▶Watch the tutorial · 6 min — LR+ and LR− worked through, plus the Fagan nomogram rule of thumb.

Where this shows up in the AKT

Judging how useful a test really is, in a single number.
Combining a test result with your clinical suspicion — the pre-test probability.
One figure that captures both sensitivity and specificity — and, like them, is independent of prevalence.

A likelihood ratio (LR) is the shove that moves you from pre-test probability to post-test probability. LR+ pushes you towards the diagnosis (how much a positive result raises the probability of disease); LR− pushes you away from it (how much a negative lowers it).

Fagan nomogram: draw a line from the pre-test probability, through the likelihood ratio, to read off the post-test probability (here 20% → LR+ 10 → ≈70%).

Two ratios, one idea

The formulas (for understanding, not memorising)

LR+ = sensitivity / (1 − specificity) — how much more likely a positive is in someone with the disease than without it.
LR− = (1 − sensitivity) / specificity — how much a negative result shifts you away from disease.

The anchor point: an LR of 1 means the result tells you nothing at all. The further from 1, the more the result should change your mind.

The golden rules

How big is big enough? The numbers worth memorising.
Value	Interpretation
LR+ >10	Strong rule-in
LR+ 5–10	Moderate rule-in
LR+ 2–5	Weak rule-in
LR ≈ 1	Useless — the test barely helps
LR− 0.2–0.5	Weak rule-out
LR− 0.1–0.2	Moderate rule-out
LR− <0.1	Strong rule-out

In short: LR+ >10 rules in; LR− <0.1 rules out; anything close to 1 barely moves the needle.

Worked example — D-dimer for PE

D-dimer has sensitivity 95% and specificity 50%. Two quick sums capture the whole test:

LR+ = 0.95 / (1 − 0.50) = 0.95 / 0.50 = 1.9 — weak; almost useless for ruling PE in.
LR− = (1 − 0.95) / 0.50 = 0.05 / 0.50 = 0.1 — strong; excellent for ruling PE out.

This is exactly why D-dimer is used as a rule-out test when the pre-test probability is low: a negative is genuinely reassuring, while a positive still needs a more specific test (imaging) to confirm.

Pre-test to post-test: the Fagan nomogram

The Fagan nomogram is a graphical tool that combines three things to give a clinical answer:

Pre-test probability — your clinical suspicion before the test, often from prevalence or a clinical score such as the Wells score.
Likelihood ratio — from the test.
Post-test probability — the answer you read off.

Draw a straight line from your pre-test probability, through the LR, and read off the post-test probability on the far side. LRs are independent of prevalence, but the post-test probability is not — it depends entirely on where you started. The same positive result means very different things in a 70-year-old smoker with chest pain (high pre-test probability) versus a 25-year-old with the same symptom (low pre-test probability). A strong test on a very unlikely diagnosis can still leave it unlikely.

Rule of thumb (no nomogram needed)

LR+ of 2 adds about 15 percentage points to the probability of disease.
LR+ of 5 adds about 30 percentage points.
LR+ of 10 adds about 45 percentage points.
LR− of 0.5 subtracts about 15 points; LR− of 0.1 subtracts about 45 points.

The headline: a likelihood ratio ties the test to the patient in front of you. The LR is fixed, but where it lands you depends on the pre-test probability you brought to it.

The traps that cost marks

Trap 1 — treating an LR near 1 as meaningful. A ratio close to 1 carries almost no information; a positive (or negative) result hardly shifts the probability.

Trap 2 — ignoring the pre-test probability. A strong ratio applied to a very unlikely diagnosis can still leave it unlikely. The LR is only half the calculation.

Trap 3 — confusing LR+ with PPV. LR+ tells you how much a positive result shifts probability; PPV tells you how likely the patient actually has the disease (and PPV depends on prevalence, LR+ does not).

AKT-style question

A test for pulmonary embolism has a sensitivity of 95% and a specificity of 50%. What is the most appropriate interpretation of its likelihood ratios?

LR+ is high, so the test is good for ruling PE in
LR− is high, so the test is poor for ruling PE out
LR+ is low, so a positive result does not increase the probability of PE much
LR− is high, so a negative result increases the probability of PE

Reveal answer

C — LR+ is low. With sensitivity 95% and specificity 50%, LR+ = 0.95 / 0.50 ≈ 1.9 — too low to rule disease in, so a positive barely raises the probability of PE. The LR− is around 0.1, which is strong for ruling out. This is why D-dimer is an excellent rule-out tool but a poor rule-in: a negative reassures, a positive needs imaging to confirm.

Lock it in

LR+ = sensitivity / (1 − specificity); LR− = (1 − sensitivity) / specificity.
LR+ >10 rules in; LR− <0.1 rules out; near 1 is useless.
Built from sensitivity and specificity, so LRs are properties of the test — independent of prevalence.
Combine the LR with your pre-test probability (Fagan nomogram, or the 2/5/10 → +15/30/45 rule) to reach a post-test probability.

Source concepts: standard diagnostic-test methodology (e.g. Centre for Evidence-Based Medicine likelihood-ratio tables; Fagan TJ, NEJM 1975). A revision summary, not a clinical protocol.

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