Diagnostic tests II: likelihood ratios and pre- to post-test probability
Convert sensitivity and specificity into likelihood ratios, move from pre-test to post-test probability with odds, and read ROC curves — the Bayesian reasoning behind every referral decision.
Reviewed and kept current
Last editorial review: 28 September 2026 · Next scheduled refresh: March 2027.
Clinical vignette
Begin with the patient. Read the case, weigh the options, then work through the unit.
Two patients with a painful, swollen calf are seen on the same afternoon. Both have a D-dimer test with a sensitivity of 96% and a specificity of 45%. Both results are negative.
For Ms D you estimated a pre-test probability of DVT of 10%. For Mr E, whose clinical picture is more worrying, you estimated 50%.
Each asks the same question: “So I don’t have a clot?” The test result is identical. The pre-test probabilities are not.
The local pathway combines the D-dimer with a clinical probability score. You must decide what to tell each patient, and whether either needs imaging.
Options to weigh
- Tell both patients that a negative D-dimer rules out a clot.
- Calculate the negative likelihood ratio, convert each pre-test probability to odds, and work out each patient’s post-test probability.
- Reassure Ms D, and arrange imaging for Mr E because his post-test probability remains too high to rule out a clot.
- Tell Mr E that the negative result means no further tests are needed.
Reflect
How would the same negative result change what you tell Ms D and what you tell Mr E, and what would you be careful not to say to either?
What this unit covers
Pocket-sized and self-contained: three AKT-style questions, the answers explained, an expanded discussion with hyperlinked guidance, a decision algorithm and a certificate.
By the end of this unit you will be able to
Five steps, in order
Answer first, then learn. Each section unlocks as you progress, and your progress is saved in this browser.
1Three single best answer questions
Commit to an answer for all three single best answer questions, then submit. Sections 2–4 open on submission; your certificate opens at 2/3 or better.
Question 1 of 3
A clinical examination sign has a sensitivity of 80% and a specificity of 90% for a diagnosis.
What is the positive likelihood ratio of the sign?
Question 2 of 3
You estimate that a patient has a 20% probability of a diagnosis before testing. A test with a positive likelihood ratio of 8 is positive.
What is the approximate post-test probability?
Question 3 of 3
Five different tests are available for the same condition.
Which result would be most useful for ruling out the condition?
2Answers explained
Question 1
Correct answer D. 8
0.22 is the negative likelihood ratio — a deliberate distractor. An LR+ of 8 means a positive finding is 8 times more likely in someone with the disease than in someone without it.
Question 2
Correct answer C. 67%
You must convert to odds before multiplying. Multiplying the probability directly (20% × 8 = 160%) is impossible — that is the trap.
Question 3
Correct answer B. A negative result with LR− of 0.05
A negative likelihood ratio close to 0 produces the largest fall in probability. LR− of 0.05 cuts the pre-test odds twenty-fold. LR− of 0.5 barely changes anything, and an LR of 1 means the test result makes no difference at all.
| LR | Effect on probability |
|---|---|
| >10 or <0.1 | Large, often conclusive |
| 5–10 or 0.1–0.2 | Moderate |
| 2–5 or 0.2–0.5 | Small, sometimes important |
| 1–2 or 0.5–1 | Rarely important |
3Expanded discussion, worked examples & guidance
1. Probability and odds
For small probabilities (below about 10%), odds and probability are nearly equal — which is also why an odds ratio approximates a relative risk when the outcome is rare (Unit 5).
2. Likelihood ratios
Because they are built from sensitivity and specificity, likelihood ratios do not depend on prevalence. They let you personalise: your estimate of this patient’s pre-test probability, times the LR, gives this patient’s post-test probability.
Try it · live calculator
Pre-test to post-test probability
3. Worked examples
Worked example 1 — ruling out
Pre-test probability of a DVT is 10%. A D-dimer with sensitivity 96% and specificity 45% is negative. What is the post-test probability?
- LR− = 0.04 / 0.45 = 0.089.
- Pre-test odds = 0.1/0.9 = 0.111.
- Post-test odds = 0.111 × 0.089 = 0.0099.
- Probability = 0.0099/1.0099.
Answer. ≈1% — low enough to stop investigating in most pathways.
Worked example 2 — the same test in a high-risk patient
Pre-test probability 50%, same negative D-dimer.
- Pre-test odds = 1.
- Post-test odds = 0.089.
- Probability = 0.089/1.089.
Answer. ≈8% — too high to rule out; image. This is why pathways combine D-dimer with a clinical probability score.
Worked example 3 — two tests in sequence
Pre-test probability 30%; test 1 positive (LR+ 4), then test 2 positive (LR+ 3). Assume the tests are independent.
- Pre-test odds = 0.3/0.7 = 0.43.
- × 4 = 1.71; × 3 = 5.14.
- Probability = 5.14/6.14.
Answer. ≈84%. Serial LRs multiply only if the tests are conditionally independent — often not true for related clinical signs.
4. ROC curves
A receiver operating characteristic curve plots sensitivity (true-positive rate) on the y-axis against 1 − specificity (false-positive rate) on the x-axis for every possible cut-off. The closer the curve hugs the top-left corner, the better the test. The area under the curve (AUC), or c-statistic, is also used to describe how well risk scores such as QRISK3 discriminate: 0.5 is no better than chance and above 0.8 is good.
- Never multiply a probability by an LR — convert to odds first.
- An LR of 1 is useless whatever the sensitivity and specificity look like.
- The x-axis of a ROC curve is 1 − specificity, not specificity.
References and further reading
Guidance checked at the last editorial review (28 September 2026). Always use the current version.
4Decision algorithm
From pre-test to post-test probability
Start
Estimate the pre-test probability (prevalence in similar patients, clinical score, or gestalt).
Action
Convert to odds: odds = p / (1 − p).
Decision
Was the test result positive?
Action
Post-test odds = pre-test odds × LR.
Action
Convert back: p = odds / (1 + odds).
Decision
Is the post-test probability above your action (treat or refer) threshold?
Outcome
Document your reasoning, the residual risk and the safety-net advice.
Revision summary; apply clinical judgement and local pathways.
5Certificate of completion
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