What a likelihood ratio is, and why clinicians prefer it
A likelihood ratio compares how often a particular test result occurs in people with the condition against how often it occurs in people without it. LR+ = Se ÷ (1 − Sp) is that ratio for a positive result; LR− = (1 − Se) ÷ Sp is the ratio for a negative one. An LR+ of 16 means a positive result is sixteen times more likely to come from someone with the condition than from someone without it.
The reason this framing matters is that likelihood ratios are prevalence-free, like sensitivity and specificity, but unlike sensitivity and specificity they are directly actionable. Sensitivity tells you about a group of patients you already know are ill, which is never the situation you are in. LR+ tells you what to do with the result in front of you: multiply your current odds by it.
They also condense two numbers into one per result, which makes tests genuinely comparable. Comparing a test with 95% sensitivity and 70% specificity against one with 80% and 92% is an argument; comparing LR+ of 3.2 against LR+ of 10.0 is an observation. And unlike predictive values, a likelihood ratio published in one country's clinic can be applied unchanged in yours — only the pre-test probability changes.
The underlying sensitivity and specificity, with their confidence intervals, come from the sensitivity and specificity calculator; this page takes them the next step.
Why the arithmetic runs on odds
Bayes' theorem is unwieldy in probabilities and trivial in odds. In odds form it reads: post-test odds = pre-test odds × likelihood ratio. That is the entire theorem. Every complication in the probability form — the denominator with four terms, the sensitivity to prevalence — is an artefact of the units.
So the procedure is three steps. Convert your pre-test probability to odds with O = p ÷ (1 − p). Multiply by LR+ if the result was positive, or by LR− if it was negative. Convert back with p = O ÷ (1 + O). A 20% pre-test probability is odds of 0.25; a positive result on a test with LR+ = 16 gives odds of 4.0; converting back gives 80%.
Two consequences follow directly from the multiplication, and both are worth internalising. First, a likelihood ratio of exactly 1 changes nothing, because multiplying odds by 1 leaves them alone. That is the definition of a useless result, and it happens whenever Se + Sp = 1. Second, the same likelihood ratio moves probability by wildly different amounts depending on where you start. An LR+ of 10 takes a 10% pre-test probability to 53% but takes a 50% pre-test probability only to 91%. The odds always multiply by ten; the probability cannot, because it is bounded at 100%. The reference table below makes that concrete.
Thomas Fagan published a nomogram for this in 1975: a three-column chart where a straight edge laid across your pre-test probability and the likelihood ratio reads off the post-test probability. This calculator does the same arithmetic without the ruler, and the chart it draws is the same relationship plotted continuously.
Worked example: a test with 80% sensitivity and 95% specificity
A patient has a clinical picture that puts the probability of the condition at roughly 20% before testing. The test in question has 80% sensitivity and 95% specificity.
- LR+. 0.80 ÷ (1 − 0.95) = 0.80 ÷ 0.05 = 16.0.
- LR−. (1 − 0.80) ÷ 0.95 = 0.20 ÷ 0.95 = 0.2105.
- Pre-test odds. 0.20 ÷ 0.80 = 0.25.
- If the result is positive. Post-test odds = 0.25 × 16.0 = 4.00. Probability = 4.00 ÷ 5.00 = 80.0%.
- If the result is negative. Post-test odds = 0.25 × 0.2105 = 0.05263. Probability = 0.05263 ÷ 1.05263 = 5.0%.
- Diagnostic odds ratio. 16.0 ÷ 0.2105 = 76.0, which is also (0.80 × 0.95) ÷ (0.20 × 0.05) = 0.76 ÷ 0.01.
Read the two answers as a pair. A positive result takes you from 20% to 80% — a fourfold rise in probability and a sixteenfold rise in odds. A negative result takes you from 20% to 5%, which is a useful but not decisive reduction: one in twenty of these patients still has the condition. Whether 5% is low enough to stop investigating is a clinical judgement about the consequences of a miss, not a statistical one. That threshold — the probability below which you stop testing — is the number the likelihood ratio framework was designed to help you argue about explicitly.
How big does a likelihood ratio need to be?
The conventional bands come from the JAMA Users' Guides to the Medical Literature. For LR+, values above 10 produce large shifts, 5 to 10 moderate, 2 to 5 small, and 1 to 2 minimal. For LR−, below 0.1 is large, 0.1 to 0.2 moderate, 0.2 to 0.5 small, and 0.5 to 1 minimal. These are conventions for describing the multiplier, not thresholds for clinical action.
Whether a shift is clinically useful depends on where it lands you relative to your treatment threshold. A test with LR+ of 3 is worthless in a patient at 2% pre-test probability, because the result lands at 5.8% and you were never going to treat at either number. The same test is decisive in a patient at 40%, taking them to 67%. The value of a test depends on the pre-test probability, and a test ordered at the wrong pre-test probability wastes money whatever its likelihood ratios.
The diagnostic odds ratio — LR+ divided by LR−, equivalently (Se × Sp) ÷ ((1 − Se) × (1 − Sp)) — compresses the two into a single figure that is useful for comparing tests and for meta-analysis. Its weakness is that it is blind to the trade-off: a test with LR+ 20 and LR− 0.5 and a test with LR+ 2 and LR− 0.05 both score 40, yet the first is a rule-in test and the second a rule-out test. Use the DOR for ranking, never for choosing.
Post-test probability for a given likelihood ratio
| Likelihood ratio | Pre-test 10% | Pre-test 25% | Pre-test 50% |
|---|---|---|---|
| 0.1 | 1.10% | 3.23% | 9.09% |
| 0.2 | 2.17% | 6.25% | 16.67% |
| 0.5 | 5.26% | 14.29% | 33.33% |
| 1 | 10.00% | 25.00% | 50.00% |
| 2 | 18.18% | 40.00% | 66.67% |
| 5 | 35.71% | 62.50% | 83.33% |
| 10 | 52.63% | 76.92% | 90.91% |
| 20 | 68.97% | 86.96% | 95.24% |
The row for LR = 1 reproduces the pre-test probability exactly, which is the definition of an uninformative result.
Pitfalls and assumptions
- Multiplying probabilities instead of odds. The most common error. A 20% probability times an LR of 16 is not 320%; convert to odds first.
- Chaining likelihood ratios from correlated tests. Multiplying LRs from two tests assumes they are conditionally independent given disease status. Two tests measuring the same biology are not, and the chained result will overstate certainty.
- Collapsing a continuous result to positive or negative. A strongly abnormal value carries far more information than a marginal one. Stratum-specific likelihood ratios by result band preserve that; a single cut-off throws it away.
- Using a population prevalence as a pre-test probability for a symptomatic patient. Symptoms have already raised the probability. Starting from the population rate systematically understates the post-test probability.
- Taking a specificity of exactly 100% at face value. It makes LR+ infinite and almost always reflects a control group too small to contain a false positive.
- Ignoring the confidence intervals on Se and Sp. A likelihood ratio inherits both, and LR+ is especially unstable because 1 − Sp appears in the denominator.
- Assuming the LR transfers between settings. It transfers far better than a predictive value, but not perfectly: spectrum differences change sensitivity, and referral filters change specificity.
Where likelihood ratios sit among the other measures
All the diagnostic measures are views of the same 2×2 table. Sensitivity and specificity read down its columns and describe the test. PPV and NPV read across its rows and describe the patient, but only for the prevalence in the study that produced them. Likelihood ratios are the bridge: they carry the column information in a form you can apply across any row.
In fact the post-test probability after a positive result that this page reports is numerically identical to the PPV that the predictive-value calculator gives for the same sensitivity, specificity and prevalence. The two pages differ only in which route through Bayes' theorem they show you. Use whichever framing makes the decision clearer.
Outside diagnostics, the same cross-product that gives the diagnostic odds ratio measures association between an exposure and an outcome — see the odds ratio calculator — and its cohort-study counterpart is the risk ratio from the relative risk calculator. Once a diagnosis has been made and treatment is being weighed, the number needed to treat calculator converts trial event rates into the number of patients you must treat to prevent one event, which is the natural next question after “how sure am I?”.
Key terms
- Odds
- p ÷ (1 − p). A probability of 0.20 is odds of 0.25; a probability of 0.80 is odds of 4.
- Pre-test probability
- Your belief that the condition is present before the result arrives. A prevalence in screening, a clinical estimate at the bedside.
- Fagan nomogram
- A three-scale chart published in 1975 that solves post-test odds = pre-test odds × LR graphically with a straight edge.
- Diagnostic odds ratio
- LR+ ÷ LR−. A single summary of test discrimination; 1 means no information, and it does not distinguish rule-in from rule-out tests.
- Stratum-specific likelihood ratio
- A separate LR for each band of a continuous result, which preserves the information that a single cut-off discards.