Concept Architecture
Concept
Theoretically, Negative Likelihood Ratio (LR?) is a diagnostic accuracy measure that quantifies how much less likely a negative test result is among individuals with a target condition than among individuals without it. It represents the evidential effect of a negative test result on the probability that the condition is present. The concept is founded on conditional probability and Bayesian inference and exists to assess the ability of a diagnostic test to rule out a condition.
Mathematically, the Negative Likelihood Ratio is represented as the false-negative proportion divided by the true-negative proportion. It combines sensitivity and specificity into a prevalence-independent relative measure and is used to transform pre-test odds into post-test odds following a negative result. Values close to zero provide stronger evidence against the presence of the condition, whereas a value of 1 indicates that the negative result provides no diagnostic information.
In practice, the Negative Likelihood Ratio is estimated from diagnostic accuracy studies by comparing an index test with a valid reference standard. It is applied to update disease probability after a negative result, evaluate rule-out performance and model the effects of diagnostic strategies on treatment decisions, resource use, costs and health outcomes.
Purpose
Used to quantify the extent to which a negative test result reduces the probability of a target condition, assess diagnostic rule-out performance and support clinical and health economic decision-making.
Mathematical Formulae
Primary Formula
LR? = (1 ? Sensitivity) / Specificity
Supporting Formulae
Sensitivity = TP / (TP + FN)
Specificity = TN / (TN + FP)
LR? = [FN / (TP + FN)] / [TN / (TN + FP)]
Pre-test Odds = Pre-test Probability / (1 ? Pre-test Probability)
Post-test Odds = Pre-test Odds ? LR?
Post-test Probability = Post-test Odds / (1 + Post-test Odds)
where:
- LR? = negative likelihood ratio
- TP = true positives
- FN = false negatives
- TN = true negatives
- FP = false positives
Related Mathematical Methods
- Likelihood Ratio
- Positive Likelihood Ratio
- Sensitivity
- Specificity
- Bayes' Theorem
- Prior Odds
- Posterior Odds
- Pre-Test Probability
- Post-Test Probability
- Diagnostic Odds Ratio
Example
A diagnostic test for deep-vein thrombosis has a sensitivity of 0.95 and a specificity of 0.80.
Negative Likelihood Ratio:
LR? = (1 ? 0.95) / 0.80 = 0.0625
The patient's pre-test probability of deep-vein thrombosis is 30%.
Pre-test Odds = 0.30 / (1 ? 0.30) = 0.4286
Post-test Odds = 0.4286 ? 0.0625 = 0.0268
Post-test Probability = 0.0268 / (1 + 0.0268) = 0.0261
A negative test result reduces the estimated probability of deep-vein thrombosis from 30% to approximately 2.6%.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| Division | =(1-B2)/C2 | Calculates the Negative Likelihood Ratio from sensitivity and specificity. |
| Division | =D2/(1-D2) | Converts pre-test probability into pre-test odds. |
| Multiplication | =E2*F2 | Updates pre-test odds following a negative test result. |
| Division | =G2/(1+G2) | Converts post-test odds into post-test probability for diagnostic decision modelling. |
| IF | =IF(H2<0.05,"Condition unlikely","Further assessment") | Applies a specified post-test probability threshold within a diagnostic pathway. |
VBA (Optional)
A VBA macro can automate Negative Likelihood Ratio calculations and update post-test probabilities across multiple diagnostic tests, patient subgroups or model scenarios.
Sources
- Deeks JJ, Altman DG. Diagnostic tests 4: likelihood ratios. BMJ. 2004;329:168?169.
- McGee S. Simplifying likelihood ratios. Journal of General Internal Medicine. 2002;17(8):646?649.
- Zhou XH, Obuchowski NA, McClish DK. Statistical Methods in Diagnostic Medicine. 2nd ed.
- Bossuyt PM, Reitsma JB, Bruns DE, et al. STARD 2015: an updated list of essential items for reporting diagnostic accuracy studies. BMJ. 2015;351:h5527.
- Briggs A, Claxton K, Sculpher M. Decision Modelling for Health Economic Evaluation.
Related Concepts (2)
Library
Publications
1
Statistical Analysis of Cost-Effectiveness Data — Willan & Briggs, 1st Edition ed., 2006 (John Wiley & Sons)
A synthesis of statistical methods for analysing cost-effectiveness data, including net-benefit regression, confidence intervals for the ICER, cost-effectiveness acceptability curves, and covariate adjustment. Part of the Wiley Statistics in Practice series.
BookView source →
Frequently Asked Questions (6)
What is a negative likelihood ratio?
A diagnostic test measure expressing how much a negative result changes the odds a patient does not have the condition tested for.
Source: Sackett et al. 1991
What does a negative likelihood ratio tell us about ruling out disease?
A negative likelihood ratio expresses how much a negative test result changes the odds that a patient does not have the condition, and by extension how well the test rules disease out. A low value, well below one, means a negative result makes the disease much less likely, so a strongly negative test can effectively exclude it. Applied to the patient's starting odds, it yields their revised odds after the negative result. Gauging how convincingly a negative test excludes disease is its role. Sackett and colleagues (1991) describe this measure.
Source: Sackett et al. 1991
How is a negative likelihood ratio calculated?
A negative likelihood ratio is calculated as one minus the sensitivity divided by the specificity, comparing the probability of a negative result in those with the condition, the false negative rate, with the probability of a negative result in those without it, the true negative rate. So a negative likelihood ratio is calculated from the test's sensitivity and specificity as the false negative rate over the true negative rate, which gives, for a negative result, the ratio of its probability in the diseased to its probability in the non-diseased, summarising how much a negative result shifts the odds of disease downward, independently of prevalence.
Source: Altman & Bland 1994
How is a negative likelihood ratio interpreted?
A negative likelihood ratio is interpreted by how far below one it lies: values close to zero indicate that a negative result strongly lowers the probability of disease and thus rules it out effectively, while values nearer one indicate that a negative result changes the probability little. So a negative likelihood ratio is interpreted as the strength with which a negative test result reduces the odds of the condition, with smaller values being more useful for ruling out disease, which helps judge whether a negative result meaningfully lowers the probability of the condition or leaves it largely unchanged.
Source: Sackett et al. 1991
How is a negative likelihood ratio used?
A negative likelihood ratio is used to update a patient's probability of disease after a negative test result, by combining the pre-test odds with the negative likelihood ratio to obtain lower post-test odds, which convert to a reduced post-test probability. A small negative likelihood ratio produces a large reduction. So a negative likelihood ratio is used in diagnostic reasoning to quantify how much a negative result should lower the estimated probability of disease, allowing the test's evidence to be applied to a patient's starting probability, and it is particularly relevant when a test is used to rule out a condition.
Source: Sackett et al. 1991
How does a negative likelihood ratio relate to a positive likelihood ratio?
A negative likelihood ratio relates to a positive likelihood ratio as the counterpart for the opposite result: the positive likelihood ratio quantifies how much a positive result raises the odds of disease, while the negative likelihood ratio quantifies how much a negative result lowers them. Both derive from the test's sensitivity and specificity and are prevalence-independent. So the two likelihood ratios together describe a test's performance for its two results, the positive one for ruling in disease and the negative one for ruling out, and each is used to update the probability of disease in the appropriate direction after the corresponding result.
Source: Sackett et al. 1991
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Verified by Dr Darrin Baines
British health economist
Professional identity: darrinbaines.org
Verification date: 9 Dec 2025
Content version: 1.0.0
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