Concept Architecture
Concept
Theoretically, Likelihood Ratio (LR) is a comparative measure that quantifies how strongly an observed result supports one hypothesis relative to another. In diagnostic testing, it represents how much more likely a test result is among individuals with the target condition than among individuals without it. The concept is founded on conditional probability and Bayesian inference and exists to convert prior odds into posterior odds after evidence is observed.
Mathematically, the Likelihood Ratio is represented as the probability of the observed evidence under one hypothesis divided by its probability under an alternative hypothesis. For binary diagnostic tests, the positive likelihood ratio combines sensitivity and specificity for a positive result, whereas the negative likelihood ratio combines them for a negative result. Values greater than 1 increase support for the condition, values below 1 decrease support, and a value of 1 provides no discriminatory information.
In practice, Likelihood Ratios are estimated from diagnostic accuracy studies using cross-tabulated test results against a valid reference standard. They are applied to update pre-test odds to post-test odds, assess diagnostic performance and estimate how test information may alter clinical pathways, resource use and health economic outcomes.
Purpose
Used to quantify the evidential value of an observed result, update disease probability following diagnostic testing, compare diagnostic performance and support clinical and health economic decision-making under uncertainty.
Mathematical Formulae
Primary Formula
LR = P(E�H?) / P(E�H?)
where:
- LR = likelihood ratio
- E = observed evidence
- H? = hypothesis of interest
- H? = alternative or null hypothesis
Supporting Formulae
LR? = Sensitivity / (1 ? Specificity)
LR? = (1 ? Sensitivity) / Specificity
Sensitivity = TP / (TP + FN)
Specificity = 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? = positive likelihood ratio
- LR? = negative likelihood ratio
- TP = true positives
- FN = false negatives
- TN = true negatives
- FP = false positives
Related Mathematical Methods
- Bayes' Theorem
- Conditional Probability
- Prior Odds
- Posterior Odds
- Pre-Test Probability
- Post-Test Probability
- Sensitivity
- Specificity
- Diagnostic Odds Ratio
Example
A diagnostic test for a cardiovascular condition has a sensitivity of 0.90 and a specificity of 0.80.
Positive likelihood ratio:
LR? = 0.90 / (1 ? 0.80) = 4.50
The patient's pre-test probability is 20%.
Pre-test Odds = 0.20 / (1 ? 0.20) = 0.25
Post-test Odds = 0.25 ? 4.50 = 1.125
Post-test Probability = 1.125 / (1 + 1.125) = 0.529
A positive test result increases the estimated probability of the condition from 20% to approximately 52.9%.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| Division | =B2/(1-C2) | Calculates the positive likelihood ratio from sensitivity and specificity. |
| 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 using the relevant likelihood ratio. |
| Division | =G2/(1+G2) | Converts post-test odds into post-test probability for diagnostic decision modelling. |
VBA (Optional)
A VBA macro can automate the calculation of positive and negative likelihood ratios and update disease probabilities across multiple diagnostic strategies.
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 likelihood ratio?
A diagnostic test measure expressing how much a given result, positive or negative, changes the odds that a patient truly has the condition.
Source: Sackett et al. 1991
How does a likelihood ratio change the odds a patient has a condition?
A likelihood ratio for a diagnostic test says how much a particular result, positive or negative, shifts the odds that a patient truly has the condition. A high positive likelihood ratio means a positive result makes the disease much more likely, while a low negative ratio means a negative result largely rules it out. Applied to a patient's pre-test odds, it yields the revised post-test odds, translating a test result into an updated probability of disease. Moving the odds with a test result is its function. Sackett and colleagues (1991) describe this measure.
Source: Sackett et al. 1991
How is a likelihood ratio used?
A likelihood ratio is used to update a patient's probability of having a condition after a test result, by combining the pre-test odds with the likelihood ratio to give the post-test odds, which can be converted back to a probability. A high positive likelihood ratio raises the probability of disease, and a low negative likelihood ratio lowers it. So a likelihood ratio is used in diagnostic reasoning to revise the estimated probability of disease in light of a test result, formalising how the result should change belief about the diagnosis, which allows a test's evidence to be applied to individual patients with different starting probabilities.
Source: Sackett et al. 1991
How is a likelihood ratio calculated?
A likelihood ratio is calculated from the test's sensitivity and specificity. The positive likelihood ratio is the sensitivity divided by one minus the specificity, comparing the chance of a positive result in those with and without the condition; the negative likelihood ratio is one minus the sensitivity divided by the specificity. So a likelihood ratio is calculated from how the test performs in those with and without the disease, giving, for each type of result, the ratio of its probability in the diseased to its probability in the non-diseased, which summarises the result's diagnostic strength independently of the disease prevalence.
Source: Altman & Bland 1994
How is a likelihood ratio interpreted?
A likelihood ratio is interpreted by its size and direction: a value above one raises the probability of disease, with larger values providing stronger evidence for the condition, while a value below one lowers it, with values closer to zero providing stronger evidence against. A likelihood ratio of one leaves the probability unchanged. So a likelihood ratio is interpreted as how much and in which direction a test result shifts the odds of disease, with values far from one being more informative, which helps judge whether a test result meaningfully changes the diagnosis or leaves the probability of disease essentially as it was before.
Source: Sackett et al. 1991
Why are likelihood ratios useful?
Likelihood ratios are useful because they summarise a test result's diagnostic value in a single number that does not depend on disease prevalence, and they can be applied directly to update any patient's pre-test probability into a post-test probability. Unlike predictive values, they are stable across settings with different prevalences. So likelihood ratios are useful for interpreting test results in individual patients and across settings, providing a prevalence-independent measure of how much a result changes the probability of disease, which makes them a practical tool for evidence-based diagnostic reasoning where the starting probability varies from patient to patient.
Source: Sackett et al. 1991
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Verified by Dr Darrin Baines
British health economist
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Verification date: 9 Dec 2025
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