Concept Architecture
Concept
Theoretically, Positive Likelihood Ratio (LR?) is a diagnostic accuracy measure that quantifies how much more likely a positive test result is among individuals with a target condition than among individuals without it. It represents the evidential value of a positive test result for confirming the presence of disease. The concept is founded on conditional probability and Bayesian inference and exists to assess the rule-in capability of diagnostic tests.
Mathematically, the Positive Likelihood Ratio is represented as the ratio of the true-positive rate to the false-positive rate. It combines sensitivity and specificity into a prevalence-independent measure that updates pre-test odds to post-test odds following a positive test result. Larger values indicate greater diagnostic evidence supporting the presence of disease, whereas a value of 1 indicates that the test provides no diagnostic information.
In practice, Positive Likelihood Ratio is estimated from diagnostic accuracy studies by comparing an index test with an appropriate reference standard. It is widely applied in clinical diagnostics, epidemiology and health economic evaluation to assess diagnostic performance, estimate post-test probabilities and evaluate the value of diagnostic technologies.
Purpose
Used to quantify the diagnostic value of a positive test result, assess rule-in performance, update disease probability following testing and support clinical and health economic decision-making.
Mathematical Formulae
Primary Formula
LR? = Sensitivity / (1 ? Specificity)
where:
- LR? = positive likelihood ratio
Supporting Formulae
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:
- TP = true positives
- FN = false negatives
- TN = true negatives
- FP = false positives
Related Mathematical Methods
- Likelihood Ratio
- Negative Likelihood Ratio
- Bayes' Theorem
- Sensitivity
- Specificity
- Pre-Test Probability
- Post-Test Probability
- Diagnostic Odds Ratio
Example
A diagnostic test for pulmonary embolism has a sensitivity of 0.90 and a specificity of 0.80.
Positive Likelihood Ratio:
LR? = 0.90 / (1 ? 0.80)
LR? = 0.90 / 0.20 = 4.50
The patient's pre-test probability of pulmonary embolism is 25%.
Pre-test Odds = 0.25 / (1 ? 0.25) = 0.333
Post-test Odds = 0.333 ? 4.50 = 1.50
Post-test Probability = 1.50 / (1 + 1.50) = 0.60
A positive test result increases the estimated probability of pulmonary embolism from 25% to 60%.
Excel Implementation
| Function | Example Formula | Health Economics Application |
|---|---|---|
| Division | =B2/(1-C2) | Calculates the Positive 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 Positive Likelihood Ratio. |
| Division | =G2/(1+G2) | Converts post-test odds into post-test probability for diagnostic decision modelling. |
| IF | =IF(H2>0.80,"High probability","Further assessment") | Applies a decision threshold based on post-test probability. |
VBA (Optional)
A VBA macro can automatically calculate Positive Likelihood Ratios, update post-test probabilities and compare the diagnostic performance of multiple testing 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 positive likelihood ratio?
A diagnostic test measure expressing how much a positive result increases the odds that a patient truly has the condition tested for.
Source: Sackett et al. 1991
What does a positive likelihood ratio tell us about confirming disease?
A positive likelihood ratio expresses how much a positive test result raises the odds that a patient truly has the condition, and so how well the test confirms disease. A high value, well above one, means a positive result makes the disease much more likely, so a strongly positive test can effectively rule it in. Applied to the patient's starting odds, it yields their revised odds after the positive result. Gauging how convincingly a positive test confirms disease is its role. Sackett and colleagues (1991) describe this measure.
Source: Sackett et al. 1991
How is a positive likelihood ratio calculated?
A positive likelihood ratio is calculated as the sensitivity divided by one minus the specificity, comparing the probability of a positive result in those with the condition, the true positive rate, with the probability of a positive result in those without it, the false positive rate. So a positive likelihood ratio is calculated from the test's sensitivity and specificity as the true positive rate over the false positive rate, which gives, for a positive result, the ratio of its probability in the diseased to its probability in the non-diseased, summarising how much a positive result shifts the odds of disease upward, independently of prevalence.
Source: Altman & Bland 1994
How is a positive likelihood ratio interpreted?
A positive likelihood ratio is interpreted by how far above one it lies: large values indicate that a positive result strongly raises the probability of disease and thus rules it in effectively, while values nearer one indicate that a positive result changes the probability little. So a positive likelihood ratio is interpreted as the strength with which a positive test result increases the odds of the condition, with larger values being more useful for ruling in disease, which helps judge whether a positive result meaningfully raises the probability of the condition or leaves it much as it was before the test.
Source: Sackett et al. 1991
How is a positive likelihood ratio used?
A positive likelihood ratio is used to update a patient's probability of disease after a positive test result, by combining the pre-test odds with the positive likelihood ratio to obtain higher post-test odds, which convert to an increased post-test probability. A large positive likelihood ratio produces a large increase. So a positive likelihood ratio is used in diagnostic reasoning to quantify how much a positive result should raise 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 confirm, or rule in, a condition.
Source: Sackett et al. 1991
How does a positive likelihood ratio relate to a negative likelihood ratio?
A positive likelihood ratio relates to a negative 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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British health economist
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Verification date: 9 Dec 2025
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