Positive Likelihood Ratio

Positive Likelihood Ratio for Ruling In Disease and Acting on a Positive Result

The positive likelihood ratio (LR+) of a test compares the chance of a positive result in people who have the target condition with the chance of a positive result in people who do not, so it equals sensitivity divided by the false positive rate. A patient's odds of disease after a positive result are the pre-test odds times LR+, which means a large value lets a positive result go a long way towards confirming the condition, while a value of 1 leaves the odds where they were. In health economics LR+ matters because, with the pre-test probability, it fixes how many people without the condition are sent for treatment on the strength of a positive result, and so the harm and cost that a quicker route to treatment trades against the confirmatory tests it avoids. Specificity carries most of the weight in LR+, and published values are easily inflated by weak study designs. The same LR+ can justify treating on a positive result in a high-risk population and fail to do so in a low-risk one.

How LR+ comes out of the two-by-two table

Deeks and Altman define the likelihood ratio of any test result as the probability of that result in people with the disease divided by its probability in people without it. A ratio above 1 points towards the presence of disease, and the further it lies from 1 the stronger the evidence. When a test is reported as positive or negative, the ratio for the positive result is LR+.

$$LR^{+} = \frac{P(T^{+} \mid D^{+})}{P(T^{+} \mid D^{-})} = \frac{Se}{1 - Sp} = \frac{TP / (TP + FN)}{FP / (FP + TN)}$$

where $T^{+}$ is a positive test result, $D^{+}$ and $D^{-}$ denote presence and absence of the target condition, $Se$ is sensitivity, $Sp$ is specificity, and $TP$, $FN$, $FP$ and $TN$ are the true positive, false negative, false positive and true negative counts against the reference standard. The numerator is the true positive rate and the denominator the false positive rate.

Deeks and Altman regard ratios above 10 as strong evidence to rule a diagnosis in, in most circumstances. Their example is a smoking history of 40 or more pack years in the diagnosis of obstructive airway disease. That history was found in 42 of 148 people with the disease and 2 of 144 without it, so LR+ is (42/148) divided by (2/144), about 20.4. A prior probability of 0.1 has odds of 0.111, which become about 2.27 after the positive finding, a post-test probability of about 0.69.

The negative likelihood ratio page deals with the rule-out side, and the general likelihood ratio page covers results with more than two levels and the combination of several tests.

Updating the probability of disease with LR+

LR+ acts on odds, so the update needs a starting probability taken from outside the accuracy study. Following Bayes' theorem, the pre-test probability is converted to odds, multiplied by LR+ and converted back.

$$O_{pre} = \frac{p}{1 - p}, \quad O_{post} = O_{pre} \times LR^{+}, \quad p_{post} = \frac{O_{post}}{1 + O_{post}}$$

where $p$ is the probability of the condition before testing, $O_{pre}$ and $O_{post}$ are the odds before and after a positive result, and $p_{post}$ is the post-test probability given a positive result. Pewsner and colleagues note that this post-test probability equals the positive predictive value in the population concerned.

Their third heart sound example shows how much the starting point matters. In general practice the sign had an LR+ of 18 for heart failure, yet at a pre-test probability of 16% a positive finding raised the probability only to 77%, short of a confident diagnosis. In a population with a pre-test probability of 50%, such as a cardiology outpatient clinic, the same result would give 95%.

Why specificity carries most of the weight in LR+

The mnemonic SpPIn holds that a positive result on a highly specific test rules the diagnosis in. Pewsner and colleagues show that the power to rule in depends on sensitivity as well. The third heart sound had a specificity of 99% but a sensitivity of only 24%, and they describe its power to rule in as eroded by the low sensitivity.

The formula also shows why specificity dominates once it is high. Because the false positive rate, 1 minus specificity, sits in the denominator, small changes near a specificity of 1 move LR+ a long way. With sensitivity fixed at 0.80, LR+ is 16 at a specificity of 0.95 and 40 at 0.98, whereas raising sensitivity to 0.95 at a specificity of 0.95 lifts LR+ only to 19. The same effect makes estimates fragile. From the study counts (10 true positives among 41 with heart failure and 3 false positives among 218 without), the third heart sound's LR+ is about 17.7, but the rounded 24% and 99% would suggest 24. When a study finds no false positives at all, LR+ has no finite value, and Pewsner and colleagues report an LR+ of infinity, with a 95% confidence interval of 21 to infinity, for auscultatory percussion in pleural effusion.

Bias and precision in studies that claim to rule in

Pewsner and colleagues state that studies with methodological flaws tend to overestimate test accuracy, and several of the tests promoted as SpPIn rested on such studies. The auscultatory percussion result came from a diagnostic case-control design, open to spectrum bias. The CAGE questionnaire result (three or more positive answers, LR+ 206 with a 95% confidence interval of 29 to 1468) came from a study that applied the reference test only to a sample of CAGE-negative patients, which can introduce partial verification bias. A study of abdominojugular reflux in heart failure used a reference standard that included the reflux sign itself, a possible incorporation bias, and its LR+ of 6.0 had a 95% confidence interval of 1.3 to 29.

Transport between settings is a separate problem. Across 23 meta-analyses (416 studies), Leeflang and colleagues found a statistically significant link between prevalence and sensitivity or specificity in 8 of them (35%), with specificity tending to be lower at higher prevalence. Since the false positive rate is the denominator of LR+, a fall in specificity in a referral population lowers LR+ and weakens the rule-in. They point to mechanisms such as patient spectrum and referral filters, which change prevalence and accuracy together.

LR+ in diagnostic economic evaluation

NICE treats test accuracy as an intermediate measure (PMG36 section 4.6.12) and asks for linked evidence modelling when direct evidence on final outcomes is missing (PMG48 section 2.3.14). In a decision tree built on such linked evidence, LR+ governs the positive branch. Together with the pre-test probability it fixes the share of test-positive patients who do not have the condition, and each of them carries the cost and QALY loss of unnecessary treatment.

A common choice is whether to treat on a positive result or to confirm it first. Per positive patient, treating without confirmation is worthwhile when the expected false positive harm, $(1 - p_{post}) H_{fp}$, is smaller than the confirmatory cost saved, which gives the following condition.

$$p_{post} > p^{*} = 1 - \frac{C_{conf}}{H_{fp}}$$

where $C_{conf}$ is the cost of the confirmatory test avoided for each positive patient, $H_{fp}$ is the monetary value of treating a person who does not have the condition (treatment cost plus QALYs lost times the value placed on a QALY), and $p^{*}$ is the post-test probability at which the two strategies tie. The form assumes a perfect confirmatory test with no harms and no delay to treatment for true positives. Because $O_{post} = O_{pre} \times LR^{+}$, the tie can also be read as a pre-test probability.

$$O_{pre}^{*} = \frac{p^{*}}{(1 - p^{*}) \times LR^{+}}$$

where $O_{pre}^{*}$ is the pre-test odds above which treating on a positive result is preferred, so a larger LR+ widens the range of populations in which treatment without confirmation pays.

Because PMG36 (section 4.7.18) warns that sensitivity and specificity are usually correlated, LR+ in a probabilistic sensitivity analysis is better derived in each iteration from a jointly sampled sensitivity and specificity pair than given its own distribution.

Worked example of treating on a positive result in two populations

All numbers in this example are illustrative. A test with sensitivity 0.80 and specificity 0.95 is used in 10,000 patients with a suspected condition, and patients who test negative are managed the same way in both strategies. The comparator sends every positive patient for confirmatory biopsy costing GBP 400, assumed perfect, before treatment. The alternative treats every positive patient without confirmation. Treating a person without the condition costs GBP 2,000 and loses 0.05 QALYs, valued at GBP 25,000 per QALY, the lower of the two values PMG36 (section 6.3.2) uses when presenting net health benefit. So $H_{fp}$ is GBP 2,000 plus GBP 1,250, or GBP 3,250.

1. Compute LR+ and the threshold. LR+ is 0.80 / 0.05 = 16. The post-test probability at which the strategies tie is 1 minus 400 / 3,250, which is 1 minus 0.1231, or about 87.7%.

2. Update the pre-test probability. At a pre-test probability of 20%, the odds are 0.20 / 0.80 = 0.25, which become 4.0 after a positive result, a post-test probability of 4.0 / 5.0 = 0.80, or 80.0%. At 40%, the odds of 0.6667 become 10.667, a post-test probability of 10.667 / 11.667, about 91.4%.

3. Count the patients.

4. Compare net monetary benefit. At 20%, treating without confirmation saves 2,000 x GBP 400 = GBP 800,000 of biopsies but treats 400 people without the condition at 400 x GBP 3,250 = GBP 1,300,000, so incremental net monetary benefit is minus GBP 500,000, or minus GBP 50 per patient tested. At 40%, the saving is 3,500 x GBP 400 = GBP 1,400,000 and the harm 300 x GBP 3,250 = GBP 975,000, so net monetary benefit is GBP 425,000, or GBP 42.50 per patient.

5. Interpret. The same LR+ of 16 gives a post-test probability below the 87.7% threshold in the first population and above it in the second, so treating on a positive result is worth adopting only in the second. Per patient tested, net monetary benefit is GBP 462.50 times the pre-test probability minus GBP 142.50, which is zero at a pre-test probability of about 30.8%. The pre-test formula gives the same figure, since the threshold odds, 0.8769 divided by 0.1231 or about 7.125, divided by 16 give 0.445, a probability of 0.445 / 1.445, about 30.8%. The two routes agree because the test is used in both strategies and its cost cancels.

A more specific test changes the answer for the lower-risk population. With specificity 0.98 and the same sensitivity, LR+ is 40, and at 20% pre-test probability there are 160 false positives among 1,760 positives, a post-test probability of about 90.9%. Treating without confirmation then saves 1,760 x GBP 400 = GBP 704,000 against GBP 520,000 of harm, a gain of GBP 184,000 per 10,000 patients, and the break-even pre-test probability falls to about 15.1%. Raising sensitivity to 0.95 at the original specificity gives LR+ of 19 and moves the break-even only to about 27.3%.

Boundaries with nearby measures

LR+ and the positive predictive value answer different questions. LR+ is estimated within each disease group and needs a pre-test probability before it says anything about a patient, whereas the predictive value is the result already combined with one study's prevalence. LR+ and LR- describe the two results of one binary test, and their ratio is the diagnostic odds ratio, which the diagnostic accuracy page discusses.

Common misreadings

Most errors with LR+ come from reading it apart from the pre-test probability or the study it was estimated in. Others come from taking a large ratio from a small or flawed study at face value.

• Reading LR+ as a probability. An LR+ of 8 does not mean an 80% chance of disease after a positive result, since the post-test probability depends on the pre-test odds.
• Treating high specificity as a rule-in on its own. Low sensitivity lowers LR+, as the third heart sound example shows.
• Trusting very large ratios from small or biased studies. A specificity near 100% makes LR+ unstable, and the CAGE estimate of 206 had a lower confidence limit of 29.
• Assuming LR+ is fixed across settings. Spectrum and referral filters can shift specificity, and with it LR+, between the study and the decision population.
• Equating a high LR+ with a cost-effective test. As the example shows, whether to act on a positive result also depends on prevalence, the cost of confirmation and the harm of treating people without the condition.

Sources

• Deeks JJ, Altman DG. Diagnostic tests 4: likelihood ratios. BMJ. 2004;329(7458):168-169. https://doi.org/10.1136/bmj.329.7458.168
• Leeflang MMG, Rutjes AWS, Reitsma JB, Hooft L, Bossuyt PMM. Variation of a test's sensitivity and specificity with disease prevalence. CMAJ. 2013;185(11):E537-E544. https://doi.org/10.1503/cmaj.121286
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). London: NICE; 2022, last updated 13 October 2026. Sections 4.6.12, 4.7.18 and 6.3.2. https://www.nice.org.uk/process/pmg36
• National Institute for Health and Care Excellence. NICE HealthTech programme manual (PMG48). London: NICE; 2025, last updated 17 December 2025. Section 2.3.14. https://www.nice.org.uk/process/pmg48/chapter/methods-for-guidance-produced-in-the-nice-healthtech-programme
• Pewsner D, Battaglia M, Minder C, Marx A, Bucher HC, Egger M. Ruling a diagnosis in or out with "SpPIn" and "SnNOut": a note of caution. BMJ. 2004;329(7459):209-213. https://doi.org/10.1136/bmj.329.7459.209