Negative Likelihood Ratio

Negative Likelihood Ratio for Ruling Out Disease and Valuing a Negative Result

The negative likelihood ratio (LR-) of a test is the probability of a negative result in people who have the target condition divided by the probability of a negative result in people who do not, which works out as (1 minus sensitivity) divided by specificity. Multiplying a patient's pre-test odds by LR- gives the odds of disease after a negative result, so a value near 0 means a negative result goes a long way towards ruling the condition out, while a value of 1 means it changes nothing. In health economics LR- matters because it sets how much disease remains among the people a test sends home, and so the number of missed cases that a cheaper or less invasive pathway trades against the confirmatory tests it saves. High sensitivity alone does not guarantee a rule-out, LR- may not transport between settings, and the same LR- can be worth adopting in one population and not in another.

From the two-by-two table to LR-

Deeks and Altman describe each test result as having its own likelihood ratio, defined as the probability of that result in people with the disease divided by its probability in people without it. A ratio below 1 is associated with absence of the disease, and the further it lies from 1 the stronger the evidence. For a test reported as positive or negative, the ratio for the negative result is LR-, calculated as follows.

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

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

Deeks and Altman state that ratios above 10 and below 0.1 are considered to provide strong evidence to rule in or rule out diagnoses respectively in most circumstances. On the odds scale an LR- of 0.1 is the mirror of an LR+ of 10, because both change the odds tenfold. The general likelihood ratio page covers multi-level results and the positive likelihood ratio has its own page.

Updating to a post-test probability with Bayes' theorem

LR- acts on odds, not probabilities, so the update needs a starting probability from outside the accuracy study. Following Bayes' theorem, the pre-test probability is converted to odds, multiplied by LR- and converted back to a probability.

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

where $p$ is the probability of the condition before testing, $O_{pre}$ and $O_{post}$ are the odds before and after a negative result, and $p_{post}$ is the post-test probability of disease given a negative result. Pewsner and colleagues note that this residual probability equals one minus the negative predictive value. The same LR- therefore leaves very different residual risks. A tenfold cut in odds takes a 5% pre-test probability to about 0.5%, but a 50% pre-test probability only to about 9.1%.

Why high sensitivity alone does not rule a diagnosis out

The mnemonic SnNOut holds that a negative result on a highly sensitive test rules the diagnosis out. Pewsner and colleagues show that the power to rule out also depends on specificity, because specificity is the denominator of LR-. In their reanalysis, clinical criteria for Alzheimer's disease had a sensitivity of 93% but a specificity of 23%, giving an LR- of only 0.3; at the study's pre-test probability of 53.6%, the probability after a negative result was still 25%. By contrast, the Ottawa ankle rules (sensitivity 99%, specificity 39%) gave an LR- of 0.04 and a post-test probability of 0.7% from a pre-test probability of 17.1%.

The reverse also holds. In Deeks and Altman's smoking example, dichotomising at 40 pack years gives a specificity of 98.6% but a sensitivity of 28.4%, so LR- is 71.6/98.6, about 0.73, and a negative result does little to exclude obstructive airway disease. Precision matters too. Pewsner and colleagues cite an ascites study with 15 cases whose LR- of 0.10 had a 95% confidence interval of 0.01 to 0.7, compatible with a post-test probability of 15.8%. Deeks and Altman note that confidence intervals for likelihood ratios can be calculated in the same way as for risk ratios.

Whether LR- transports between settings

Applying a published LR- to a new prevalence assumes that sensitivity and specificity stay fixed. Leeflang and colleagues reanalysed 23 meta-analyses (416 studies) and found a statistically significant association between prevalence and sensitivity or specificity in 8 of them (35%). Across the reanalysed meta-analyses, specificity tended to be lower at higher prevalence, with no systematic effect for sensitivity. They attribute this to mechanisms such as patient spectrum and referral filters that affect prevalence and accuracy together.

Since specificity is the denominator of LR-, lower specificity in a higher-prevalence setting raises LR- and weakens the rule-out. Study design adds to the problem, and Pewsner and colleagues describe how spectrum bias in diagnostic case-control studies, partial verification bias and incorporation bias can overstate accuracy. An analyst importing an LR- into a model should therefore take it from studies whose population and care pathway match the decision setting.

LR- in diagnostic economic evaluation

NICE's technology appraisal manual (PMG36, 2022, last updated 13 October 2026) treats diagnostic test accuracy statistics as intermediate measures that can be used in models as predictors of future health outcomes. The NICE HealthTech programme manual (PMG48) states that most outcomes of testing follow from treatment or preventive measures being started, modified or stopped, and that a linked evidence modelling approach should be used when direct evidence on final outcomes is missing. LR- sits on the negative branch of such a decision tree. Together with the pre-test probability it fixes the share of discharged patients who carry undetected disease, and each of those patients brings the costs and QALY losses of delayed treatment.

For a rule-out decision, discharging a person with a negative result instead of sending them for confirmation is worthwhile when the expected loss from a missed case is smaller than the confirmatory cost saved, which gives the following condition.

$$p_{post} < p^{*} = \frac{C_{conf}}{L_{miss}}$$

where $C_{conf}$ is the cost of the confirmatory test avoided, $L_{miss}$ is the monetary value of a missed case (QALYs lost multiplied by the cost-effectiveness threshold, plus extra treatment costs), and $p^{*}$ is the residual risk at which discharge and confirmation tie. This simple form assumes a perfect confirmatory test with no harms of its own, and it governs only the discharge decision after testing, so the triage test's own cost must also be recovered.

For uncertainty, PMG36 states that sensitivity and specificity are usually correlated and may vary with how a test is used or interpreted, so point estimates with distributions are not usually appropriate. In probabilistic sensitivity analysis LR- is therefore better computed inside each iteration from jointly sampled sensitivity and specificity than given an independent distribution of its own.

Worked example of one rule-out test in two referral populations

All numbers in this example are illustrative. A triage test costing GBP 20 is offered to 10,000 referred patients, with sensitivity 0.95 and specificity 0.50. Patients who test negative are discharged; the rest receive confirmatory imaging costing GBP 300, assumed perfect. The comparator images everyone. A missed case loses 0.4 QALYs and adds GBP 3,000 of later treatment cost, valued at an illustrative GBP 20,000 per QALY, so $L_{miss}$ is GBP 11,000.

1. Compute LR- and the threshold. LR- is (1 minus 0.95) / 0.50 = 0.10. The residual risk at which discharge breaks even is 300 / 11,000, about 2.73%.

2. Update the pre-test probability. At a pre-test probability of 15%, the odds are 0.15 / 0.85 = 0.1765; after a negative result they are 0.01765, a post-test probability of about 1.73%. At 30%, the odds of 0.4286 fall to 0.04286, a post-test probability of about 4.11%.

3. Count the patients.

4. Compare net monetary benefit. At 15%, imaging avoided saves 4,325 x GBP 300 = GBP 1,297,500; less GBP 200,000 of triage tests, the net saving is GBP 1,097,500. The 75 missed cases cost 75 x GBP 11,000 = GBP 825,000, so incremental net monetary benefit is GBP 272,500, or GBP 27.25 per patient. At 30%, the saving is 3,650 x GBP 300 minus GBP 200,000 = GBP 895,000, and 150 missed cases cost GBP 1,650,000, so net monetary benefit is minus GBP 755,000, or minus GBP 75.50 per patient.

5. Interpret. The same LR- of 0.10 leaves a residual risk below the 2.73% threshold in the lower-risk population and above it in the higher-risk one, so the triage strategy is worth adopting only in the first. Because every patient pays for the GBP 20 triage test, net monetary benefit per patient is GBP 130 minus GBP 6.85 for each percentage point of pre-test probability, reaching zero at about 19.0%, below the 21.9% at which residual risk reaches 2.73%. A test with sensitivity 0.975 and the same specificity has an LR- of 0.05; at 30% it leaves 75 false negatives among 3,575 discharged (about 2.10%) and gains GBP 47,500 per 10,000 patients. The value of a lower LR- thus depends on prevalence, the cost of confirmation and the harm of a missed case.

Boundaries with nearby measures

LR- and the negative predictive value answer different questions. LR- conditions on disease status and is combined with a pre-test probability, while the predictive value is the result already combined with one study's prevalence. LR- and the positive likelihood ratio describe the two results of one test, and their ratio, LR+ divided by LR-, is the diagnostic odds ratio, which the diagnostic accuracy page discusses. Sensitivity and specificity are the inputs, LR- the summary used for updating.

Common misreadings

Most errors with LR- come from reading it apart from the pre-test probability or the setting it was estimated in. Each of the following can change the cost-effectiveness of a diagnostic pathway.

• Reading LR- as a probability. An LR- of 0.2 does not mean a 20% chance of disease after a negative result; the residual risk depends on the pre-test odds.
• Treating high sensitivity as a rule-out on its own. Low specificity raises LR-, as the Alzheimer's criteria example shows.
• Assuming LR- is fixed across settings. Spectrum and referral filters can shift specificity, and so LR-, between the study and the decision population.
• Multiplying ratios from two negative tests. Chaining LR- values assumes conditional independence, which correlated tests violate (see the likelihood ratio page).
• Equating a low LR- with a cost-effective test. As the example shows, adoption also depends on prevalence, confirmatory costs and the harm of missed cases.

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 and 4.7.18. https://www.nice.org.uk/process/pmg36/chapter/economic-evaluation-2
• National Institute for Health and Care Excellence. NICE HealthTech programme manual (PMG48). London: NICE; 2025, updated 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