Negative Predictive Value

Negative Predictive Value in Diagnostic Rule-Out Decisions

When a diagnostic or screening test comes back negative, the negative predictive value says how often that reassurance is correct among the people tested in a given setting. It is read from the test-negative row of a test accuracy table and equals one minus the risk of disease that remains after a negative result. In health economics it governs the negative branch of a diagnostic model, where people are discharged, reassured or left untreated, and its complement fixes how many missed cases carry the costs and QALY losses of delayed treatment. Because it depends on the prevalence of the condition where the test is used, a figure taken from one study can be badly wrong in another population, and a very high value may add little to what was known before testing. The abbreviation NPV is also used for net present value, a discounting measure with its own page, but here it always means the diagnostic measure.

The test-negative row of the two-by-two table

An accuracy study cross-classifies each index test result against a reference standard. The Cochrane Handbook for Diagnostic Test Accuracy Reviews defines predictive values as measures conditional on the index test result and gives the negative predictive value formally as the probability of being non-diseased given a negative result, a conditional probability estimated from the counts as follows.

$$NPV = P(D^{-} \mid T^{-}) = \frac{TN}{TN + FN}$$

where $D^{-}$ denotes absence of the target condition, $T^{-}$ a negative index test result, $TN$ the number of true negatives and $FN$ the number of false negatives.

Pewsner and colleagues note that the negative predictive value equals one minus the post-test probability of disease given a negative result. That complement, the residual risk among people sent home, is the quantity a model most often needs. Sensitivity and specificity condition the other way, on true disease status, and the positive predictive value reads the test-positive row. The diagnostic accuracy page covers the full set of measures.

How prevalence moves the negative predictive value

Leeflang and colleagues state that predictive values directly depend on disease prevalence and so cannot be carried over directly between settings. Bayes' theorem shows why, by writing the measure in terms of sensitivity, specificity and the prevalence in the population tested.

$$NPV = \frac{Sp \times (1 - p)}{Sp \times (1 - p) + (1 - Se) \times p}$$

where $Se$ is sensitivity, $Sp$ is specificity and $p$ is the prevalence, or pre-test probability, in the setting where the test is used. The numerator is the share of all people tested who are true negatives and the denominator is the share who test negative. As $p$ falls, false negatives shrink towards zero and the measure rises towards one, whatever the quality of the test.

A useful check compares the measure with the probability of being free of the condition before testing, which is one minus $p$. Subtracting that from the expression above gives the gain from testing.

$$NPV - (1 - p) = \frac{p (1 - p)(Se + Sp - 1)}{Sp \times (1 - p) + (1 - Se) \times p}$$

where the terms are as defined above and $Se + Sp - 1$ is Youden's index. The Cochrane Handbook describes a Youden's index of zero as equivalent to uninformed guessing, and the identity shows that such a test leaves the negative predictive value equal to the pre-test probability of no disease. A test is informative for ruling out only to the extent that it lifts the measure above one minus $p$. The odds form of the same update, using the negative likelihood ratio, is set out on that page.

Where estimates come from and when they transfer

A negative predictive value can be read straight from a study only when the proportion of people with the condition in the study matches the intended population, as in a consecutive cross-sectional series from that setting. Leeflang and colleagues excluded case-control studies from their analysis because disease prevalence cannot be estimated from them, the prevalence in the study group being set by design. Mercaldo, Lau and Zhou note that because predictive values are functions of both test accuracy and prevalence, constructing their confidence intervals for a particular patient is not straightforward, and they developed estimates and intervals, including logit-transformed ones, which they applied to two case-control studies.

Verification of negatives is a second issue, since in many screening and triage studies only some people who test negative receive the reference standard. Arifin and Yusof note that predictive values in verified patients are unbiased under a missing-at-random assumption, such as verification that depends only on the test result, which is why Begg and Greenes and imputation corrections for verification bias work through them before correcting sensitivity and specificity.

Pooling is a third. Chapter 10 of the Cochrane Handbook advises against pooling predictive values across studies in a meta-analysis because they depend on prevalence, which is likely to vary between studies. It adds that between-study variation in prevalence may induce extra heterogeneity, and that an average predictive value would relate to use of the test at some average but unknown prevalence. Reviews can therefore pool sensitivity and specificity and recompute predictive values at a stated prevalence.

The negative branch of a diagnostic decision model

NICE's technology appraisal manual (PMG36) treats diagnostic test accuracy statistics as intermediate measures that, when incorporated into models, can be used as predictors of future health outcomes. The NICE HealthTech programme manual (PMG48) adds that most outcomes come after testing, through treatment or preventive measures being started, modified or stopped. In a decision tree for a rule-out test, the negative predictive value splits the people sent home into those correctly reassured and those whose disease is missed, and the missed group carries the later costs and health losses.

The number of missed cases follows from the measure and the number of people who test negative, as below.

$$M = N_{neg} \times (1 - NPV) = N \times p \times (1 - Se)$$

where $M$ is the number of missed cases, $N$ is the number of people tested and $N_{neg}$ is the number who test negative. The right-hand form shows that missed cases depend on prevalence and sensitivity alone. Specificity changes how many people are sent home, and so the negative predictive value, but not how many cases are missed. Comparing two tests by their predictive values alone can therefore mislead, because a less specific test sends fewer people home and spares fewer confirmatory tests.

For this reason models are usually parameterised with sensitivity, specificity and prevalence, and the negative predictive value is derived inside the model. In probabilistic sensitivity analysis it is then recomputed in each iteration from jointly sampled accuracy, since PMG36 notes that sensitivity and specificity are usually correlated.

Worked example of a case-control estimate carried into two settings

All numbers in this example are illustrative. A case-control accuracy study of a rule-out test enrols 200 people with the condition and 200 without. The test gives 180 true positives, 20 false negatives, 160 true negatives and 40 false positives. People who test negative can be discharged instead of receiving a confirmatory scan costing GBP 250, assumed perfectly accurate and free of harms. A missed case loses 0.5 QALYs, valued at GBP 25,000 per QALY, the lower of the two values NICE's PMG36 uses for net health benefit, and adds GBP 2,000 of later treatment cost, so each missed case costs GBP 12,500 + GBP 2,000 = GBP 14,500.

1. Read the study. Sensitivity is 180 / 200 = 0.90 and specificity is 160 / 200 = 0.80. The negative predictive value observed in the study is 160 / (160 + 20) = 0.889, but this reflects the 50% prevalence fixed by the design.

2. Recompute at primary care prevalence. Assuming the study's sensitivity and specificity hold in primary care, although Pewsner and colleagues class case-control studies as vulnerable to spectrum bias, the Bayes form at a prevalence of 2% gives the following.

$$NPV_{2\%} = \frac{0.80 \times 0.98}{0.80 \times 0.98 + 0.10 \times 0.02} = \frac{0.784}{0.786} = 0.9975$$

where $NPV_{2\%}$ is the negative predictive value at 2% prevalence. The residual risk after a negative result is 1 minus 0.9975, or 0.25%. The gain over not testing is 0.9975 minus 0.98, about 1.75 percentage points, since 98% of these patients were free of the condition before the test.

3. Recompute at secondary care prevalence. At 20%, the same formula gives the following.

$$NPV_{20\%} = \frac{0.80 \times 0.80}{0.80 \times 0.80 + 0.10 \times 0.20} = \frac{0.64}{0.66} = 0.9697$$

where $NPV_{20\%}$ is the negative predictive value at 20% prevalence, a residual risk of about 3.03% and a gain of about 17 percentage points over the pre-test 0.80.

4. Count people per 10,000 tested.

The last row multiplies the people testing negative by 1 minus 0.889, which is 20 / 180, and at 2% that gives 7,860 × 20 / 180 = 873.3 missed cases where the correct count is 20.

5. Value the decision to discharge negatives. The comparison here is between discharging people who test negative and scanning them anyway, with the triage test already done. At 2% prevalence, discharge saves 7,860 × GBP 250 = GBP 1,965,000 of scans and loses 20 × GBP 14,500 = GBP 290,000 through missed cases, a net monetary benefit of GBP 1,675,000 per 10,000 tested. Using the case-control figure of 0.889 instead, the missed-case loss becomes 873.3 × GBP 14,500 = GBP 12,663,333 and the net benefit minus GBP 10,698,333, so the model would wrongly reject discharge. At 20% prevalence, discharge saves 6,600 × GBP 250 = GBP 1,650,000 but loses 200 × GBP 14,500 = GBP 2,900,000, a net benefit of minus GBP 1,250,000.

6. Interpret. Discharge pays when the residual risk is below 250 / 14,500, about 1.72%. The correctly recomputed value meets that bar at 2% prevalence (0.25% residual risk) and fails it at 20% (3.03%), while the uncorrected study value of 0.889 fails it everywhere. The 99.75% figure also needs reading against the 98% pre-test probability of no disease before it is quoted as evidence of a strong test.

Boundaries with nearby measures

Specificity and the negative predictive value are often confused. Specificity is the share of people without the condition who test negative, while the predictive value is the share of people testing negative who are without the condition, and prevalence enters only the second by definition, although Leeflang and colleagues found that specificity tended to be lower in studies with higher prevalence. The negative likelihood ratio is the more portable summary that is combined with a pre-test probability, and the negative predictive value is the result of that combination in one population. Its counterpart for positive results is the positive predictive value, and the two move in opposite directions as prevalence changes.

Common misreadings in appraisal and modelling

Most errors with the negative predictive value come from forgetting the prevalence behind it. Each of the following can change the estimated missed cases, and so the costs and QALYs, of a rule-out pathway.

• Reading a high value as proof of a good test. At low prevalence almost any test scores highly, so the value should be compared with the pre-test probability of no disease.
• Importing a value from another setting. A figure from a case-control study, or from a population with different prevalence, has to be recomputed from sensitivity and specificity at the decision population's prevalence.
• Treating one minus the value as the test's miss rate. One minus sensitivity is the share of cases missed, while one minus the negative predictive value is the share of negatives who have the condition.
• Ranking tests by the value alone. Missed cases depend on sensitivity and prevalence, and the savings from discharge depend on how many people test negative.

Sources

• Arifin WN, Yusof UK. Partial verification bias correction using inverse probability bootstrap sampling for binary diagnostic tests. Diagnostics (Basel). 2022;12(11):2839. https://doi.org/10.3390/diagnostics12112839
• 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
• Macaskill P, Gatsonis C, Deeks JJ, Harbord RM, Takwoingi Y. Chapter 10: Analysing and presenting results. In: Deeks JJ, Bossuyt PM, Gatsonis C, editors. Cochrane Handbook for Systematic Reviews of Diagnostic Test Accuracy. Version 1.0. The Cochrane Collaboration; 2010. Sections 10.2.3.1, 10.2.3.2 and 10.4.2. https://methods.cochrane.org/sites/methods.cochrane.org.sdt/files/uploads/Chapter%2010%20-%20Version%201.0.pdf
• Mercaldo ND, Lau KF, Zhou XH. Confidence intervals for predictive values with an emphasis to case-control studies. Statistics in Medicine. 2007;26(10):2170-2183. https://doi.org/10.1002/sim.2677
• National Institute for Health and Care Excellence. NICE technology appraisal and highly specialised technologies guidance: the manual (PMG36). London: NICE; 31 January 2022, last updated 13 October 2026. Sections 4.2.16, 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; 14 July 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