Positive Predictive Value

Positive Predictive Value in Rule-In and Confirmation Decisions

When a diagnostic or screening test comes back positive, the positive predictive value says how often that result is right in a given setting. It equals the risk of disease after a positive result, so its complement is the share of positives who go on to confirmation or treatment without having the condition. In health economics it governs the positive branch of a diagnostic or screening model, where the number of false positives fixes the cost of work-up and any harm from treating people who are well. Because the measure depends on how common the condition is, one test can score high in a specialist clinic and low in population screening. The counterpart for negative results is the negative predictive value, which carries the cost of missed cases.

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

A diagnostic 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 positive predictive value formally as the probability of being diseased given a positive result, a conditional probability estimated from the counts as follows.

$$PPV = P(D^{+} \mid T^{+}) = \frac{TP}{TP + FP}$$

where $D^{+}$ denotes presence of the target condition, $T^{+}$ a positive index test result, $TP$ the number of true positives and $FP$ the number of false positives.

Pewsner and colleagues note that the positive predictive value equals the post-test probability of disease given a positive result. Sensitivity and specificity condition the other way, on true disease status. One minus specificity is the share of people without the condition who test positive, whereas one minus the positive predictive value is the share of positive results that are false.

Why the positive predictive value falls with prevalence

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

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

where $Se$ is sensitivity, $Sp$ is specificity and $p$ is the prevalence, or pre-test probability, where the test is used. The numerator is the share of everyone tested who are true positives and the denominator is the share who test positive. As $p$ falls, false positives come from an ever larger group without the condition, and the measure falls towards zero however sensitive the test is.

Deeks and Altman describe the same update in odds form, with post-test odds equal to pre-test odds multiplied by the likelihood ratio.

$$\frac{PPV}{1 - PPV} = LR^{+} \times \frac{p}{1 - p}, \quad LR^{+} = \frac{Se}{1 - Sp}$$

where $LR^{+}$ is the positive likelihood ratio and the other terms are as defined above. Because one minus specificity is the denominator of $LR^{+}$, specificity carries most of the weight once it is high. A test with sensitivity 0.90 and specificity 0.95 has an $LR^{+}$ of 18, and at 1% prevalence the pre-test odds of 1 in 99 become post-test odds of about 0.18, a predictive value of about 15.4%. Raising specificity to 0.99 lifts the predictive value to about 47.6%, while raising sensitivity to 1.00 lifts it only to about 16.8%.

Where estimates come from and when they transfer

A positive predictive value can be read directly from a study only when the share of people with the condition in the study matches the intended population. Chapter 10 of the Cochrane Handbook advises against pooling predictive values in a meta-analysis because they depend on prevalence, which is likely to vary between studies, so reviews pool sensitivity and specificity and recompute the predictive value at a stated prevalence.

Recomputing is approximate, since Leeflang and colleagues found that specificity tended to be lower in studies with higher prevalence, with no such systematic effect for sensitivity. As specificity drives the false positive count, recomputing for a screening setting from accuracy measured in referred patients may misstate the false positives.

Arifin and Yusof note that when only some patients receive the reference standard, usually those testing positive, predictive values in verified patients remain unbiased if verification depends only on the test result, which is why the Begg and Greenes and multiple imputation corrections for verification bias work through them.

Case-finding algorithms in electronic health care data are judged by the same measures, and Chubak, Pocobelli and Weiss argue that a high positive predictive value is important for identifying a cohort of persons with a condition of interest. In a cost study built on such a cohort, one minus the predictive value is the share of the cohort without the condition, whose costs are then wrongly attributed to it.

The positive 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, the positive predictive value splits the people who test positive into true cases, who gain from treatment, and false positives, who incur confirmation costs and any harm from work-up.

When every positive result is confirmed by a further test, the confirmation spend per case found follows directly from the measure.

$$K = \frac{C}{PPV}$$

where $K$ is the confirmation cost per true case found and $C$ is the cost of one confirmatory test. At a predictive value of 0.20, five people are worked up for each case found.

Models are usually parameterised with sensitivity, specificity and prevalence, deriving the predictive value inside the model. In probabilistic sensitivity analysis the derived value is recomputed in each iteration from jointly sampled accuracy, because PMG36 notes that sensitivity and specificity are usually correlated and asks for correlation between parameters to be reflected in the probabilistic analysis. The rest of a screening model is covered under screening economics.

Raising the predictive value with a second test

One way to lift a low predictive value is to retest people positive on the first test and send forward only those positive on both. The first-stage predictive value then becomes the pre-test probability for the second test. Assuming the errors of the two tests are independent within each disease group, the combined accuracy of the positive-on-both rule is as follows.

$$Se_{12} = Se_1 \times Se_2, \quad Sp_{12} = 1 - (1 - Sp_1)(1 - Sp_2)$$

where $Se_1$ and $Sp_1$ are the sensitivity and specificity of the first test, $Se_2$ and $Sp_2$ those of the second, and $Se_{12}$ and $Sp_{12}$ those of the combined rule. Specificity rises and sensitivity falls, so the predictive value improves at the price of extra missed cases. If the two tests tend to err on the same people, the gain is smaller than the formula implies.

Worked example of adding a triage test to a screening pathway

All numbers in this example are illustrative. A screening test with sensitivity 0.90 and specificity 0.95 is offered to 10,000 people at 1% prevalence. Strategy A sends every positive for a confirmatory test costing GBP 300, assumed perfectly accurate. Strategy B first gives positives a second test costing GBP 40, with sensitivity and specificity of 0.90 and errors independent of the first test, and confirms only those positive again. Finding and treating a case is worth GBP 10,000 against missing it, for example 0.5 QALYs valued at GBP 25,000 per QALY, the lower of the two values NICE's PMG36 uses for net health benefit, less GBP 2,500 of treatment cost. Missed cases fare as they would without screening, false positives cost only the tests they receive, and the first test is common to both strategies, so its cost is left out.

1. Read the first test. Of 100 people with the condition, 90 test positive, and of 9,900 without, 5% or 495 test positive. The predictive value is 90 / 585, about 0.1538, so 6.5 people are confirmed for each case found.

2. Cost strategy A. Confirmation costs 585 × GBP 300, which is GBP 175,500, or GBP 1,950 per case found. Its net monetary benefit is 90 × GBP 10,000 minus GBP 175,500, which is GBP 724,500.

3. Apply the second test. Of the 90 true positives, 81 test positive again, and of the 495 false positives, 10% or 49.5 do. Among the 130.5 people sent for confirmation the predictive value is 81 / 130.5, as the Bayes form confirms.

$$PPV_2 = \frac{0.90 \times 0.1538}{0.90 \times 0.1538 + 0.10 \times 0.8462} = \frac{0.1384}{0.2230} = 0.621$$

where $PPV_2$ is the predictive value after both tests and 0.8462 is one minus 0.1538.

4. Cost strategy B. Second tests cost 585 × GBP 40, which is GBP 23,400, and confirmations 130.5 × GBP 300, which is GBP 39,150. With 81 cases found, net monetary benefit is GBP 810,000 minus GBP 62,550, which is GBP 747,450, so B beats A by GBP 22,950 per 10,000 screened.

5. Repeat at 10% prevalence. Now 900 of 1,000 cases and 450 of 9,000 others test positive, a first-stage predictive value of 900 / 1,350, about 0.667. Strategy A gives 900 × GBP 10,000 minus 1,350 × GBP 300, which is GBP 8,595,000. Strategy B finds 810 cases and confirms 855 people, giving GBP 8,100,000 minus GBP 54,000 of second tests and GBP 256,500 of confirmations, which is GBP 7,789,500. Strategy A is now better by GBP 805,500.

6. Find the break-even predictive value. Per first-stage positive, the gain from adding the second test is given below.

$$\Delta = C \left[ q (1 - Se_2) + (1 - q) Sp_2 \right] - c_2 - q (1 - Se_2) G = 230 - 1240 q$$

where $\Delta$ is the net monetary benefit of B minus A per first-stage positive, $q$ is the first-stage positive predictive value, $C$ is the GBP 300 confirmation cost, $c_2$ is the GBP 40 second test cost and $G$ is the GBP 10,000 value of a case found. The gain is zero at 230 / 1,240, a first-stage predictive value of about 0.185. At 0.1538 it is about GBP 39.23 per positive, matching the GBP 22,950 gain across 585 positives.

7. Interpret. The second test pays when fewer than about 18.5% of first-stage positives have the condition, because most confirmations are then spent on people without it. Strategy B has the higher predictive value in both settings, about 0.621 at 1% and about 0.947 at 10%, yet loses heavily at the higher prevalence, because that value was bought with 9 extra missed cases per 10,000 at 1% and 90 at 10%.

Boundaries with nearby measures

Sensitivity is the share of people with the condition who test positive, the predictive value the share of positives who have the condition, and prevalence enters only the second by definition. The positive likelihood ratio is the more portable summary combined with a pre-test probability, and the positive predictive value is the result of that combination in one population. For an informative test the positive and negative predictive values move in opposite directions as prevalence changes.

Common misreadings in screening and appraisal

Most errors with the positive predictive value come from forgetting the prevalence behind it. Each one below distorts the false positives a pathway is credited with.

• Reading one minus specificity as the share of positives that are false. In the example a specificity of 0.95 at 1% prevalence still left about 85% of positives false.
• Importing a value from another setting. A figure from a case-control study or a referral clinic has to be recomputed at the prevalence of the decision population, and even then specificity may differ between settings.
• Choosing a pathway by its predictive value alone. A higher value bought with a second test loses cases, as the 10% setting shows.
• Sampling the value as an independent parameter. In a probabilistic analysis this breaks its link with sensitivity, specificity and prevalence.

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
• Chubak J, Pocobelli G, Weiss NS. Tradeoffs between accuracy measures for electronic health care data algorithms. Journal of Clinical Epidemiology. 2012;65(3):343-349.e2. https://doi.org/10.1016/j.jclinepi.2011.09.002
• 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
• 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 10.2.3.2 and 10.4.2. https://methods.cochrane.org/sites/methods.cochrane.org.sdt/files/uploads/Chapter%2010%20-%20Version%201.0.pdf
• 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, 4.7.18 and 4.7.19. 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