Post-Test Probability

Post-Test Probability in Diagnostic Decisions and Test Evaluation

Post-test probability is the chance that a patient has the target condition once a particular test result is known. It is the figure a clinician acts on, and in a diagnostic decision model it is the probability attached to each test-result branch, so it fixes how many people are treated or reassured correctly and how many wrongly. Its value depends as much on where the patient started as on the accuracy of the test, so the same test can settle a diagnosis in one clinic and not in another. In health economics it links accuracy evidence to costs and QALYs, and comparing it with a treatment threshold shows whether a test can change what happens next.

Updating the probability on the odds scale

The update takes a pre-test probability and the likelihood ratio of the observed result. Deeks and Altman define that ratio as the probability of the specific result in people with the disease divided by its probability in people without it, and multiply the pretest odds by it, an application of Bayes' theorem.

$$O_{pre} = \frac{p}{1 - p}, \qquad O_{post} = O_{pre} \times LR_{r}, \qquad p_{post} = \frac{O_{post}}{1 + O_{post}}$$

where $p$ is the probability of the condition before the test, $O_{pre}$ and $O_{post}$ are the odds before and after the result, $LR_{r}$ is the likelihood ratio for the result $r$ actually observed, and $p_{post}$ is the post-test probability. For a binary test the result is either positive, with the positive likelihood ratio, or negative, with the negative likelihood ratio.

The same update can be written directly in terms of sensitivity and specificity.

$$p_{post}^{+} = \frac{Se \, p}{Se \, p + (1 - Sp)(1 - p)}, \qquad p_{post}^{-} = \frac{(1 - Se) \, p}{(1 - Se) \, p + Sp \, (1 - p)}$$

where $Se$ is sensitivity, $Sp$ is specificity, $p_{post}^{+}$ is the probability of the condition after a positive result and $p_{post}^{-}$ the probability after a negative result. The first expression is the positive predictive value in a population whose prevalence equals $p$, and the second is one minus the negative predictive value in that population.

In Deeks and Altman's smoking example, also used on the positive likelihood ratio page, 40 or more pack years has a likelihood ratio of about 20.4 and takes a prior probability of 0.1 to about 0.69, and Fagan's nomogram gives the same answer without the conversions.

Each test result has its own likelihood ratio, so a test reported in several categories gives a separate post-test probability for each category. Deeks and Altman note that forcing such results into positive and negative may discard useful diagnostic information. In their table, smoking 20 to 40 pack years had a likelihood ratio of 1.01, which leaves the probability unchanged.

How far one result can move the probability

Because the update is multiplicative on the odds scale, the same likelihood ratio shifts the probability by different amounts depending on the starting point. A ratio of 10 takes a pre-test probability of 1% only to about 9.2%, but takes 40% to about 87.0%. Deeks and Altman regard ratios above 10 and below 0.1 as strong evidence to rule a diagnosis in or out in most circumstances, yet in a low-prevalence screening setting even a strong positive result can leave the condition less likely than not.

McGee offers a bedside approximation that avoids odds. A likelihood ratio of 2 raises the probability by about 15 percentage points, 5 by about 30 and 10 by about 45, and the reciprocals 0.5, 0.2 and 0.1 lower it by the same amounts. He reports that these estimates are accurate to within 10% of the calculated answer for pre-test probabilities between 10% and 90%, with an average error of 4%, because probability is nearly linear in log odds over that band.

Likelihood ratios carry sampling error, and the post-test probability inherits it. The 95% confidence interval that Deeks and Altman report for the smoking ratio runs from 5.04 to 82.8. Applied to the same prior of 0.1, those limits give post-test odds of 0.56 and 9.2, a post-test probability anywhere between about 36% and 90%.

Thresholds that turn a probability into an action

A post-test probability matters only through the decision it supports. Pauker and Kassirer's therapeutic, testing and test-treatment thresholds, set out on the clinical decision analysis page, decide whether a given post-test probability leads to treatment.

The treatment threshold follows from comparing expected outcomes. Treating is preferred when the expected gain in people with the condition outweighs the expected loss in people without it.

$$p \, B > (1 - p) \, H \quad \Longleftrightarrow \quad p > p^{*} = \frac{H}{H + B}$$

where $p$ is the current probability of the condition, $B$ is the net benefit of treating a person who has the condition compared with not treating them, $H$ is the net harm of treating a person who does not, and $p^{*}$ is the treatment threshold. In an economic evaluation $B$ and $H$ can be expressed as net monetary benefit, QALYs valued at a stated amount per QALY minus treatment cost, so the threshold moves when the value placed on health or the price of treatment changes.

A test can change management only if one of its results lands on the other side of $p^{*}$. Ignoring the test's own cost and harm, that happens when the positive result lifts the probability above the threshold and the negative result drops it below. On the odds scale the condition is $O^{*} / LR^{+} < O_{pre} < O^{*} / LR^{-}$, where $O^{*}$ is the threshold expressed as odds. Once the cost and harm of testing are counted, the band narrows to the range between Pauker and Kassirer's testing and test-treatment thresholds.

Post-test probabilities in diagnostic decision models

In a decision tree for a test strategy, post-test probabilities split each result branch into people with and without the condition, and treatment costs and harms attach to those groups. The NICE manual (PMG36, section 4.6.12) treats diagnostic test accuracy statistics as intermediate measures that, when incorporated into models, can predict future health outcomes, and asks that their uncertainty be quantified and presented. Section 4.6.13 adds that the benefits of diagnostic testing generally arise from the treatment or prevention that follows, so the model follows patients through treatment and monitoring.

The usual way to carry uncertainty is to recompute post-test probabilities in each iteration of a probabilistic sensitivity analysis. PMG36 (section 4.7.18) 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. Jointly sampled accuracy values and a sampled pre-test probability then give post-test probabilities that keep that correlation. Where they straddle the treatment threshold across iterations, value of information analysis can show whether a better accuracy study would be worth its cost.

Worked example of a test before treatment in two populations

All numbers in this example are illustrative. For a suspected condition, treating a person with the condition gains 0.30 QALYs and costs GBP 1,500, and treating a person without it loses 0.02 QALYs at the same cost. QALYs are valued at GBP 25,000, the lower of the two values the NICE manual (PMG36, section 6.3.2) uses when presenting net health benefit. A test with sensitivity 0.90 and specificity 0.80 costs GBP 100 and has no direct harm.

1. Treatment threshold. $B$ is 0.30 × 25,000 less 1,500, or GBP 6,000, and $H$ is 0.02 × 25,000 plus 1,500, or GBP 2,000. The threshold is 2,000 / (2,000 + 6,000) = 0.25.

2. Likelihood ratios. LR+ is 0.90 / 0.20 = 4.5 and LR- is 0.10 / 0.80 = 0.125.

3. Post-test probabilities at a pre-test probability of 40%. The pre-test odds are 0.40 / 0.60 = 0.667. A positive result gives odds of 0.667 × 4.5 = 3.0 and a probability of 3.0 / 4.0 = 0.75. A negative result gives odds of 0.667 × 0.125 = 0.0833 and a probability of 0.0833 / 1.0833, about 0.077. The positive result sits above 0.25 and the negative below it, so the test decides treatment.

4. Expected net monetary benefit per patient at 40%, against no treatment. Treating everyone yields 0.40 × 6,000 less 0.60 × 2,000, or GBP 1,200. True positives make up 0.90 × 0.40 = 0.36 of patients and false positives 0.20 × 0.60 = 0.12, so testing and treating positives yields GBP 2,160 less GBP 240 less the GBP 100 test, or GBP 1,820. The post-test route agrees, since 48% test positive and each positive patient is worth 0.75 × 6,000 less 0.25 × 2,000, or GBP 4,000, giving 0.48 × 4,000 = 1,920 before the test cost and GBP 1,820 after it. Testing is preferred by GBP 620 per patient over treating everyone.

5. The same test at a pre-test probability of 80%. The odds are 4.0. A positive result gives 18.0, a probability of about 0.947, and a negative result gives 0.5, a probability of 0.5 / 1.5, about 0.333. Both results now sit above the 0.25 threshold, so treatment follows whatever the result. Treating everyone yields GBP 4,800 less GBP 400, or GBP 4,400, while testing yields 0.72 × 6,000 less 0.04 × 2,000 less 100, or GBP 4,140. Treating without testing is preferred by GBP 260 per patient.

6. The band in which testing pays. The threshold odds are 0.25 / 0.75 = 0.333. Ignoring the test cost, a result can cross the threshold when the pre-test odds lie between 0.333 / 4.5 = 0.0741 and 0.333 / 0.125 = 2.667, a pre-test probability between about 6.9% and 72.7%. With the GBP 100 cost included, the expected net benefit of testing is 5,800 times the pre-test probability minus 500, which equals zero at about 8.6%, and that of treating everyone is 8,000 times the pre-test probability minus 2,000, which equals the testing line at 1,500 / 2,200, about 68.2%. These are the testing and test-treatment thresholds.

7. Interpretation. The test's accuracy is identical in both populations, yet it is worth using only in the first, because only there do its two post-test probabilities fall on opposite sides of the treatment threshold by enough to repay its cost.

Sequential tests and the independence assumption

When tests are done in sequence, the post-test probability after the first becomes the pre-test probability for the second. Multiplying likelihood ratios in this way assumes the tests are conditionally independent, so that within each disease group the second result is unrelated to the first. Two positive results from tests that each have an LR+ of 4.5 would take 40% to odds of 0.667 × 4.5 × 4.5 = 13.5, about 93.1%, only if the tests were independent. Tests that respond to the same feature of the disease count the same evidence twice.

Novielli, Sutton and Cooper developed a meta-analytic framework to estimate the accuracy of two tests used in combination while allowing for their likely nonindependence, applied to the D-dimer test and the Wells score for deep vein thrombosis. Their results showed the importance of allowing for it, so a model of a testing sequence is better built on joint evidence than on a product of single-test ratios.

Boundaries with nearby measures

Pre-test and post-test probability are the before and after of one update. Predictive values are post-test probabilities tied to one study's prevalence, and Deeks and Altman note that they can rarely be generalised beyond the study sample, whereas likelihood ratios can be combined with prior probabilities from different settings. A posterior distribution applies the same Bayesian logic to an unknown parameter such as a treatment effect, while the post-test probability concerns whether a patient with a given result has the condition.

Common misreadings

Most errors come from separating the post-test probability from its inputs. Others lose sight of the decision it is meant to support.

• Reading a likelihood ratio as the post-test probability. An LR+ of 20 gives about 69% from a prior of 0.1 and about 99.4% from a prior of 0.9.
• Borrowing accuracy from a different setting. McGee shows that tachypnoea discriminated pneumonia in children whose symptoms had lasted 6 days or more (LR+ 3.5, LR- 0.1) but had no significant diagnostic value in those ill for less than 6 days.
• Using a bedside approximation in a model. McGee's rule is inaccurate below 10% and above 90%, the region where many screening and rule-out decisions lie.
• Equating a high post-test probability with a worthwhile test. In the worked example a positive result at 80% pre-test probability gave about 95%, yet testing gave GBP 260 less net monetary benefit per patient than treating everyone, because neither result changed treatment.

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
• McGee S. Simplifying likelihood ratios. Journal of General Internal Medicine. 2002;17(8):646-649. https://doi.org/10.1046/j.1525-1497.2002.10750.x
• 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, 4.6.13, 4.7.18 and 6.3.2. https://www.nice.org.uk/process/pmg36
• Novielli N, Sutton AJ, Cooper NJ. Meta-analysis of the accuracy of two diagnostic tests used in combination: application to the D-dimer test and the Wells score for the diagnosis of deep vein thrombosis. Value in Health. 2013;16(4):619-628. https://doi.org/10.1016/j.jval.2013.02.007
• Pauker SG, Kassirer JP. Therapeutic decision making: a cost-benefit analysis. New England Journal of Medicine. 1975;293(5):229-234. https://doi.org/10.1056/NEJM197507312930505
• Pauker SG, Kassirer JP. The threshold approach to clinical decision making. New England Journal of Medicine. 1980;302(20):1109-1117. https://doi.org/10.1056/NEJM198005153022003