How does a test result change what you already suspected?

Start here

Why does the same positive result mean different things in different patients?

A test result does not give a probability by itself. It changes the probability you already had before the test.

Picture 1,000 patients like yours, of whom 200 have the disease. Test A picks up 80% of those with the disease and wrongly flags 10% of those without it. Now count the positive results.

Doctors often find this hard when the numbers are given as percentages. In 1978, Casscells and colleagues asked 60 students, house officers and physicians at Harvard Medical School: if a test for a disease with a prevalence of 1 in 1,000 has a false-positive rate of 5%, what is the chance that a person with a positive result has the disease? Only 11 answered correctly, and the most common answer, 95%, was given by 27. In natural frequencies the answer is visible: of 1,000 people, 1 has the disease and about 50 of the other 999 test positive, so about 1 in 51, or 2%, if the test detects every case. Hoffrage and colleagues (2000) showed that presenting the same information as natural frequencies instead of probabilities considerably reduces these errors.

Sensitivity, specificity, predictive values and likelihood ratios are explained in the diagnostic tests chapter. This chapter uses them to update probabilities, combine tests and make decisions.

Bayes' rule

How do you turn the probability of a result into the probability of disease?

Studies tell you how often the test is positive in people with the disease. Your patient needs the reverse: how often people with a positive test have the disease. Bayes' rule converts one into the other.

P(D | +) = P(+ | D) × P(D) ÷ P(+)

P(+) = P(+ | D) × P(D) + P(+ | no D) × P(no D)

P(D | +) = Se × p ÷ [Se × p + (1 − Sp) × (1 − p)] = 0.80 × 0.20 ÷ (0.16 + 0.08) = 0.667

P(D | +)

probability of disease given a positive result (the post-test probability, equal to the PPV in this population)

p

pre-test probability of disease, 0.20

Se

sensitivity, P(+ | D) = 0.80

1 − Sp

false-positive rate, P(+ | no D) = 0.10

The same rule written with odds is shorter, and it is the form used at the bedside:

post-test odds = pre-test odds × likelihood ratio → 0.25 × 8 = 2 → 2 ÷ (1 + 2) = 0.667

LR+ = Se ÷ (1 − Sp) = 0.80 ÷ 0.10 = 8. For a negative result, LR− = (1 − Se) ÷ Sp = 0.20 ÷ 0.90 = 0.22, and the probability falls to 5.3%. Reading P(positive | disease) as if it were P(disease | positive) is the error in the Casscells question.

Pre-test probability

Where does the probability before the test come from?

Every post-test probability depends on the pre-test probability you put in. It is an estimate for this patient, not a fixed number for the disease.

1. Prevalence in your setting: how common the disease is among patients like this one who present to your clinic, ward or programme.
2. History and examination: each finding raises or lowers that starting probability. Findings are themselves tests with likelihood ratios.
3. Validated clinical prediction scores: where a score has been tested in patients like yours, it gives a probability directly.
4. Your own estimate: if nothing else is available, state a range (for example 10% to 30%) and check whether the decision changes across it.

How much does the test add?

The change from the pre-test to the post-test probability is the gain from the test. With test A (LR+ 8, LR− 0.22), the gain is largest when the pre-test probability is in the middle range, and small when you are already nearly sure.

Teaching values. At 80%, a negative result still leaves almost a 1 in 2 chance of disease, so it cannot rule the disease out.

Combining results

Can the result of one test become the starting point for the next?

Yes. The post-test probability after test A becomes the pre-test probability for test B. With odds, you multiply by each likelihood ratio in turn.

post-test odds = pre-test odds × LR_A × LR_B = 0.25 × 8 × 18 = 36 → 97.3%

This multiplication is valid only if the tests are conditionally independent: among patients with the disease, a positive A does not make a positive B more likely, and the same holds among patients without the disease. In practice, tests that detect the same abnormality, or react to the same interfering condition, tend to make the same mistakes. Then test B's likelihood ratio in patients who are already positive on A is smaller than its published value, and multiplying overstates the post-test probability.

What to do: use a likelihood ratio for the combination, measured in studies where both tests were done on the same patients, or report the chained result as an upper estimate and say that independence was assumed.

Interactive

Make the two tests share their errors and watch the post-test probability fall

Two tests in a pathway

Should you ask for both tests to be positive, or either?

Serial testing calls a patient positive only if both tests are positive. Parallel testing calls a patient positive if either test is positive. The choice changes the combined sensitivity and specificity.

Serial: Se = Se_A × Se_B = 0.72 · Sp = 1 − (1 − Sp_A)(1 − Sp_B) = 0.995

Parallel: Se = 1 − (1 − Se_A)(1 − Se_B) = 0.98 · Sp = Sp_A × Sp_B = 0.855

• Serial testing raises specificity and PPV (97.3% here) and lowers sensitivity and NPV (93.4%). It suits confirming a diagnosis before a treatment that is risky, expensive or lifelong, and it saves tests: B is needed in only 240 patients.
• Parallel testing raises sensitivity and NPV (99.4% here) and lowers specificity and PPV (62.8%). It suits urgent settings where missing the disease is dangerous and both tests can be done at once.
• Order in a serial pathway does not change the combined sensitivity and specificity, but it changes how many second tests are needed. Starting with B would need test A in 220 patients instead of test B in 240. Put the cheaper or safer test first.
• Both sets of formulas assume conditional independence. With related tests, serial testing gains less specificity and parallel testing gains less sensitivity than the formulas show.

Interactive

Send 1,000 patients through one test, two tests in series, or two in parallel, and watch PPV and NPV

Comparison

Serial or parallel testing: which does what?

Decisions

When is a test worth doing at all?

A test is useful only if its result could change what you do. If you would treat whatever the result, or would not treat whatever the result, the test adds cost and risk without changing the decision.

In 1975 Pauker and Kassirer described a treatment threshold: the probability of disease above which treating is better than not treating. It depends on the benefit of treating a patient who has the disease (B) and the harm of treating a patient who does not (H):

treatment threshold = H ÷ (H + B) = 1 ÷ (1 + 9) = 10%

In 1980 they added two more thresholds for when a test is available. Below the testing threshold, withhold treatment without testing. Above the test-treatment threshold, treat without testing. Between the two, test and let the result decide. Setting the expected values of the three strategies equal gives, for a test with no risk of its own:

testing threshold = (1 − Sp) × H ÷ [(1 − Sp) × H + Se × B] = 0.10 ÷ (0.10 + 7.2) = 1.4%

test-treatment threshold = Sp × H ÷ [Sp × H + (1 − Se) × B] = 0.90 ÷ (0.90 + 1.8) = 33.3%

Benefit and harm must be in the same units: deaths prevented, quality-adjusted life years, or a value agreed with the patient. The thresholds are only as good as those estimates, so check whether the decision changes across a plausible range.

Interactive

Change the benefit, harm and test accuracy, and watch the testing and test-treatment thresholds move

Decision analysis

How do you compare strategies whose outcomes have different values?

Draw a decision tree. Each strategy branches into its possible outcomes with their probabilities and values. The expected value of a strategy is the sum of probability × value over its outcomes.

1. Decision node (square): the choice you control, such as treat, test first or wait.
2. Chance nodes (circles): events you do not control, each with a probability. The probabilities leaving a node add up to 1.
3. End points: each outcome gets a value, such as survival, quality-adjusted life years, cost, or a utility from 0 (death) to 1 (full health).
4. Calculate from the right: at each chance node, multiply each outcome's value by its probability and add them. At the decision node, choose the branch with the highest expected value.
5. Sensitivity analysis: change each uncertain input over its plausible range and see whether the best strategy changes. The thresholds above are the points where it does.

EV(test A first) = 0.16 × 9 + 0.04 × 0 + 0.08 × (−1) + 0.72 × 0 = 1.36

Expected value describes the average over many patients like this one. The values placed on outcomes are judgements, and they should reflect what matters to patients.

Bayesian statistics

What does a Bayesian analysis of a trial add to a p-value?

A p-value is the probability of data at least this extreme if the treatment had no effect. A Bayesian analysis answers the question clinicians usually ask: given these data, how probable is it that the treatment works?

It uses the same logic as the diagnostic test, with the treatment effect in place of the disease:

• Prior: what was believed about the effect before the trial, written as a probability distribution. A sceptical prior is centred on no effect, an enthusiastic prior on a benefit, and a non-informative (flat) prior adds almost nothing to the data.
• Likelihood: what the trial data say about each possible effect.
• Posterior: the prior updated by the data. From it you read a point estimate, a 95% credible interval (the range with a 95% probability of containing the effect, given the prior and the model), and the posterior probability of benefit, such as P(risk ratio < 1) or P(risk ratio < 0.90).

ICH E9 defines Bayesian approaches as analyses that give a posterior probability distribution for a parameter such as the treatment effect, derived from the observed data and a prior distribution.

A simple calculation with the normal approximation

On the log risk-ratio scale, a normal prior and a normal likelihood give a normal posterior. Its precision (1 ÷ variance) is the sum of the two precisions, and its mean is their precision-weighted average.

1 ÷ SD_post² = 1 ÷ SD_prior² + 1 ÷ SE²

mean_post = (mean_prior ÷ SD_prior² + log RR ÷ SE²) × SD_post²

Made-up trial: risk ratio for 30-day death 0.80, standard error of log RR 0.13 (95% confidence interval 0.62 to 1.03, p = 0.086). With a sceptical prior centred on RR 1 (SD 0.15 on the log scale):

precision 44.4 + 59.2 = 103.6 · mean = (0 × 44.4 + (−0.223) × 59.2) ÷ 103.6 = −0.127 · SD = 0.098

posterior RR 0.88, 95% credible interval 0.73 to 1.07 · P(RR < 1) = 90.3%

Uses in trials: reading a result with a non-significant p-value as a probability of benefit and of clinically important benefit, adding information from earlier studies through the prior, and writing decision rules for interim analyses in adaptive designs as posterior probabilities. In January 2026 the US FDA published a draft guidance on using Bayesian methods for the primary analysis of the main trials that support the effectiveness and safety of drugs and biological products.

Cautions: the prior must be chosen and justified before the data are seen, and results should be shown under a range of priors, from sceptical to enthusiastic. With little data, the prior dominates the result.

Interactive

Choose a prior and move the trial result, and watch the probability of benefit change

Comparison

Credible interval or confidence interval: what does each 95% mean?

The confidence interval is explained in the hypothesis testing chapter.

Decide

Which approach answers your question?

Start from the decision you face.

The formula

Worked example: the patient at 20%, step by step

All numbers are made-up teaching values, calculated in Python.

1. Pre-test probability 20%, so pre-test odds = 0.20 ÷ 0.80 = 0.25.
2. Test A: sensitivity 80%, specificity 90%. LR+ = 0.80 ÷ 0.10 = 8. LR− = 0.20 ÷ 0.90 = 0.222.
3. Test A positive: odds 0.25 × 8 = 2, probability 2 ÷ 3 = 66.7%. Test A negative: odds 0.25 × 0.222 = 0.056, probability 5.3%.
4. Test B: sensitivity 90%, specificity 95%, LR+ = 0.90 ÷ 0.05 = 18. Both positive, assuming independence: odds 2 × 18 = 36, probability 36 ÷ 37 = 97.3%.
5. If the tests were strongly related (50% of the largest possible overlap in the interactive element), B's likelihood ratio after a positive A would be 3.5, and the probability after both would be 87.4%.
6. Serial pathway in 1,000 patients: B done in 240, 148 called positive, combined sensitivity 72%, specificity 99.5%, PPV 97.3%, NPV 93.4%.
7. Parallel pathway: 312 called positive, sensitivity 98%, specificity 85.5%, PPV 62.8%, NPV 99.4%.
8. Thresholds with benefit 9 and harm 1: treatment threshold 10%, testing threshold 1.4%, test-treatment threshold 33.3%. At 20%, test first.
9. Expected values at 20%: no treatment 0, treat everyone 0.20 × 9 − 0.80 × 1 = 1.00, test A first 1.36.

P(D | A+, B+) = 1 ÷ [1 + (1 − p) ÷ (p × LR_A × LR_B)] = 1 ÷ (1 + 0.8 ÷ 28.8) = 97.3%

In a real paper

Bayesian reanalysis and post-test probability in published studies

Software

How to run it

Most of the calculations in this chapter need only arithmetic. Software helps with confidence intervals for test accuracy and with Bayesian models.

R

# Post-test probability from a pre-test probability and a likelihood ratio
pre     <- 0.20
lr_pos  <- 0.80 / (1 - 0.90)            # LR+ = 8
odds    <- pre / (1 - pre) * lr_pos     # 2
odds / (1 + odds)                       # 0.667

# Sensitivity, specificity, PPV, NPV and likelihood ratios with 95% CIs
# epiR: dat = c(a, b, c, d) = c(TP, FP, FN, TN)
library(epiR)
epi.tests(c(160, 80, 40, 720), method = "exact", conf.level = 0.95)

# Normal-approximation posterior for a trial risk ratio
y   <- log(0.80); se <- 0.13            # trial estimate and SE of log RR
m0  <- 0;         s0 <- 0.15            # sceptical prior on log RR
w   <- c(1 / s0^2, 1 / se^2)
post_sd   <- sqrt(1 / sum(w))
post_mean <- (m0 * w[1] + y * w[2]) / sum(w)
exp(post_mean + c(-1, 1) * qnorm(0.975) * post_sd)   # 95% credible interval
pnorm(0, mean = post_mean, sd = post_sd)             # P(RR < 1) = 0.903

Functions checked against the documentation: epi.tests(dat, method = "exact", digits = 2, conf.level = 0.95) in the epiR package (version 2.0.96), where a, b, c and d are test-positive/outcome-positive, test-positive/outcome-negative, test-negative/outcome-positive and test-negative/outcome-negative counts. pnorm and qnorm are in base R (stats). Full Bayesian models, such as the one in the PROPPR reanalysis, can be fitted with the brms package.

SPSS

IBM SPSS Statistics (version 25 and later) has Bayesian procedures for continuous outcomes. For two independent groups: Analyze > Bayesian Statistics > Independent Samples Normal. Under Bayesian Analysis choose Characterize Posterior Distribution, Estimate Bayes Factor or Use Both Methods, and set the credible interval percentage and the priors in the dialog.

Reporting

How to report it

State where every input came from, what was assumed, and the result with its uncertainty.

Post-test probability and combined tests

“We estimated the pre-test probability of disease as 20% from the prevalence in our clinic. A positive result on test A (likelihood ratio 8.0) raised the probability to 67%. A second positive result on test B (likelihood ratio 18) raised it to 97%, assuming that the two tests are conditionally independent. In a serial pathway (test B only after a positive test A), the combined sensitivity was 72% and the combined specificity 99.5%.”

Bayesian analysis of a trial

“In a pre-specified Bayesian analysis with a sceptical prior centred on no effect (normal on the log risk-ratio scale, SD 0.15), the posterior risk ratio for 30-day mortality was 0.88 (95% credible interval 0.73 to 1.07). The posterior probability that the drug reduced mortality was 90.3%, and the probability of at least a 10% relative reduction was 58.9%. With a non-informative prior these probabilities were 95.7% and 81.7%.”

• Report the prior (distribution, centre, spread), why it was chosen, and results under at least one other prior.
• Report the model and software, and for full Bayesian models the sampling settings and convergence checks.
• Keep the frequentist estimate with its 95% confidence interval alongside: CONSORT 2025 item 26 asks for the estimated effect size and its precision for each outcome, and for binary outcomes both absolute and relative effect sizes.

By specialty

Bayes' rule and combined tests by specialty

Common myths

Common mistaken beliefs

• ✗Myth: A positive result on a test with 90% sensitivity means a 90% chance of disease.

  ✓Sensitivity is P(positive | disease). The chance of disease after a positive result depends on the pre-test probability and the false-positive rate.

• ✗Myth: Two positive tests always multiply their likelihood ratios.

  ✓Only if the tests are conditionally independent. Related tests add less information, and multiplying overstates the probability.

• ✗Myth: Parallel testing is always better because it misses fewer cases.

  ✓It also lowers specificity and PPV and needs more tests. In the teaching example PPV fell from 97.3% (serial) to 62.8%.

• ✗Myth: A serial pathway keeps the sensitivity of the first test.

  ✓Combined sensitivity is Se_A × Se_B, lower than either test: 72% from tests of 80% and 90%.

• ✗Myth: Every available test should be done.

  ✓Below the testing threshold or above the test-treatment threshold, no result would change management.

• ✗Myth: A 93% posterior probability of benefit proves the treatment works.

  ✓It depends on the prior and the model, and a post hoc analysis is exploratory. Look at the probability of a clinically important benefit and at other priors.

Why this matters to you

Five consequences of misusing Bayes' rule and combined tests

1. Overconfident diagnosis

   Chaining related tests gives a post-test probability that is too high, and further investigation stops too early.

2. Unnecessary tests

   Tests ordered below the testing threshold or above the test-treatment threshold add cost and risk without changing management.

3. A common viva error

   Answering "90%" when asked how likely disease is after a positive result on a test with 90% sensitivity.

4. Misreading trials

   A non-significant trial is called negative, or a Bayesian probability from a flat prior is presented as independent confirmation.

5. Harm to patients

   A parallel pathway used for confirmation labels many healthy patients as diseased. A serial pathway used for exclusion misses cases.

History

History of Bayesian reasoning in medicine

1. 1763

   Thomas Bayes and Richard Price

   "An essay towards solving a problem in the doctrine of chances" by the late Rev. Thomas Bayes was communicated by Richard Price and published in the Philosophical Transactions of the Royal Society.

2. 1959

   Robert Ledley and Lee Lusted

   Published "Reasoning foundations of medical diagnosis" in Science, describing how symbolic logic, probability and value theory can explain how physicians reason.

3. 1975

   Stephen Pauker and Jerome Kassirer

   Described a therapeutic threshold: above it, treating is preferable, below it, withholding treatment is preferable.

4. 1978

   Ward Casscells and colleagues

   Asked 60 students, house officers and physicians at Harvard Medical School to calculate the chance of disease after a positive result. 11 answered correctly, and the most common answer was 95%.

5. 1980

   Pauker and Kassirer

   Published "The threshold approach to clinical decision making", with a testing threshold and a test-treatment threshold.

6. 1995

   Gerd Gigerenzer and Ulrich Hoffrage

   Published "How to improve Bayesian reasoning without instruction: frequency formats" in Psychological Review.

7. 1998

   ICH E9

   The statistical principles guideline for clinical trials stated that Bayesian and other approaches may be considered when the reasons for their use are clear and the conclusions do not depend on the particular assumptions and methods chosen.

8. 2014

   Arjun Manrai and colleagues

   Repeated the Casscells question in Boston. 14 of 61 respondents (23%) answered correctly, and 95% was again the most common answer.

9. 2026

   US Food and Drug Administration

   Published the draft guidance "Use of Bayesian Methodology in Clinical Trials of Drug and Biological Products" (notice of availability, 12 January 2026).

Guidelines

Bayesian methods and test results in guidelines

Reading a paper

Five checks when a paper uses Bayes' rule, combined tests or a Bayesian analysis

• Is the pre-test probability stated, and does it match patients like yours?
• When two tests are combined, were they studied together in the same patients, or were separate accuracies multiplied under an independence assumption?
• For a serial or parallel pathway, are the combined sensitivity, specificity and number of tests reported, not only one of them?
• In a Bayesian analysis, is the prior stated, justified before the data, and varied in a sensitivity analysis?
• Is the probability of a clinically important effect reported, and not only the probability of any effect?

Why is it called that?

Word surgery

Exam answers

Model answer outlines

Past papers

TNMGRMU questions on bayes' rule and combining tests

Questions from MD, MS, DM, MCh and Ph.D. papers that this chapter helps you answer. Tap a sitting to open the university's own question paper.

1. ×1Bayesian methods and its application in trialsMCh Surgical Oncology · Feb 2025
2. ×1Post-test probabilityDM Neonatology · Aug 2024
3. ×1Bayesian methods in statisticsDM Medical Oncology · Aug 2021
4. ×1Describe Bayes theorem and its clinical applicationsDM Cardiology · Aug 2012