Post-Test Probability Calculator
For a positive result, enter LR+. For a negative result, enter LR−. An LR of 1 means the test provides no diagnostic information.
Run a calculation in either of the two tabs first, then return here to see the complete step-by-step Bayesian update.
No data yet — enter values in the Likelihood Ratio or Sensitivity + Specificity tab first.
What Is Post-Test Probability?
Post-test probability is the estimated probability that a condition is present after combining a pre-test probability with the result of a diagnostic test. It answers the practical question: given that a patient tested positive (or negative), what is the updated probability of the condition?
Before any test is run, a clinician or researcher starts with a pre-test probability — their best estimate of how likely the condition is. That estimate may come from disease prevalence in the population, clinical findings, patient history, or prior test results. The diagnostic test then produces a result, and Bayes' theorem provides the mathematical framework for updating that starting probability.
A positive result generally raises the probability; a negative result generally lowers it. How much the probability changes depends on two things: where it started (the pre-test probability) and how informative the test is (the likelihood ratio).
Bayes' Theorem and Diagnostic Testing
Bayes' theorem provides the foundation for calculating post-test probability. In diagnostic testing, it relates the probability of a condition given a test result to the probability of the test result given the condition:
Bayes' Theorem (Probability Form)
P(D|T) = [P(T|D) × P(D)] ÷ P(T)
Bayes' Theorem (Odds Form)
Post-test odds = Pre-test odds × LR
Where D = condition present, T = positive test result, P(D) = pre-test probability, P(D|T) = post-test probability, and P(T|D) relates to test sensitivity.
The odds form is far more practical for diagnostic reasoning. Rather than tracking separate conditional probabilities, you simply multiply the pre-test odds by the relevant likelihood ratio. The calculator above uses this exact method.
The Bayesian Update Process — Step by Step
Post-test probability requires three steps: convert probability to odds, multiply by the likelihood ratio, then convert odds back to probability. Here is the full method:
Estimate how likely the condition is before the test. For example, a condition affects 10% of the population being tested, so the pre-test probability = 10% = 0.10.
Pre-test odds = P ÷ (1 − P) = 0.10 ÷ 0.90 = 0.1111. Odds express the ratio of the probability the condition is present to the probability it is absent.
If the test has sensitivity = 90% and specificity = 95%, then LR+ = 0.90 ÷ (1 − 0.95) = 0.90 ÷ 0.05 = 18. For a positive result, use LR+.
Post-test odds = Pre-test odds × LR+ = 0.1111 × 18 = 2.000.
Post-test probability = 2.000 ÷ (1 + 2.000) = 2.000 ÷ 3.000 = 66.7%. The test raised the probability from 10% to 66.7%.
Result (hypothetical educational example): Pre-test probability = 10%, Sensitivity = 90%, Specificity = 95%, LR+ = 18, Pre-test odds = 0.111, Post-test odds = 2.00, Post-test probability = 66.7%. You can verify this using the Sensitivity + Specificity tab of the calculator above.
Likelihood Ratios: LR+ and LR−
A likelihood ratio (LR) quantifies how much a test result changes the pre-test probability of a condition. It is the single most useful summary of a diagnostic test's performance for clinical decision-making because it directly translates into a probability shift via the odds form of Bayes' theorem.
Positive Likelihood Ratio (LR+)
LR+ = Sensitivity ÷ (1 − Specificity)
LR+ describes how much more likely a positive result is among people with the condition than among those without it. A test with sensitivity = 90% and specificity = 95% has LR+ = 0.90 ÷ 0.05 = 18. This means a positive result is 18 times more likely in someone with the condition than in someone without it.
Negative Likelihood Ratio (LR−)
LR− = (1 − Sensitivity) ÷ Specificity
LR− describes how much more likely a negative result is among people with the condition compared to those without it. Using the same test: LR− = (1 − 0.90) ÷ 0.95 = 0.10 ÷ 0.95 = 0.105. A negative result is about 0.1 times as likely in someone with the condition as in someone without it — meaning it substantially reduces the probability.
Table: Likelihood Ratio Interpretation Reference
| LR+ Value | Diagnostic Shift | Practical Interpretation |
|---|---|---|
| > 10 | Large | Positive result provides strong evidence for condition; shifts probability substantially |
| 5 – 10 | Moderate | Meaningful increase in probability; clinically relevant in moderate pre-test scenarios |
| 2 – 5 | Small | Modest increase; may be relevant when pre-test probability is already elevated |
| ~1 | None | Test result provides essentially no diagnostic information |
| LR− Value | Diagnostic Shift | Practical Interpretation |
|---|---|---|
| < 0.1 | Large | Negative result provides strong evidence against condition; substantially reduces probability |
| 0.1 – 0.2 | Moderate | Meaningful reduction in probability; clinically useful when pre-test probability is moderate |
| 0.2 – 0.5 | Small | Modest reduction; may still leave residual probability in high pre-test scenarios |
| ~1 | None | Negative result provides no diagnostic information |
Note: These thresholds are widely cited reference points, not absolute clinical rules. The clinical relevance of any probability shift depends on the specific condition, treatment threshold, and patient context. See McGee, S.R. (2002) in the National Library of Medicine for evidence-based likelihood ratio interpretation.
Sensitivity and Specificity: What They Actually Measure
Sensitivity and specificity describe how a test performs within known groups — they do not tell you the probability that a positive result is correct. Understanding this distinction prevents the most common errors in interpreting diagnostic tests.
Sensitivity (True Positive Rate)
Sens = TP ÷ (TP + FN)
P(Test + | Condition Present)
Specificity (True Negative Rate)
Spec = TN ÷ (TN + FP)
P(Test − | Condition Absent)
Sensitivity answers: among people who actually have the condition, what fraction test positive? A sensitivity of 90% means 90% of true cases are detected; 10% are missed (false negatives).
Specificity answers: among people who do not have the condition, what fraction test negative? A specificity of 95% means 95% of true non-cases test negative; 5% test positive falsely (false positives).
Both sensitivity and specificity are properties of the test itself and do not change with disease prevalence. This is what makes them useful for characterizing a test's intrinsic performance. However, neither tells you the probability that a positive test result reflects a true case — that depends on prevalence, which is why post-test probability (and PPV) are prevalence-dependent.
Pre-Test Probability: What It Is and How It Is Estimated
Pre-test probability is the estimated probability of a condition before a diagnostic test is applied. It is the starting point for every Bayesian update in diagnostic reasoning.
Pre-test probability often reflects disease prevalence in the relevant population, but it is not always the same as prevalence. An individual's pre-test probability may be higher or lower than the population prevalence based on clinical context:
- Presenting symptoms and their specificity for the condition
- Patient age, sex, and demographics
- Known risk factors and medical history
- Physical examination findings
- Results of prior diagnostic tests
- Clinical prediction rules and validated scores
The same diagnostic test can produce very different post-test probabilities depending on where the pre-test probability starts. A test with LR+ = 10 will raise a pre-test probability of 1% to approximately 9%, but the same test applied to a 30% pre-test probability produces a post-test probability of around 81%. This is why applying population-level test results to individual patients without considering context can be misleading.
PPV, NPV, and Post-Test Probability: How They Relate
Positive predictive value (PPV) and negative predictive value (NPV) are specific versions of post-test probability calculated for a defined population. They share the same Bayesian logic but differ in scope.
Table: Sensitivity, Specificity, PPV, NPV — Comparison
| Measure | Question Answered | Depends on Prevalence? | Fixed Property of Test? |
|---|---|---|---|
| Sensitivity | Among those with condition, how many test positive? | No | Yes |
| Specificity | Among those without condition, how many test negative? | No | Yes |
| PPV | Given a positive test, what is the probability of the condition? | Yes | No |
| NPV | Given a negative test, what is the probability of absence? | Yes | No |
| Post-Test Probability | What is the probability of condition after this specific test result? | Yes (via pre-test probability) | No |
When PPV is calculated using population prevalence as the pre-test probability, it equals the post-test probability for a positive result. When an individual's clinical pre-test probability differs from population prevalence, the personalized post-test probability will differ from the population-level PPV. This is why the likelihood ratio method — which works from any starting pre-test probability — is more flexible than looking up PPV tables.
Probability vs. Odds: A Practical Distinction
Probability and odds express the same underlying frequency in different scales. The calculation uses odds as an intermediate step because likelihood ratios are designed to multiply odds directly.
Probability to Odds
Odds = P ÷ (1 − P)
Example: P = 0.20
Odds = 0.20 ÷ 0.80 = 0.25
Odds to Probability
P = Odds ÷ (1 + Odds)
Example: Odds = 0.25
P = 0.25 ÷ 1.25 = 0.20
Table: Probability vs. Odds — Side-by-Side Comparison
| Probability | Odds | Plain English |
|---|---|---|
| 1% (0.01) | 0.0101 | 1 in 100 chance |
| 10% (0.10) | 0.1111 | 1 in 10 chance |
| 20% (0.20) | 0.25 | 1 in 5 chance |
| 50% (0.50) | 1.00 | Even odds |
| 75% (0.75) | 3.00 | 3 times as likely to occur as not |
| 90% (0.90) | 9.00 | 9 times as likely to occur as not |
📊 Worked Examples
All examples below are hypothetical and are provided for educational purposes only. They do not represent advice about any specific clinical situation.
Example 1 — Positive Test with Low Pre-Test Probability
LR+ = 0.85 ÷ (1 − 0.92) = 0.85 ÷ 0.08 = 10.625
Odds = 0.05 ÷ 0.95 = 0.0526
0.0526 × 10.625 = 0.5591
0.5591 ÷ 1.5591 = 35.9%
Interpretation (hypothetical example): Despite a positive result from a test with good performance, the starting probability of 5% limits the post-test probability to approximately 36%. This illustrates why the base rate matters: even a strong positive test cannot overcome a very low prior probability without more confirmatory evidence.
Example 2 — Negative Test with Moderate Pre-Test Probability
LR− = (1 − 0.90) ÷ 0.95 = 0.10 ÷ 0.95 = 0.1053
Odds = 0.30 ÷ 0.70 = 0.4286
0.4286 × 0.1053 = 0.04513
0.04513 ÷ 1.04513 = 4.3%
Interpretation (hypothetical example): A negative result from this test lowers the probability from 30% to approximately 4.3%. This is a meaningful reduction, though a residual probability of ~4% remains. Whether this is low enough for clinical purposes depends entirely on the specific condition, treatment implications, and clinical context.
Example 3 — Effect of Pre-Test Probability on the Same Test
Table: Post-Test Probability for Three Pre-Test Probabilities (LR+ = 18, hypothetical example)
| Pre-Test Probability | Pre-Test Odds | Post-Test Odds | Post-Test Probability |
|---|---|---|---|
| 1% | 0.0101 | 0.182 | 15.4% |
| 10% | 0.1111 | 2.000 | 66.7% |
| 30% | 0.4286 | 7.714 | 88.5% |
This table illustrates that the same LR+ = 18 produces dramatically different post-test probabilities depending on where you start. The pre-test probability is not just an input — it is the dominant factor determining how much clinical weight to place on the result.
Pre-Test Probability vs. Post-Test Probability
Table: Pre-Test vs. Post-Test Probability — Key Differences
| Feature | Pre-Test Probability | Post-Test Probability |
|---|---|---|
| When measured | Before the diagnostic test | After the test result is known |
| Uses test result? | No | Yes |
| Influenced by | Prevalence, symptoms, history, risk factors | Pre-test probability + test performance + result |
| Synonyms | Prior probability, baseline probability | Posterior probability, revised probability |
| Role in Bayes | The prior | The posterior |
| Becomes the next | Starting point for Bayesian update | Can become pre-test probability for a second test |
Sequential Bayesian Updating and Multiple Tests
When multiple diagnostic tests are available, Bayes' theorem can be applied sequentially: the post-test probability from one test becomes the pre-test probability for the next. This is the formal basis for sequential diagnostic reasoning.
However, sequential updating with multiple likelihood ratios is valid only when the tests are conditionally independent — meaning the result of one test does not affect the result of the other for any individual with or without the condition. When tests measure related biological processes or share systematic sources of error, their likelihood ratios cannot be multiplied directly. Doing so would overestimate the combined diagnostic value.
Common Misconceptions About Diagnostic Tests
Post-Test Probability: Complete Entity Reference
The table below defines every key concept associated with post-test probability and Bayesian diagnostic reasoning, formatted for quick reference.
| Term | Symbol / Formula | Definition |
|---|---|---|
| Pre-Test Probability | P(D) | Estimated probability of the condition before the diagnostic test is applied |
| Post-Test Probability | P(D|T) | Updated probability of the condition after incorporating the test result via Bayes' theorem |
| Prior Probability | P(D) | Synonymous with pre-test probability in Bayesian terminology |
| Posterior Probability | P(D|T) | Synonymous with post-test probability; the result of a Bayesian update |
| Likelihood Ratio | LR | Factor by which the pre-test odds are multiplied to obtain post-test odds |
| Positive Likelihood Ratio | LR+ = Sens ÷ (1−Spec) | How much more likely a positive test result is in those with the condition vs. those without |
| Negative Likelihood Ratio | LR− = (1−Sens) ÷ Spec | How much more likely a negative test result is in those with the condition vs. those without |
| Sensitivity | TP ÷ (TP+FN) | Proportion of true cases that test positive; the true positive rate |
| Specificity | TN ÷ (TN+FP) | Proportion of true non-cases that test negative; the true negative rate |
| Pre-Test Odds | P ÷ (1−P) | The ratio of the probability the condition is present to the probability it is absent, before testing |
| Post-Test Odds | Pre-odds × LR | Pre-test odds updated by the likelihood ratio; converted back to probability for the final answer |
| PPV | TP ÷ (TP+FP) | Positive predictive value; probability that a positive test result reflects a true case |
| NPV | TN ÷ (TN+FN) | Negative predictive value; probability that a negative test result reflects a true non-case |
| Prevalence | Cases ÷ Population | Proportion of a population with the condition at a given time; often used to estimate pre-test probability |
| True Positive (TP) | — | Condition present AND test positive; a correct positive result |
| True Negative (TN) | — | Condition absent AND test negative; a correct negative result |
| False Positive (FP) | — | Condition absent BUT test positive; an incorrect positive result |
| False Negative (FN) | — | Condition present BUT test negative; an incorrect negative result (missed case) |
| Bayes' Theorem | P(D|T) = [P(T|D)×P(D)] ÷ P(T) | Mathematical rule for updating probability based on new evidence; the basis of all post-test probability calculations |
References and Further Reading
- Sackett, D.L. et al. Clinical Epidemiology: A Basic Science for Clinical Medicine, 2nd ed. Little, Brown and Company, 1991. Foundational text on likelihood ratios and diagnostic reasoning.
- McGee, S.R. (2002). Simplifying likelihood ratios. Journal of General Internal Medicine, 17(8), 647–650. PubMed Central
- National Library of Medicine. Sensitivity and Specificity. StatPearls. ncbi.nlm.nih.gov
- Deeks, J.J. & Altman, D.G. (2004). Diagnostic tests 4: likelihood ratios. BMJ, 329, 168–169. BMJ
- Pewsner, D., Jäger, C. & Häniger, M. (2004). Summarising diagnostic tests: accuracy in clinical practice. BMJ. BMJ
- OpenStax. Introductory Statistics, Chapter on Probability. openstax.org
- Penn State STAT 415. Introduction to Mathematical Statistics — Bayesian Estimation. online.stat.psu.edu
Frequently Asked Questions
Post-test probability is the updated estimate of how likely a condition is after a diagnostic test result has been obtained. It combines the pre-test probability (how likely the condition was before the test) with the test result's likelihood ratio using Bayes' theorem. A positive test result raises the probability; a negative result lowers it. The magnitude of the shift depends on both the starting probability and the strength of the likelihood ratio.
First compute the likelihood ratio: LR+ = Sensitivity ÷ (1 − Specificity) for a positive result, or LR− = (1 − Sensitivity) ÷ Specificity for a negative result. Then convert the pre-test probability to odds: Odds = P ÷ (1 − P). Multiply the pre-test odds by the LR to get post-test odds. Finally, convert back: Post-test probability = Post-test odds ÷ (1 + Post-test odds). The Sensitivity + Specificity tab of the calculator above performs all these steps automatically.
A likelihood ratio quantifies how much a test result changes the probability of a condition. The positive likelihood ratio (LR+) = Sensitivity ÷ (1 − Specificity) tells you how much more likely a positive result is in someone with the condition versus someone without it. The negative likelihood ratio (LR−) = (1 − Sensitivity) ÷ Specificity tells you how much more likely a negative result is in someone with the condition. An LR of 1 means the test provides no diagnostic information; LR+ > 10 or LR− < 0.1 generally indicate a large diagnostic shift.
Pre-test probability is the estimated probability of a condition before a diagnostic test is applied — it is based on prevalence, symptoms, risk factors, and other clinical information. Post-test probability is the updated estimate after incorporating the test result. Pre-test probability is the prior in Bayesian terms; post-test probability is the posterior. The update between them is driven by the likelihood ratio of the test result.
Sensitivity is a fixed property of the test: P(Test Positive | Condition Present). It does not change with disease prevalence. PPV (positive predictive value) is a property of the test result in a particular population: P(Condition Present | Test Positive). PPV depends heavily on prevalence. In a low-prevalence population, a test with high sensitivity can still have low PPV because false positives outnumber true positives. This is why PPV alone cannot replace the likelihood ratio approach when pre-test probability varies.
Yes. Post-test probability is calculated from pre-test probability, and pre-test probability often reflects disease prevalence. The same test applied to two populations with different prevalence rates will produce different post-test probabilities, even if the sensitivity and specificity are identical. A test with LR+ = 10 applied to a 1% pre-test probability gives a post-test probability of approximately 9%, while the same LR+ applied to a 30% pre-test probability gives approximately 81%.
Odds = P ÷ (1 − P). For example, if P = 0.20, then Odds = 0.20 ÷ 0.80 = 0.25. To convert odds back to probability: P = Odds ÷ (1 + Odds). For Odds = 0.25: P = 0.25 ÷ 1.25 = 0.20. The odds form of Bayes' theorem uses this conversion because likelihood ratios are designed to multiply odds directly, making the calculation simpler than applying the full conditional probability formula.
An LR of exactly 1 means the test result provides no diagnostic information at all. When LR = 1, post-test odds = pre-test odds × 1 = pre-test odds, so the post-test probability equals the pre-test probability. The test neither raises nor lowers the probability of the condition. In practice, tests rarely have LR = 1 for both positive and negative results, but some tests have LR values very close to 1, indicating limited diagnostic usefulness.
Yes, in principle: the posterior from one test becomes the prior for the next, and each test result is applied via its likelihood ratio. However, this sequential approach is valid only if the tests are conditionally independent — meaning the result of one test does not influence the result of the other among people with or without the condition. Tests that measure the same underlying biological process or share systematic biases are not independent, and multiplying their likelihood ratios overestimates the combined diagnostic value.
PPV is a specific form of post-test probability calculated using population prevalence as the pre-test probability. When PPV is calculated for a general population and a specific patient's clinical pre-test probability equals that prevalence, PPV and post-test probability are numerically equivalent. When a patient's individual pre-test probability differs from population prevalence — because of symptoms, risk factors, or prior results — the personalized post-test probability will differ from the population PPV, and the likelihood ratio method gives the more accurate individual estimate.
A high LR− (close to 1) means that a negative result does not substantially change the probability of the condition — the test is not good at ruling out. Conversely, a low LR− (close to 0) means the negative result substantially reduces the probability, making the test useful for ruling out the condition. LR− < 0.1 is generally considered a large, clinically meaningful diagnostic shift downward.
Bayes' theorem is a mathematical rule for updating probabilities in light of new evidence. In diagnostic testing, it is expressed as: P(Disease | Test Result) = [P(Test Result | Disease) × P(Disease)] ÷ P(Test Result). The odds form — Post-test odds = Pre-test odds × Likelihood Ratio — is practically simpler and is the standard method used in clinical diagnostic reasoning and this calculator.