Bayes' Theorem Calculator
Apply Bayes' theorem to update probabilities with new evidence — compute posterior probability, likelihood ratios, and confusion matrices for medical tests.
Post-Test Probability Calculator
Calculate post-test probability from sensitivity, specificity, and prevalence with likelihood ratios, confusion matrix, sequential testing, and PPV/NPV sensitivity analysis.
True positive rate
True negative rate
Number of sequential positive tests
Post-test Prob (+ result)⧉
48.65%
PPV: If test is positive, 48.65% truly have the condition
Post-test Prob (− result)⧉
0.55%
NPV: 99.45% truly disease-free if negative
LR+⧉
18.00
✅ Strong rule-in
LR−⧉
0.1053
Good rule-out
Accuracy⧉
94.75%
(TP + TN) / Total
Youden Index⧉
0.8500
Sens + Spec − 1 (0 = useless, 1 = perfect)
Probability Shift
Pre-test: 5.00%
Post-test (+): 48.65%
Post-test (−): 0.55%
Confusion Matrix (per 10,000)
| Disease + | Disease − | Total | |
|---|---|---|---|
| Test + | 450 (TP) | 475 (FP) | 925 |
| Test − | 50 (FN) | 9025 (TN) | 9075 |
| Total | 500 | 9500 | 10,000 |
PPV / NPV vs Prevalence
| Prevalence | PPV | NPV | PPV Bar |
|---|---|---|---|
| 0.1% | 1.77% | 99.99% | |
| 0.5% | 8.29% | 99.95% | |
| 1% | 15.38% | 99.89% | |
| 2% | 26.87% | 99.79% | |
| 5% | 48.65% | 99.45% | |
| 10% | 66.67% | 98.84% | |
| 20% | 81.82% | 97.44% | |
| 30% | 88.52% | 95.68% | |
| 50% | 94.74% | 90.48% | |
| 70% | 97.67% | 80.28% |
Planning notes, formulas, and examples
About the Post-Test Probability Calculator
The post-test probability calculator shows how a diagnostic test result changes the probability of disease. Starting from a pre-test probability, it combines sensitivity and specificity to estimate the probability after a positive or negative result.
This is a direct application of Bayes' theorem to medical testing. It shows why positive predictive value and negative predictive value depend heavily on prevalence: the same test can look strong in one setting and weak in another.
Enter the test characteristics and prevalence to see the confusion matrix, likelihood ratios, sequential-testing effect, and how predictive values change across different prevalence levels.
When This Page Helps
This calculator is useful when the question is not just whether a test is positive, but what that result actually means in context. Prevalence can dominate the answer, so the same sensitivity and specificity can produce very different post-test probabilities.
Showing the confusion matrix, likelihood ratios, and predictive values together makes it easier to see whether a test result is genuinely informative or just looks persuasive on the surface.
How to Use the Inputs
- Enter the test's sensitivity (true positive rate) as a percentage.
- Enter the test's specificity (true negative rate) as a percentage.
- Enter the pre-test probability or disease prevalence.
- Set the number of sequential tests to see how repeated positives increase certainty.
- Review the confusion matrix per 10,000 people to understand false positive/negative counts.
- Study the PPV/NPV vs prevalence table to see prevalence effects.
- Use presets for common medical tests.
Formula used
LR+ = Sensitivity / (1 − Specificity). LR− = (1 − Sensitivity) / Specificity. Post-test odds = Pre-test odds × LR. PPV = (Sens×Prev) / (Sens×Prev + (1−Spec)×(1−Prev)).Example Calculation
Result: PPV = 48.6%, NPV = 99.4%, LR+ = 18
At 5% prevalence, a positive result raises probability from 5% to 48.6%. A negative result lowers it to 0.55%. LR+ of 18 is a strong diagnostic tool — each positive multiplies the odds by 18.
Tips & Best Practices
- LR+ > 10 strongly rules in disease; LR− < 0.1 strongly rules out. Between 1 and 10 the test has modest diagnostic power.
- PPV depends heavily on prevalence — a great test performs poorly in low-prevalence screenings due to many false positives.
- Sequential testing dramatically increases certainty — two positive results with LR+ of 18 multiply odds by 18² = 324.
- The Youden index (Sens + Spec − 1) ranges from 0 (useless) to 1 (perfect) — a quick summary of test quality.
- NPV is almost always high in low-prevalence settings because most people truly don't have the disease.
- Consider the clinical context: a SnNOut rule (sensitive test, negative result rules out) vs SpPIn rule (specific test, positive result rules in).
The Base Rate Fallacy in Screening
Mass screening for rare diseases suffers from the base rate fallacy. Even with a 99% sensitive and 99% specific test, screening for a disease with 0.1% prevalence produces 10× more false positives than true positives (PPV ≈ 9%). This is why targeted testing based on clinical risk factors is preferred over universal screening.
ROC Curves and Optimal Thresholds
The ROC curve plots sensitivity vs (1−specificity) across all possible test thresholds. The area under the ROC curve (AUC) summarizes overall test performance. This calculator evaluates a single point on the ROC curve — the chosen threshold that determines the specific sensitivity/specificity trade-off.
Sequential and Parallel Testing Strategies
Sequential testing (test again if positive) maximizes specificity with each step but may miss cases. Parallel testing (confirm if either positive) maximizes sensitivity. The optimal strategy depends on the cost of false positives vs false negatives in the clinical context.
Sources & Methodology
Last updated:
Frequently Asked Questions
At 1% prevalence, 99% of people are disease-free. Even a 95% specific test produces false positives in 5% of 99 = ~5 people, while catching 1% of 1 person with disease. Most positives are false — hence low PPV.
Sensitivity asks: of those WITH disease, what fraction tests positive? PPV asks: of those who TEST positive, what fraction has disease? Sensitivity is a test property; PPV depends on prevalence.
LR transforms pre-test odds to post-test odds. Convert probability to odds (p/(1−p)), multiply by LR, convert back. LR+ applies to positive results, LR− to negative results.
Yes! The same framework applies to any binary classifier: spam detection, quality control, fraud detection. Sensitivity = recall, PPV = precision in machine learning terminology.
If the two tests are independent and measure the result separately, sequential testing multiplies likelihood ratios. If it is just the same test repeated, the independence assumption is weaker.
A graphical tool connecting pre-test probability, likelihood ratio, and post-test probability on three scales. Drawing a line from pre-test through LR gives post-test probability. It gives the same information numerically.
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