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Fagan NomogramFree in-browser calculator

Bayes Calculator.

Interactive Fagan nomogram for Bayesian post-test probability. Enter pretest probability and likelihood ratio — see the update instantly. Data never leaves your browser.

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Validated2026-04-05
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When to use

  • Convert pretest probability to post-test probability using a known likelihood ratio
  • Visualize Bayesian diagnostic reasoning on an interactive Fagan nomogram
  • Determine how much a positive or negative test result shifts disease probability
  • Quantify diagnostic information gain in bits using Shannon entropy
  • Compute post-test probability from sensitivity, specificity, and a test result

Do not use for

  • Computing sensitivity, specificity, or likelihood ratios from raw data — use the Diagnostic Test Calculator
  • Finding the optimal threshold for a continuous-score test — use the ROC/AUC Calculator
  • Comparing two measurement methods — use the Method Comparison Analyzer

A positive test does not mean you have the disease

A positive result shifts probability by a factor determined by the likelihood ratio. A highly specific test (LR+ = 20) applied to a low-prevalence condition (pretest = 1%) still yields a modest post-test probability of about 17%. Always anchor to pretest probability.

Odds form makes Bayesian updates simple

Post-test odds = pretest odds x LR. This multiplicative form is far simpler than the full Bayes formula with normalizing denominators. Convert back to probability only at the end: probability = odds / (1 + odds).

An LR of 1 means the test is useless

A likelihood ratio of 1 means the test result is equally likely in diseased and non-diseased patients — it provides zero diagnostic information. LR+ > 10 or LR- < 0.1 represents strong evidence.

Sequential testing multiplies likelihood ratios

When applying multiple independent tests, multiply their LRs: post-test odds = pretest odds x LR1 x LR2. This is how clinicians combine evidence from history, exam, and lab tests in a stepwise fashion.

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Method

Post-test probability is computed using Bayes' theorem in odds form: post-test odds = pretest odds ×\times likelihood ratio, then converted back to probability. The Fagan nomogram is rendered as an interactive SVG with draggable anchor points. Shannon entropy H(p)=−p⋅log⁡2(p)−(1−p)⋅log⁡2(1−p)H(p) = -p \cdot \log_2(p) - (1-p) \cdot \log_2(1-p) quantifies diagnostic uncertainty before and after testing. All calculations run client-side in the browser.

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Validated

Last validated 2026-04-05. Calculations are designed for planning and documentation support; verify procurement decisions against manufacturer specifications or institutional SOPs.

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How to cite

How to Cite

ConductScience Bayes Calculator (v1.0). ConductScience, Inc. 2026. Available at: https://conductscience.com/tools/bayes-calculator

Fagan TJ. Letter: Nomogram for Bayes theorem. N Engl J Med. 1975;293(5):257. doi:10.1056/NEJM197507312930513

Bayesian Reasoning in Diagnostics

Bayes' theorem connects prior beliefs to updated beliefs through evidence. In diagnostic testing:

Pretest probability — the probability of disease before the test, based on prevalence, clinical presentation, and prior tests.
Likelihood ratio — how much more likely the test result is in diseased vs non-diseased patients.
Post-test probability — the updated probability of disease after incorporating the test result.

The key insight: a positive test does NOT mean you have the disease. It shifts the probability by a factor determined by the likelihood ratio. A highly specific test (LR+ = 20) applied to a low-prevalence condition (pretest = 1%) still yields a modest post-test probability (~17%).

Information Theory Perspective

Shannon entropy H(p)=−p⋅log⁡2(p)−(1−p)⋅log⁡2(1−p)H(p) = -p \cdot \log_2(p) - (1-p) \cdot \log_2(1-p) measures diagnostic uncertainty in bits. Before testing, uncertainty is H(pretest). After testing, uncertainty drops to H(posttest).

The information gain — the drop in entropy — quantifies how useful the test is. A test that moves probability from 50% to 90% reduces uncertainty by more bits than one that moves it from 90% to 95%, even though the absolute change is larger in the second case.

This provides a principled way to compare tests: the test with higher expected information gain is more diagnostically useful, regardless of whether it is used for ruling in or ruling out.

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