Post Test Probability Calculator - Fagan-style Bayes Update

The post test probability calculator takes a pre-test probability and a test likelihood ratio and returns the Bayes-rule post-test probability for that single test result. The same tool shows pre-test odds and post-test odds so the Fagan-style intermediate step stays visible, and it accepts the likelihood ratio directly or derives it from sensitivity and specificity when the test summary is what you have on hand.

• • Reading a single screening result: plug in the prevalence and the test sensitivity and specificity, then read the post-test probability for a positive or negative screen.
• • Updating a class prior: use a chosen prior probability and a published likelihood ratio to walk through the Bayes-rule update during a statistics lesson.
• • Comparing a strong and a weak test: swap the likelihood ratio in and out to see how a high-LR test and a low-LR test move the pre-test probability at the same prevalence.
• • Working through a homework problem: use the same calculator for a homework set so the intermediate odds, the LR, and the final probability all match the textbook Bayes-rule update.

The result panel lists the post-test probability, the pre-test odds, the post-test odds, the likelihood ratio, and a one-line reading. The same Bayes rule update is taught in epidemiology, evidence-based medicine, and introductory statistics, and it drives the Fagan nomogram used in clinical teaching.

When you want to work the same update from a 2x2 table of true positives, false positives, false negatives, and true negatives, the Bayes Theorem Calculator handles the cell counts and reads the posterior probability from there.

The post test probability calculator converts the pre-test probability into pre-test odds, multiplies by the likelihood ratio for the observed test result, and converts the post-test odds back to a probability. Sensitivity and specificity only matter when the LR source is the test summary, in which case the calculator derives LR+ or LR- first.

• prevalence: Pre-test probability of the condition, expressed as a percentage between 0.01% and 99.99%.
• testDirection: Positive test uses LR+; negative test uses the absolute value of LR-.
• sensitivity: Test sensitivity as a percentage. Used only when the LR source is sensitivity and specificity.
• specificity: Test specificity as a percentage. Used only when the LR source is sensitivity and specificity.
• likelihoodRatio: LR+ for a positive test or absolute value of LR- for a negative test when entered directly.

Pre-test odds sit next to post-test odds, and the likelihood ratio is repeated so the multiplier that drove the update is never hidden.

According to Wikipedia likelihood ratios in diagnostic testing, the positive likelihood ratio is sensitivity divided by 1 minus specificity, and the negative likelihood ratio is 1 minus sensitivity divided by specificity, with both restricted to non-negative values.

For a different statistics-for-students update that pairs naturally with a Bayes-rule lesson, the T-Test Calculator turns a sample mean and a sample standard deviation into a t-statistic and a p-value.

Four concepts carry the weight of a post test probability result. Naming them keeps the Bayes rule from being read as a black box.

A 90/90 test on a 1% prevalence gives a post-test probability of about 8%, a low absolute probability even after a positive screen.

According to Wikipedia pre- and post-test probability, post-test probability is calculated from pre-test probability by converting pre-test probability to pre-test odds, multiplying by the likelihood ratio, and converting post-test odds back to a probability.

When the published sensitivity or specificity comes with a confidence interval, the Confidence Interval Calculator is a useful companion for reading the uncertainty around the point estimate before you trust the LR.

Six steps cover both the sensitivity-and-specificity workflow and the direct likelihood ratio workflow. The same result panel works for both.

1. 1 Set the pre-test probability: enter the population prevalence or the individual prior as a percentage. The default 10% is a useful starting point for a quick worked example.
2. 2 Pick the observed test direction: choose positive test (use LR+) or negative test (use LR-). The same Bayes rule works in both directions, but the likelihood ratio is different.
3. 3 Choose the LR source: select sensitivity and specificity when the test summary is what you have, or likelihood ratio when the literature already reports an LR+ or LR- for the test.
4. 4 Enter sensitivity and specificity or the LR: type the percentage values for sensitivity and specificity, or type the LR+ (positive test) or absolute value of LR- (negative test) on the direct entry path.
5. 5 Read the post-test probability and the odds: treat the post-test probability, the pre-test odds, and the post-test odds as a set. The result panel also shows the LR the calculator used so the multiplier is never hidden.
6. 6 Frame the result as a screen: use the post-test probability to decide whether to repeat the test, order a different test, or discuss further workup.

A reader working a textbook example enters prevalence = 1%, test direction = positive, sensitivity = 90%, specificity = 90%. The calculator derives LR+ = 9, pre-test odds = 0.0101, post-test odds = 0.0909, post-test probability 8.33%.

When a classroom example also needs a test of independence from a 2x2 table of test results and disease status, the Chi-Square Calculator takes the same cells and returns the chi-square statistic and the p-value.

The Bayes rule update is the same in every evidence-based-medicine textbook, and a calculator that mirrors the textbook steps is faster than the pen-and-paper alternative.

• • Bayes-rule update in one form: the form follows the textbook sequence: pre-test probability, test direction, LR source, then the result.
• • Sensitivity and specificity support: enter sensitivity and specificity and the calculator derives LR+ or LR- for you, so the LR never has to be hand-computed from a textbook.
• • Direct LR entry: switch the LR source to enter an LR+ or LR- directly when the literature already reports the LR for the test.
• • Odds visible alongside the probability: the result panel shows pre-test odds and post-test odds next to the post-test probability so the multiplication step is never hidden.
• • Positive and negative test support: the same form handles a positive test and a negative test, with the correct likelihood ratio chosen for each.

The odds, the LR, and the probability are all listed in the same place so the result panel stays readable on a phone or printed page.

For another teaching-grade summary statistic that often sits next to a Bayes-rule lesson in a statistics-for-clinicians course, the Standard Deviation Calculator reads the standard deviation from a small list of numbers.

Four factors drive the post-test probability. The same calculator shows the pre-test odds, the LR, and the post-test odds, so a sensitivity analysis is a matter of changing the inputs.

• • The post test probability calculator is a teaching tool, not a clinical decision support system. A clinician or pharmacist comparing the result with the patient history and biomarkers is the usual next step.
• • Sensitivity and specificity are often reported with a confidence interval, and a borderline value can move LR+ or LR- across a clinical threshold. Treat the result as a screen, not a single answer.
• • A population prevalence used as an individual prior can underestimate or overestimate the post-test probability for a particular patient.

A flat LR near 1 leaves the odds close to their pre-test value, while a strong LR (10 or 0.1) moves the odds by a factor of 10.

According to CDC public health site, screening test results are read as a post-test probability, and a positive screen is a starting point for a clinical evaluation rather than a stand-alone diagnosis.

For a single test with a known sensitivity and a fixed number of trials, the Binomial Distribution Calculator shows the binomial probability of observing that many positives, which is a useful sanity check on the LR-based update.