Post-test probability calculator

The formula

pre-test odds = p ÷ (1 − p) post-test odds = pre-test odds × LR post-test probability = post-test odds ÷ (1 + post-test odds)

With sensitivity and specificity instead of an LR, use LR+ = sensitivity ÷ (1 − specificity) after a positive result and LR− = (1 − sensitivity) ÷ specificity after a negative one.

Worked example

The default: a patient with a 25% pre-test probability has a positive result on a test with LR+ = 6.

• Pre-test odds = 0.25 ÷ 0.75 = 1/3 (one to three).
• Post-test odds = 1/3 × 6 = 2 (two to one).
• Post-test probability = 2 ÷ 3 = 66.67%, a rise of 41.67 percentage points.

Switch to sensitivity and specificity mode with the defaults of 90% and 85%. LR+ = 0.9 ÷ 0.15 = 6, so a positive result gives the same 66.67%. Choose a negative result instead: LR− = 0.1 ÷ 0.85 = 0.1176, the post-test odds are 1/3 × 0.1176 = 0.0392, and the probability falls to 3.77%.

Reading the nomogram

The left scale runs from 0.1% at the top to 99.9% at the bottom; the right scale runs the other way. Both are on the log-odds scale, so each is evenly spaced in logit(p) = ln(p ÷ (1 − p)). The middle scale is ln(LR) ÷ 2. Because ln(post-test odds) = ln(pre-test odds) + ln(LR), the three marked points always fall on one straight line.

How to interpret the result

Likelihood ratios have rough rules of thumb. LR+ above 10 or LR− below 0.1 usually changes a decision; 5 to 10 or 0.1 to 0.2 moderately; 2 to 5 or 0.2 to 0.5 a little; close to 1, hardly at all. But the effect also depends on where you start. The same LR+ of 6 moves a 1% pre-test probability only to about 5.7%, while it moves 25% to 67%. Tests are most useful when the pre-test probability is in the middle range, where the result can push it across a treatment or testing threshold.

Where the likelihood ratio comes from

Published studies of a test usually report sensitivity and specificity, sometimes with LR+ and LR−. If you have the study's 2×2 table, the sensitivity and specificity calculator gives both likelihood ratios with confidence intervals. Using the ends of those intervals here shows how uncertain the post-test probability is.

Common mistakes

• Multiplying the probability, not the odds, by the LR. 25% × 6 = 150% is impossible; the odds method keeps the answer between 0 and 1.
• Using LR+ after a negative result. A negative result needs LR−, which is below 1 and lowers the probability.
• Chaining dependent tests as if independent. The second test's LR applies only if it adds information the first did not.
• Using a pre-test probability of 0% or 100%. No test result can change a certainty, so the calculator asks for a value strictly between.