Ask Mallard

Pre-test probability and likelihood ratios

A test does not tell you whether the patient has the condition; it revises the probability you already had.

The arithmetic that surprises people

Take a condition present in 1 in 1000, and a test with 99 percent sensitivity and 99 percent specificity. Among 100,000 people, 100 have it and about 99 test positive. Of the 99,900 without it, about 999 also test positive. So a positive result means roughly a 9 percent chance of having the condition, not 99 percent. Nothing is wrong with the test; the base rate dominates.

The arithmetic, as natural frequencies

A natural frequency tree. Of 1,000 people, 20 have the condition and 980 do not. Of the 20, 18 test positive. Of the 980, 49 test positive. So of 67 people who test positive, only 18 have the condition, a positive predictive value of 27 percent, even though the test is 90 percent sensitive and 95 percent specific.A rare condition and a good test: prevalence 2%, sensitivity 90%, specificity 95%1,000 people20 have the condition980 do not18 test positive2 test negative49 test positive931 test negativeOf 67 people who test positive, 18 have the condition: a positive predictive value of 27%.The same test in a population where the condition is common would give a far higher predictive value,which is why a predictive value quoted without its prevalence does not travel.
Worked arithmetic. The inputs are round illustrative numbers, not results from any study; the outputs are calculated from them.

Likelihood ratios do the work

  • The positive likelihood ratio is sensitivity divided by one minus specificity: how much more likely a positive result is in someone with the condition.
  • The negative likelihood ratio is one minus sensitivity divided by specificity.
  • Multiply the pre-test ODDS by the likelihood ratio to get the post-test odds, then convert back to a probability.
  • As a rough guide, a likelihood ratio of 10 raises the probability by about 45 percentage points in the middle of the range, 5 by about 30, and 2 by about 15.
  • They are stable across settings in a way predictive values are not, which is why they travel.

Where the pre-test probability comes from

From the prevalence in the setting, adjusted for the history and examination that led to the test being ordered. That is a clinical estimate and it should be made explicitly, because the whole interpretation rests on it. A test ordered without any prior suspicion has a very low pre-test probability by construction, which is why indiscriminate testing generates false positives.

The consequences of a false positive are part of the calculation

A false positive is not a neutral event: it leads to further investigation, sometimes invasive, and to a period of being treated as a patient. Whether a test is worth ordering depends on the pre-test probability, the likelihood ratios and what each kind of error costs.

What to watch out for

Sensitivity and specificity are properties of the test; predictive values are properties of the test AND the population, so a positive predictive value quoted without its prevalence is uninterpretable and usually misleading when carried to a different setting. The base-rate error that follows is the single most consequential misreading in diagnosis: with a rare condition, a highly specific test can still produce more false positives than true ones, and clinicians asked to estimate this get it wrong by an order of magnitude.

Clinical example

A D-dimer with a high negative likelihood ratio usefully excludes pulmonary embolism in a low-risk patient and does almost nothing in a high-risk one, where a negative result still leaves a probability that warrants imaging. The same test, the same numbers, a different pre-test probability.

Code for this method
Loading…
Recent studies that used this
Searching PubMed…
Related Learn pages
Sources & further reading
  1. Diagnostic tests 4: likelihood ratios (opens in a new tab) · BMJ · doi 10.1136/bmj.329.7458.168
  2. Receiver operating characteristic curve in diagnostic test assessment (opens in a new tab) · J Thorac Oncol · doi 10.1097/JTO.0b013e3181ec173d
Last source review: 2026-09-06
AI-generated educational content
Mallard uses reputable methodological and reporting sources, but this material may contain errors or omit important context. Verify important methodological decisions with the cited sources and, when appropriate, a qualified research methodologist or statistician.