LR+ = 9 and a positive result still leaves you at 8.3333%. Eleven of every twelve positives are false, and nothing about the test is wrong — the base rate is doing the work.
LR+ = 9.0000, LR− = 0.1111
A positive result: 1.0000% → 8.3333%
A negative result takes the same patient to 0.1121%. The positive gains 7.33 percentage points and the negative removes 0.89. Even after a positive, the condition is still the less likely of the two — 91.67% of positives at this base rate are false.
LR+
9.00000
sens / (1 − spec)
LR−
0.11111
(1 − sens) / spec
Post-test, positive
8.333%
from 1.000%
Post-test, negative
0.112%
pre-test odds 0.01010
The same test across every base rate
Sensitivity and specificity held fixed. Only the population being tested changes.
Post-test probability after a positive and a negative result at eleven pre-test probabilities
Pre-test
After a positive
Gain
After a negative
What a positive means
0.10%
0.893%
+0.79 pp
0.011%
still improbable
0.50%
4.327%
+3.83 pp
0.056%
still improbable
1.00%
8.333%
+7.33 pp
0.112%
still improbable
2.00%
15.517%
+13.52 pp
0.226%
worth pursuing
5.00%
32.143%
+27.14 pp
0.581%
worth pursuing
10.00%
50.000%
+40.00 pp
1.220%
more likely than not
20.00%
69.231%
+49.23 pp
2.703%
more likely than not
30.00%
79.412%
+49.41 pp
4.545%
more likely than not
50.00%
90.000%
+40.00 pp
10.000%
near-certain
70.00%
95.455%
+25.45 pp
20.588%
near-certain
90.00%
98.780%
+8.78 pp
50.000%
near-certain
The gain column peaks in the middle and collapses at both ends. A test is least informative exactly where you were already confident — and the answer to “is this test any good?” depends on who you point it at.
LR+ above 10 or LR− below 0.1 shifts a diagnosis meaningfully LR = 1 changes nothing at any base rate Odds in, odds out — probabilities are converted at both ends
What this tool shows
A 90%/90% test has LR+ = 9, and at a 1% pre-test probability a positive result reaches 8.3333%. Eleven of every twelve positives are false, with nothing wrong with the test. Move the same test to a 30% pre-test probability and a positive reaches 79.4118% — a gain of 49.41 percentage points against 7.33. How informative a test is depends on who you point it at, and the sweep table shows exactly how much.
LR+ and LR− from sensitivity and specificity
Post-test probability after a positive AND after a negative result
The percentage-point gain each result produces, which peaks in the middle
The same test swept across eleven pre-test probabilities
Pre-test and post-test odds, so the odds-in-odds-out arithmetic is visible
A rule-out example where the negative result is the valuable one
LR+ and LR− Post-test probability Base-rate sweep Odds arithmetic
Not medical advice. The arithmetic does not know your patient.
Updated 12 September 2026 · Works in any browser, no installation
A likelihood ratio says how much more often a result appears in people with the condition than in people without it, and it converts a pre-test probability into a post-test one by multiplication — in odds. LR+ = sensitivity ÷ (1 − specificity); LR− = (1 − sensitivity) ÷ specificity. Unlike predictive values, a likelihood ratio is a property of the test alone — but what it produces depends entirely on what you multiplied it by.
At a glance
Formula shown
Pre-test odds = p/(1−p). Post-test odds = pre-test odds × LR. Post-test probability = odds/(1+odds). LR+ = sens/(1−spec) and LR− = (1−sens)/spec. Because the multiplication happens in odds rather than probability, a large LR barely moves a probability that started very small: 0.01 becomes odds of 0.0101, times 9 is 0.0909, back to a probability of 0.0833.
Scenario support
Interpreting a diagnostic or screening result for a specific patient, deciding whether a test is worth ordering at all, chaining several tests together, setting referral thresholds, and explaining to anyone why a positive screening result at population prevalence is usually a false alarm.
Educational estimate
Planning support from the values you enter — not professional advice.
The same test is a different test at a different base rate
Sensitivity and specificity are fixed properties. What a result means is not, and the gap between those two facts is where most misreadings of test results live.
90% sensitivity and 90% specificity give LR+ = 9 at every base rate. The number does not move. What it produces moves enormously.
At a 0.1% pre-test probability a positive result reaches 0.8929% — a gain of 0.79 percentage points. At 1% it reaches 8.3333%; at 5%, 32.1429%; at 10%, exactly 50%; at 30%, 79.4118%.
The gain peaks at 49.41 percentage points around a 30% pre-test probability and falls away on both sides — to 17.30 at 80% and to almost nothing below 1%. A test tells you least when you were already sure, in either direction.
Which is the argument for testing people with symptoms rather than everyone. It is not that the test performs worse in a healthy population; it performs identically. The multiplication it does simply starts from a number too small for the result to land anywhere useful.
The multiplication happens in odds, which is the whole trick
“Nine times more likely” sounds like it should multiply a probability by nine. It does not, and the difference is what makes low base rates so unforgiving.
A 1% probability is odds of 1:99, or 0.010101. Times nine is 0.090909, which converted back is 8.3333% — not 9%, and certainly not the 90% people often expect.
At small probabilities odds and probabilities are nearly equal, so the multiplication looks almost like multiplying the probability — 1% times 9 is close to 8.33%. That approximation is why the intuition survives at all.
At large probabilities they diverge completely. An 80% probability is odds of 4; times nine is 36; back to a probability that is 97.2973%. Multiplying 80% by 9 would have given a nonsense above 1.
This is also Bayes’ theorem, written in its most usable form. The odds formulation exists precisely so that updating is one multiplication instead of a ratio of sums.
LR− is the half that gets ignored
Most discussion of test accuracy is about finding disease. A great deal of clinical work is about excluding it, and that is a different number.
LR− = (1 − sensitivity) ÷ specificity, and a value below 0.1 is the conventional mark for a result that meaningfully rules a condition out.
A sensitive but unspecific test is a rule-out test. The tool’s fourth preset has 95% sensitivity and only 60% specificity: LR+ is a weak 2.375, but LR− is 0.0833 and a negative result drops a patient from 20% to 2.0408%.
That test is close to useless for confirming and genuinely valuable for excluding, which a single accuracy figure would hide entirely.
Which is why both ratios belong in any report of a test. “85% accurate” collapses two different capabilities into one number that describes neither, and the mix depends on prevalence in the validation sample.
Chaining tests multiplies the ratios, if they are independent
The odds formulation makes sequential testing arithmetically trivial, which is exactly why the assumption underneath it is so easy to forget.
Post-test odds from one test become pre-test odds for the next, so two positive results with LR+ of 9 each give a combined LR of 81 — and at a 1% base rate that reaches 45.0% rather than 8.3%.
That multiplication requires conditional independence. The two tests must be independent given the true status, which two tests measuring the same underlying biology usually are not.
When they are correlated the combined LR is smaller than the product, sometimes far smaller, because the second test largely repeats the first rather than adding to it.
Two X-rays read by two radiologists are not two independent tests. Two results from genuinely different modalities are closer to it. The arithmetic cannot tell the difference, and the person doing it has to.
Where the inputs come from matters more than the arithmetic
The computation here is exact. Everything uncertain about the answer is upstream of it.
Published sensitivity and specificity come from validation studies whose patients usually differ from yours — often sicker, since spectrum bias inflates both figures when a study compares clear cases against healthy controls.
Those figures also have confidence intervals, and a sensitivity of 90% from 40 patients spans roughly 76% to 97%. The LR+ that follows spans 3.2 to 32.
The pre-test probability is usually a judgement, not a measurement. It is the number the whole calculation is most sensitive to, and it is the one least often written down.
Many tests are not binary either. A continuous marker has a different likelihood ratio at every level, and collapsing it to positive/negative at one cutoff discards information — which is the case for looking at the whole ROC curve instead.
Reporting a post-test probability
Four things, and the first is the one that makes the rest checkable.
State the pre-test probability you used and where it came from. Population prevalence, a clinical prediction rule, or a judgement are different footings, and the answer inherits whichever you chose.
Give both likelihood ratios, not accuracy. They are properties of the test that do not move with prevalence, which is what makes them portable between settings.
Give the post-test probability after a negative as well as a positive. A decision is usually about what to do with either result, and reporting only one answers half the question.
And carry the uncertainty in the inputs through. A post-test probability quoted to two decimals from a sensitivity known to within fifteen points is precision that does not exist.
Sources and methodology
References for likelihood ratios and post-test probability.
Method. The update is performed in odds and converted back at the end, rather than by a probability-space shortcut, so the arithmetic is exact at both extremes where the shortcut fails. The sweep table recomputes the full update at each pre-test probability rather than interpolating, which is what makes the shape of the gain column a measurement: it peaks at 49.41 percentage points near a 30% pre-test probability for a 90/90 test and falls to 0.79 at 0.1%. Infinite likelihood ratios from a perfect specificity or sensitivity are carried through as infinities and converted to a post-test probability of 1 rather than producing NaN. Inputs outside 0 to 1, and the degenerate case of zero sensitivity with perfect specificity, return no result. That engine is verified on every change against 115 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Sensitivity and SpecificitySensitivity, specificity, PPV, NPV, likelihood ratios and MCC from a 2×2 table, with predictive values recomputed across the prevalence range — a 99%/99% test has a PPV of 50% at 1% prevalence and 9% at 0.1%.
Bayes' TheoremPosterior probability from a prior, sensitivity and specificity — plus the true and false positives per 100,000, because a 99% accurate test for a 1-in-10,000 condition is right 0.98% of the time and the percentage alone does not make that believable.
ROC Curve & AUCBuilds the curve from raw scores with every threshold enumerated, and computes the AUC twice — trapezoid and Mann-Whitney U — which agree to 1.11e-16 across 300 datasets.
Number Needed to TreatNNT from the absolute risk reduction, with the relative figure beside it — two trials reporting the identical “50% reduction” have NNTs of 7 and 1,000, and the common shortcut says 2 for both.
Relative RiskRisk ratio and odds ratio from one table with the divergence between them plotted: they agree to half a percent at a 1% baseline, and at an 80% baseline the odds ratio is exactly half the risk ratio.
Odds RatioOdds ratio, relative risk, risk difference and number needed to treat from one 2x2 table — because an odds ratio of 6.00 can describe a relative risk of 1.50.
An educational tool, not medical advice. Published sensitivity and specificity carry confidence intervals and come from study populations that may differ from yours, and the pre-test probability is usually a judgement — so treat the output as an illustration of how much a result should move your thinking rather than as a diagnosis.
Published a post-test probability tool that performs the update in odds and converts back, which is exact at both extremes where the probability-space shortcut fails.
Built the page around a measurement rather than a caution: a 90/90 test has LR+ of 9 at every base rate, and it moves a patient 0.79 percentage points at a 0.1% pre-test probability, 7.33 at 1%, 49.41 at 30% and 17.30 at 80%. How informative a test is depends on who you point it at, and the sweep table shows by exactly how much.
Made the low-prevalence result concrete: at a 1% pre-test probability a positive reaches 8.3333%, so eleven of every twelve positives are false with nothing at all wrong with the test.
Gave LR- equal billing, with a rule-out preset where LR+ is a weak 2.375 and LR- is 0.0833 — a test close to useless for confirming and genuinely valuable for excluding, which a single accuracy figure would hide.
Verified the odds-form update against direct Bayes across 300 random tests, agreeing to within 1e-12, so the arithmetic shortcut the page teaches is checked rather than asserted.
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