Likelihood Ratio Calculator
Likelihood Ratio Calculator
Turn sensitivity and specificity into LR+ and LR−, then apply one to the probability you already had. Likelihood ratios are the only test property that combines directly with a pre-test probability.
Likelihood Ratio
Sens, spec, pre-test → post-testSensitivity 90%, specificity 90%, pre-test probability 20%, positive result
From test properties to one patient's probability
LR− = (1 − sensitivity) ÷ specificity
pre-test odds = p ÷ (1 − p)
post-test odds = pre-test odds × LR
post-test probability = odds ÷ (1 + odds)
- LR+
- how many times more often a positive result occurs in someone with the disease than in someone without it. Above 10 is conventionally a large shift; between 0.5 and 2 is barely a shift at all
- LR−
- how many times more often a negative result occurs in someone with the disease than in someone without it. Below 0.1 is a large shift downward. A good rule-out test has a very small LR−, which needs high sensitivity
- odds
- the detour that makes this work. Probabilities cannot be multiplied by a likelihood ratio; odds can. Convert to odds, multiply, convert back — that is the whole of Bayes's theorem in this setting
- pre-test probability
- yours to estimate, from local prevalence, a validated prediction rule or experience. It is the least precise input and usually the most influential, which is an argument for stating it out loud rather than for avoiding the calculation
- why not PPV
- PPV answers the same question but only for the population it was measured in. A likelihood ratio is a property of the test, so it can be carried to your patient and applied to your own pre-test probability. That is the entire reason the form exists
Worked example
Sensitivity 90%, specificity 90%, pre-test probability 20%, positive result
LR+ = 0.90 ÷ (1 − 0.90) = 9.0
Pre-test odds = 0.20 ÷ 0.80 = 0.25
Post-test odds = 0.25 × 9.0 = 2.25
Post-test probability = 2.25 ÷ 3.25 = 69.2%
The same test coming back negative gives LR− = 0.10 ÷ 0.90 = 0.111, post-test odds 0.25 × 0.111 = 0.0278, and a post-test probability of 2.7%
So one test takes a patient from 20% to either 69% or 2.7%. Notice that the negative result is far more decisive here than the positive one, which is what an LR− of 0.11 against an LR+ of 9.0 means in practice
How much a likelihood ratio moves a probability
| LR | Conventional reading | 10% pre-test becomes | 50% pre-test becomes |
|---|---|---|---|
| 10 | Large, often decisive increase | 53% | 91% |
| 5 | Moderate increase | 36% | 83% |
| 2 | Small increase | 18% | 67% |
| 1 | No information whatsoever | 10% | 50% |
| 0.5 | Small decrease | 5.3% | 33% |
| 0.2 | Moderate decrease | 2.2% | 17% |
| 0.1 | Large, often decisive decrease | 1.1% | 9.1% |
What each figure is good for
| Figure | Belongs to | Can you carry it to your patient? |
|---|---|---|
| Sensitivity | The test | Only indirectly — it does not by itself say what a result means |
| Specificity | The test | Same |
| PPV / NPV | The test and the study population | No, unless your population matches theirs |
| LR+ and LR− | The test | Yes — multiply your own pre-test odds by it |
| Post-test probability | This patient | It is the answer, and it is only as good as the pre-test estimate you fed it |
The one test property you can apply to your own patient
Sensitivity and specificity are properties of a test, but neither answers the question actually being asked at the bedside, which is what this result means for this person. Predictive values answer that question, but only for the population in which they were measured, and if that population is not yours they mislead. Likelihood ratios sit in between: they are stable properties of the test, and they combine in a single multiplication with whatever probability you had before testing.
The mechanism is Bayes's theorem, and the only awkward part is that probabilities cannot be multiplied by a likelihood ratio — odds can. So the arithmetic goes in three steps: convert the pre-test probability to odds, multiply by the likelihood ratio, convert back. A 20% pre-test probability is odds of 0.25. A positive result on a 90%-sensitive, 90%-specific test carries an LR+ of 9, giving post-test odds of 2.25 and a post-test probability of 69%. A negative result on the same test carries an LR− of 0.11 and takes the same patient down to 2.7%.
The conventional readings are worth memorising because they let you judge a test at a glance. An LR above 10, or below 0.1, produces a large and often conclusive change in probability. Between 2 and 5, or between 0.2 and 0.5, the change is small. Between 0.5 and 2 the result is very nearly uninformative, and an LR of exactly 1 means the finding was just as likely in disease as in health — it has told you nothing and should not change what you do. A surprising number of investigations ordered daily sit in that range.
Two cautions. The first is that a positive and a negative result on the same test rarely carry equal weight: a highly sensitive test rules out well and rules in poorly, and the reverse holds for a highly specific one, which is why LR+ and LR− must be quoted as a pair. The second is that the post-test probability inherits every weakness of the pre-test estimate. That estimate is a judgement, and the honest response is to state it explicitly and see whether a plausible range of values changes the decision. If it does not, the imprecision did not matter; if it does, that is worth knowing before the result comes back.
Frequently asked questions
What is a good likelihood ratio?
Conventionally, an LR+ above 10 or an LR− below 0.1 produces a large and often decisive change in probability. Ratios between 5 and 10, or 0.1 and 0.2, are moderate. Anything between 0.5 and 2 barely moves the probability and rarely justifies the test.
How do I calculate post-test probability from a likelihood ratio?
Convert the pre-test probability to odds by dividing it by one minus itself, multiply those odds by the likelihood ratio, then convert back with odds divided by one plus odds. A 20% pre-test probability and an LR+ of 9 give 0.25 × 9 = 2.25, which is a 69% post-test probability.
Why use likelihood ratios instead of predictive values?
Because predictive values belong to the population they were measured in and change with prevalence, while likelihood ratios are properties of the test. That lets you take the LR from a published study and apply it to your own patient's pre-test probability.
What does a likelihood ratio of 1 mean?
That the result carries no information. It occurred equally often in people with and without the disease, so the post-test probability is identical to the pre-test probability and the test has changed nothing.
How accurate does the pre-test probability need to be?
Accurate enough that the decision does not flip. It is an estimate, so try the calculation at the top and bottom of a plausible range. If the management is the same across that range, the imprecision is harmless; if it is not, no single number should be trusted and you need better information first.
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References
- Fagan TJ. Nomogram for Bayes's theorem. N Engl J Med. 1975;293(5):257.
- Deeks JJ, Altman DG. Diagnostic tests 4: likelihood ratios. BMJ. 2004;329(7458):168–169.
- Jaeschke R, Guyatt GH, Sackett DL. Users' guides to the medical literature III: how to use an article about a diagnostic test. JAMA. 1994;271(9):703–707.
Medical Disclaimer: The tools and content provided here are for educational and reference purposes only. They are not intended to substitute for professional medical advice, diagnosis, or treatment. Clinical decisions should always be based on the comprehensive assessment of a qualified healthcare professional.