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Likelihood Ratio Calculator

The Likelihood Ratio Calculator computes positive likelihood ratio (LR+) and negative likelihood ratio (LR-) from a diagnostic test's sensitivity and specificity. Evaluate how much a test result shifts the probability of disease for evidence-based clinical decisions.

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    Ratio Simplifier
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        What is a Likelihood Ratio?

        A likelihood ratio quantifies how much a test result changes the probability of having a condition. The positive likelihood ratio (LR+) measures how much a positive test increases disease probability. The negative likelihood ratio (LR-) measures how much a negative test decreases it. LR+ above 10 or LR- below 0.1 are considered strong evidence.

        Likelihood ratios combine sensitivity and specificity into a single metric that is independent of disease prevalence. Clinicians use LRs with pre-test probability (Bayesian reasoning) to calculate post-test probability, making them more useful than sensitivity/specificity alone for clinical decision-making.

        Formulas & Equations Used

        This Likelihood Ratio Calculator uses the following core equations:

        1 Positive Likelihood Ratio ▼
        LR+ = Sensitivity / (1 - Specificity)

        Sensitivity 95%, Specificity 90%: LR+ = 0.95 / (1 - 0.90) = 0.95 / 0.10 = 9.5.

        2 Negative Likelihood Ratio ▼
        LR- = (1 - Sensitivity) / Specificity

        Sensitivity 95%, Specificity 90%: LR- = (1-0.95) / 0.90 = 0.05 / 0.90 = 0.056.

        3 Post-Test Odds (Fagan Nomogram) ▼
        Post-Test Odds = Pre-Test Odds × Likelihood Ratio

        Pre-test probability 20% → odds = 0.25. LR+ = 9.5 → Post-test odds = 0.25 × 9.5 = 2.375 → probability = 70.4%.

        How to Use This Likelihood Ratio Calculator

        Follow these 3 simple steps:

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        Enter Your Values

        Type the known values into the input fields above. The Likelihood Ratio Calculator accepts any positive numbers.

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        Choose Calculation Mode

        Select Solve, Simplify, or Scale mode in the calculator. Each applies different equations to your inputs.

        3

        View Results

        Click Calculate to see your answer with a visual ratio bar, pie chart, and step-by-step solution breakdown.

        Example Problems & Step-by-Step Solutions

        Here are 3 worked examples using this Likelihood Ratio Calculator:

        Example 1 Test with 90% sensitivity, 85% specificity
        1 LR+ = 0.90 / (1-0.85) = 0.90/0.15 = 6.0
        2 LR- = (1-0.90) / 0.85 = 0.10/0.85 = 0.118
        3 LR+ of 6.0: moderate evidence for disease when positive
        4 LR- of 0.12: moderate evidence against disease when negative
        ✓ LR+ = 6.0, LR- = 0.12 — moderately useful test
        Example 2 Calculate post-test probability
        1 Pre-test probability: 30% → odds = 0.30/0.70 = 0.429
        2 Test positive, LR+ = 10
        3 Post-test odds = 0.429 × 10 = 4.29
        4 Post-test probability = 4.29/(1+4.29) = 81.1%
        ✓ Positive test raises probability from 30% to 81%
        Example 3 Highly sensitive test: 99% sensitivity, 50% specificity
        1 LR+ = 0.99 / 0.50 = 1.98 (weak positive evidence)
        2 LR- = 0.01 / 0.50 = 0.02 (very strong negative evidence)
        3 A negative result almost rules out disease
        4 A positive result barely changes probability
        ✓ LR+ = 2.0 (weak), LR- = 0.02 (excellent rule-out)

        Frequently Asked Questions

        What is a good likelihood ratio? ▼

        LR+ > 10 is strong evidence for disease. LR+ 5-10 is moderate. LR+ 2-5 is weak. LR- < 0.1 strongly rules out disease. LR- 0.1-0.2 is moderate. LR values near 1.0 provide no useful information.

        Why are likelihood ratios better than sensitivity/specificity? ▼

        LRs combine both metrics into a single number and are independent of disease prevalence. They can be applied directly to individual patients using pre-test probability. Sensitivity and specificity are prevalence-independent but don't directly give post-test probability.

        How do I use the Fagan nomogram? ▼

        Draw a line from your pre-test probability (left axis) through the likelihood ratio (middle axis) to find the post-test probability (right axis). This visual tool quickly converts LRs into clinically useful probability changes.

        Can likelihood ratios be used for multi-level test results? ▼

        Yes. Interval likelihood ratios can be calculated for different result ranges (e.g., low, moderate, high values) rather than just positive/negative. Each interval has its own LR for more nuanced interpretation.

        What is pre-test probability? ▼

        Pre-test probability is your estimated chance of disease before conducting the test, based on prevalence, symptoms, and clinical judgment. It's the starting point for Bayesian reasoning with likelihood ratios.

        Learn About Ratios

        What is a ratio?

        A ratio is a comparison between two or more quantities showing the relative size of one to another. Written as A : B, it means 'for every A units of the first quantity, there are B units of the second.' For example, a ratio of 3 : 4 means for every 3 parts of A, there are 4 parts of B. Ratios are used in cooking, construction, finance, science, and everyday life.

        How do I solve a proportion?

        A proportion is an equation that says two ratios are equal: A : B = C : D. To solve for a missing value, use cross-multiplication. If D is unknown: D = (B × C) / A. This works because in equal ratios, the cross products are always equal: A × D = B × C. Our Proportion Solver does this automatically — just enter any 3 values and it finds the 4th.

        How do I simplify a ratio?

        To simplify a ratio, find the Greatest Common Divisor (GCD) of both numbers and divide each by it. For example, 24 : 36 — the GCD of 24 and 36 is 12. So 24 ÷ 12 = 2 and 36 ÷ 12 = 3, giving the simplified ratio 2 : 3. Our Simplifier automatically finds the GCD and reduces your ratio to its lowest terms.

        What is ratio scaling and when is it useful?

        Scaling a ratio means multiplying both parts by the same factor to create an equivalent, larger (or smaller) ratio. For instance, scaling 2 : 5 by a factor of 3 gives 6 : 15. This is extremely useful for recipes (tripling a recipe), construction (scaling blueprints), mixing solutions, or any scenario where you need to maintain the same proportion at a different magnitude.

        What's the difference between a ratio and a fraction?

        A ratio A : B compares two quantities to each other (part-to-part), while a fraction A/B typically represents a part-to-whole relationship. However, any ratio can be expressed as a fraction: 3 : 4 is equivalent to 3/4 = 0.75. The key difference is context — ratios compare quantities side-by-side, while fractions represent a portion of a total.