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Likelihood ratios compare the probability of a test result among people with a condition with the probability of the same result among those without it.
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Formula
Se and Sp are fractions from 0 to 1 · LR+ and LR− are dimensionless
| Symbol | Meaning | Unit |
|---|---|---|
| Likelihood ratio for a positive test | dimensionless | |
| Likelihood ratio for a negative test | dimensionless | |
| Sensitivity measured against a defined reference standard and target population | fraction | |
| Specificity measured against the same reference standard and target population | fraction | |
| Pretest odds of the condition | odds | |
| Post-test odds after applying the result-specific likelihood ratio | odds |
LR^+=\frac{Se}{1-Sp},\quad LR^-=\frac{1-Se}{Sp}LR+ equals sensitivity divided by the false-positive rate; LR− equals the false-negative rate divided by specificity. Multiplying pretest odds by the likelihood ratio for the observed test result gives post-test odds; convert to probability with p=odds/(1+odds). Likelihood ratios depend on threshold, reference standard, disease spectrum and study design. They do not remove bias or replace a population-specific prior probability. Sensitivity and specificity are not positive or negative predictive values.
Sensitivity and specificity use the same threshold, reference standard and relevant target population.
LR+ and LR− compare test-result probabilities in affected and unaffected groups.
Rearrange for pretest odds
Use the likelihood ratio matching the observed positive or negative result.
Se=0.90; Sp=0.80; pretest probability=10%. Find LR+ and positive post-test probability.
LR+=4.5.
Prior odds=1/9; posterior odds=0.5, so probability is 33.3%.
Reporting sensitivity as the post-test probability after a positive result.
Combine the likelihood ratio with pretest odds; predictive value also depends on prevalence.
Formula cluster
LR+ above one raises odds and LR− below one lowers them after a negative test.
With Se=0.90 and Sp=0.80, LR+=0.90/0.20=4.5 and LR−=0.10/0.80=0.125. A 10% pretest probability gives odds 1/9; a positive result yields odds 0.5 and probability 0.5/1.5=33.3%. Check: a positive result raises probability but does not make it 90%.
Combine diagnostic test results with a known pretest probability
Here is how to work through a typical Diagnostic Likelihood Ratios (LR+ = Se/(1−Sp); LR− = (1−Se)/Sp) task step by step:
Task
Se=0.90; Sp=0.80; pretest probability=10%. Find LR+ and positive post-test probability.
Solution path
LR+=4.5.
Prior odds=1/9; posterior odds=0.5, so probability is 33.3%.
how strongly a test result changes the odds of a condition. Its output is dimensionless LR+ for positive and LR− for negative results.. LR+ above one raises odds and LR− below one lowers them after a negative test. The quantity answers a specific biological or statistical question, not automatically the full cause of an observed pattern. Tie it to the population, sample or model level from which inputs came. A number without that reference can be misleading across studies. Keep description separate from explanation: the metric summarizes data but does not by itself establish the underlying mechanism.
Use Se is the true-positive fraction and Sp the true-negative fraction against a named standard. The units are enter Se and Sp from zero to one; likelihood ratios have no units.. Check that numerator and denominator refer to the same time interval, target population and scale. Proportions entered as percentages usually need conversion to values between zero and one. Write units and definitions beside the observations. This helps catch swapped groups, mismatched age intervals or a wrong denominator before they distort the result. Also document measurement definitions and preprocessing, because studies can operationalize the same symbols differently.
The central assumption is: threshold, reference standard and test performance match the target population and disease spectrum. pretest probability comes from a relevant population rather than an assumed prevalence. If reference errors or spectrum effects shift sensitivity and specificity., interpretation changes or the formula no longer fits. State sampling, measurement method and biological scale. A model is useful because it simplifies a question, but its limits belong in the report. Do not automatically transfer parameters to different species, populations, laboratories or sampling conditions. Report these assumptions with the result so readers can judge which populations and conditions the conclusion can cover.
In the example, Se=0.90 and Sp=0.80 give LR+=4.5. The check is: a 10% prior probability rises to about 33% after a positive result, not 90%. Substitute calculated values back into the original relation and check whether ranges and units are biologically plausible. Rounded data can create a small discrepancy, but should not change direction or order of magnitude. Also test a simple limiting case, such as zero, complete detection or equal groups. This separates arithmetic errors from model limitations. Round intermediate values only at the end, and check the order of magnitude against the biological question.
A nearby quantity is positive and negative predictive values. The substantive distinction is that LRs describe test evidence against a standard; predictive values also depend on pretest probability. Both can appear in the same biological analysis, but they use different inputs and answer different questions. Choose by the requested output and verify the data definition before comparing values. A related name does not imply identical units or interchangeable interpretation. This matters when a descriptive statistic is reported alongside a model parameter. Before comparing values, align definitions, populations, observation periods and analysis rules; otherwise the difference may be methodological.
Quanta exam set
Curated exam set for "Diagnostic Likelihood Ratios":
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Which formula describes Diagnostic Likelihood Ratios?
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How do you rearrange Diagnostic Likelihood Ratios for Rearrange for pretest odds?
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Which common mistake happens with Diagnostic Likelihood Ratios?
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Create a curated FSRS exam set for LR+ = Se/(1−Sp); LR− = (1−Se)/Sp: formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.
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