In evidence-based medicine, likelihood ratios are used for assessing the value of performing a diagnostic test. They combine sensitivity and specificity into a single metric that indicates how much a test result shifts the probability that a condition (such as a disease) is present. The first description of the use of likelihood ratios for decision rules was made at a symposium on information theory in 1954. In medicine, likelihood ratios were introduced between 1975 and 1980. There is a multiclass version of these likelihood ratios.
Calculation Two versions of the likelihood ratio exist, one for positive and one for negative test results. Respectively, they are known as the positive likelihood ratio (LR+, likelihood ratio positive, likelihood ratio for positive results) and negative likelihood ratio (LR–, likelihood ratio negative, likelihood ratio for negative results). The positive likelihood ratio is calculated as
LR + = sensitivity 1 − specificity {\displaystyle {\text{LR}}+={\frac {\text{sensitivity}}{1-{\text{specificity}}}}}
which is equivalent to
LR + = Pr ( T + ∣ D + ) Pr ( T + ∣ D − ) {\displaystyle {\text{LR}}+={\frac {\Pr({T+}\mid D+)}{\Pr({T+}\mid D-)}}}
or "the probability of a person who has the disease testing positive divided by the probability of a person who does not have the disease testing positive." Here "T+" or "T−" denote that the result of the test is positive or negative, respectively. Likewise, "D+" or "D−" denote that the disease is present or absent, respectively. So "true positives" are those that test positive (T+) and have the disease (D+), and "false positives" are those that test positive (T+) but do not have the disease (D−). The negative likelihood ratio is calculated as
LR − = 1 − sensitivity specificity {\displaystyle {\text{LR}}-={\frac {1-{\text{sensitivity}}}{\text{specificity}}}}
which is equivalent to
LR − = Pr ( T − ∣ D + ) Pr ( T − ∣ D − ) {\displaystyle {\text{LR}}-={\frac {\Pr({T-}\mid D+)}{\Pr({T-}\mid D-)}}}
or "the probability of a person who has the disease testing negative divided by the probability of a person who does not have the disease testing negative." The calculation of likelihood ratios for tests with continuous values or more than two outcomes is similar to the calculation for dichotomous outcomes; a separate likelihood ratio is simply calculated for every level of test result and is called interval or stratum specific likelihood ratios. The pretest odds of a particular diagnosis, multiplied by the likelihood ratio, determines the post-test odds. This calculation is based on Bayes' theorem. (Note that odds can be calculated from, and then converted to, probability.)
Application to medicine Pretest probability refers to the chance that an individual in a given population has a disorder or condition; this is the baseline probability prior to the use of a diagnostic test. Post-test probability refers to the probability that a condition is truly present given a positive test result. For a good test in a population, the post-test probability will be meaningfully higher or lower than the pretest probability. A high likelihood ratio indicates a good test for a population, and a likelihood ratio close to one indicates that a test may not be appropriate for a population. For a screening test, the population of interest might be the general population of an area. For diagnostic testing, the ordering clinician will have observed some symptom or other factor that raises the pretest probability relative to the general population. A likelihood ratio of greater than 1 for a test in a population indicates that a positive test result is evidence that a condition is present. If the likelihood ratio for a test in a population is not clearly better than one, the test will not provide good evidence: the post-test probability will not be meaningfully different from the pretest probability. Knowing or estimating the likelihood ratio for a test in a population allows a clinician to better interpret the result. Research suggests that physicians rarely make these calculations in practice, however, and when they do, they often make errors. A randomized controlled trial compared how well physicians interpreted diagnostic tests that were presented as either sensitivity and specificity, a likelihood ratio, or an inexact graphic of the likelihood ratio, found no difference between the three modes in interpretation of test results.
Estimation table This table provide examples of how changes in the likelihood ratio affects post-test probability of disease.
*These estimates are accurate to within 10% of the calculated answer for all pre-test probabilities between 10% and 90%. The average error is only 4%. For polar extremes of pre-test probability >90% and <10%, see Estimation of pre- and post-test probability section below.
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