Likelihood ratios in diagnostic testing

In evidence-based medicine, likelihood ratios are used for assessing the value of performing a diagnostic test. They use the sensitivity and specificity of the test to determine whether a test result usefully changes the probability that a condition (such as a disease state) exists. 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.

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

which is equivalent to

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

which is equivalent to

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.)

...
Wikipedia