Likelihood ratio

In diagnostic tests, the likelihood ratio is the likelihood that a clinical sign is in a patient with disease as compared to a patient without disease.

${\text{Likelihood ratio}}={\frac {\mbox{probability of test result with disease}}{\mbox{probability of same result without disease}}}$

${\text{Likelihood ratio}}={\frac {\mbox{sensitivity}}{1-{\mbox{specificity}}}}$

To calculate probabilities of disease using a likelihood ratio:

${\text{Post-test odds}}={\text{Pre-test odds}}*{\text{Likelihood}}\ {\text{ratio}}$

This is a form of Bayes' theorem from probability theory. In this form the theorem is called Bayes' rule.

Comparing likelihoods (or odds) is different than comparing percentages. (or probabilities).

${\text{Odds}}={\frac {\text{probability}}{(1-{\text{probability}})}}$

The likelihood ratio is an alternative to sensitivity and specificity for the numeric interpretation of diagnostic tests. In a randomized controlled trial that compared the two methods, physicians were able to use both similarly although the physicians had trouble with both methods.

In mathematical statistics, the likelihood ratio is the ratio of the probabilities, or probability densities, of given data, under two different probability models. In probability theory the likelihood ratio goes by the name of Radon-NIkodym derivative.

In Bayesian statistics the likelihood ratio is often called the Bayes' factor.

Calculations

Likelihood ratios are related to sensitivity and specificity.

The positive likelihood ratio (LR+) measures the likelihood of a finding being present in patient with the disease. A large LR+, for example a value more than 10, helps rule in disease.

${\text{LR+}}={\frac {\text{sensitivity}}{(1-{\text{specificity}})}}$

The negative likelihood ratio (LR-) measures the likelihood of a finding being absent in patient with the disease. A small LR-, for example a value less than 0.1, helps rule out disease.

${\text{LR-}}={\frac {(1-{\text{sensitivity}})}{\text{specificity}}}$

Facilitating interpretation

Interpreting likelihood ratios by physicians is difficult and likelihood ratios do not improve upon sensitivity and specificity for helping physicians.

Categorizing likelihood ratios based on strength may help.

Describing likelihood ratios in non-technical language such as "A positive result is observed approximately 10 times more frequently in people with the disease than in people without the disease", may help. However, a graphic display of predictive values is even better.