Likelihood Ratio

A quantity used to test Nested Hypotheses. Let $H'$ be a Nested Hypothesis with $n'$ Degrees of Freedom within $H$ (which has $n$ Degrees of Freedom), then calculate the Maximum Likelihood of a given outcome, first given $H'$, then given $H$. Then

$${\rm LR} = {\hbox{[likelihood H']}\over\hbox{[likelihood H]}}.$$

Comparison of this ratio to the critical value of the Chi-Squared Distribution with $n-n'$ Degrees of Freedom then gives the Significance of the increase in Likelihood.

The term likelihood ratio is also used (especially in medicine) to test nonnested complementary hypotheses as follows,

$${\rm LR} = {\hbox{[true positive rate]}\over\hbox{[false pos... ...rate]}} = {\hbox{[sensitivity]}\over 1-\hbox{[specificity]}}.$$

See also Negative Likelihood Ratio, Sensitivity, Specificity