Likelihood-ratio test

In statistics, the likelihood-ratio test is a hypothesis test that involves comparing the goodness of fit of two competing statistical models, typically one found by maximization over the entire parameter space and another found after imposing some constraint, based on the ratio of their likelihoods. If the more constrained model (i.e., the null hypothesis) is supported by the observed data, the two likelihoods should not differ by more than sampling error. Thus the likelihood-ratio test tests whether this ratio is significantly different from one, or equivalently whether its natural logarithm is significantly different from zero.

The likelihood-ratio test, also known as Wilks test, is the oldest of the three classical approaches to hypothesis testing, together with the Lagrange multiplier test and the Wald test. In fact, the latter two can be conceptualized as approximations to the likelihood-ratio test, and are asymptotically equivalent. In the case of comparing two models each of which has no unknown parameters, use of the likelihood-ratio test can be justified by the Neyman–Pearson lemma. The lemma demonstrates that the test has the highest power among all competitors.

Definition

General

Suppose that we have a statistical model with parameter space

Θ

{\displaystyle \Theta }

. A null hypothesis is often stated by saying that the parameter

θ

{\displaystyle \theta }

lies in a specified subset

Θ

0

{\displaystyle \Theta _{0}}

of

Θ

{\displaystyle \Theta }

. The alternative hypothesis is thus that

θ

{\displaystyle \theta }

lies in the complement of

Θ

0

{\displaystyle \Theta _{0}}

, i.e.