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. in Θ ∖ Θ 0 {\displaystyle \Theta ~\backslash ~\Theta _{0}} , which is denoted by Θ 0 c {\displaystyle \Theta _{0}^{\text{c}}} . The likelihood ratio test statistic for the null hypothesis H 0 : θ ∈ Θ 0 {\displaystyle H_{0}\,:\,\theta \in \Theta _{0}} is given by:
λ LR = − 2 ln [ sup θ ∈ Θ 0 L ( θ ) sup θ ∈ Θ L ( θ ) ] {\displaystyle \lambda _{\text{LR}}=-2\ln \left[{\frac {~\sup _{\theta \in \Theta _{0}}{\mathcal {L}}(\theta )~}{~\sup _{\theta \in \Theta }{\mathcal {L}}(\theta )~}}\right]}
where the quantity inside the brackets is called the likelihood ratio. Here, the sup {\displaystyle \sup } notation refers to the supremum. As all likelihoods are positive, and as the constrained maximum cannot exceed the unconstrained maximum, the likelihood ratio is bounded between zero and one and the likelihood ratio test statistic between 0 and infinity. Often the likelihood-ratio test statistic is expressed as a difference between the log-likelihoods
λ LR = − 2 [ ℓ ( θ 0 ) − ℓ ( θ ^ ) ] {\displaystyle \lambda _{\text{LR}}=-2\left[\ell (\theta _{0})-\ell ({\hat {\theta }})\right]}
where
ℓ ( θ ^ ) ≡ ln [ sup θ ∈ Θ L ( θ ) ] {\displaystyle \ell ({\hat {\theta }})\equiv \ln \left[\,\sup _{\theta \in \Theta }{\mathcal {L}}(\theta )\,\right]}
is the logarithm of the maximized likelihood function L {\displaystyle {\mathcal {L}}} , and ℓ ( θ 0 ) {\displaystyle \ell (\theta _{0})} is the maximal value in the special case that the null hypothesis is true (but not necessarily a value that maximizes L {\displaystyle {\mathcal {L}}} for the sampled data) and
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