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.

Suppose that we have a statistical model with parameter space ${\displaystyle \Theta }$. A null hypothesis is often stated by saying that the parameter ${\displaystyle \theta }$ lies in a specified subset ${\displaystyle \Theta _{0}}$ of ${\displaystyle \Theta }$. The alternative hypothesis is thus that ${\displaystyle \theta }$ lies in the complement of ${\displaystyle \Theta _{0}}$, i.e. in ${\displaystyle \Theta ~\backslash ~\Theta _{0}}$, which is denoted by ${\displaystyle \Theta _{0}^{\text{c}}}$. The likelihood ratio test statistic for the null hypothesis ${\displaystyle H_{0}\,:\,\theta \in \Theta _{0}}$ is given by:

$${\displaystyle \lambda _{\text{LR}}=-2\ln \left[{\frac {~\sup _{\theta \in \Theta _{0}}{\mathcal {L}}(\theta )~}{~\sup _{\theta \in \Theta }{\mathcal {L}}(\theta )~}}\right]}$$

where the quantity inside the brackets is called the likelihood ratio. Here, the ${\displaystyle \sup }$ 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.

Often the likelihood-ratio test statistic is expressed as a difference between the log-likelihoods $${\displaystyle \lambda _{\text{LR}}=-2\left[\ell (\theta _{0})-\ell ({\hat {\theta }})\right]}$$ where $${\displaystyle \ell ({\hat {\theta }})\equiv \ln \left[\,\sup _{\theta \in \Theta }{\mathcal {L}}(\theta )\,\right]}$$ is the logarithm of the maximized likelihood function ${\displaystyle {\mathcal {L}}}$, and ${\displaystyle \ell (\theta _{0})}$ is the maximal value in the special case that the null hypothesis is true (but not necessarily a value that maximizes ${\displaystyle {\mathcal {L}}}$ for the sampled data) and $${\displaystyle \theta _{0}\in \Theta _{0}\qquad {\text{ and }}\qquad {\hat {\theta }}\in \Theta ~}$$ denote the respective arguments of the maxima and the allowed ranges they're embedded in. Multiplying by −2 ensures mathematically that (by Wilks' theorem) ${\displaystyle \lambda _{\text{LR}}}$ converges asymptotically to being χ²-distributed if the null hypothesis happens to be true. The finite-sample distributions of likelihood-ratio statistics are generally unknown.

The likelihood-ratio test requires that the models be nested – i.e. the more complex model can be transformed into the simpler model by imposing constraints on the former's parameters. Many common test statistics are tests for nested models and can be phrased as log-likelihood ratios or approximations thereof: e.g. the Z-test, the F-test, the G-test, and Pearson's chi-squared test; for an illustration with the one-sample t-test, see below.

If the models are not nested, then instead of the likelihood-ratio test, there is a generalization of the test that can usually be used: for details, see relative likelihood.