LikelihoodRatio

Likelihood ratio is a statistic used in statistical inference to compare how well two competing hypotheses explain observed data. It is defined from the likelihood function, which measures the probability of the data given a set of model parameters. Given data x and a parameter value theta, the likelihood is L(theta; x). The likelihood ratio compares two scenarios, typically two hypotheses or two parameter values, via the ratio of their likelihoods. In hypothesis testing, the generalized form is Lambda(x) = sup{L(theta0; x)} / sup{L(theta; x): theta in Θ}, where the numerator is the maximum likelihood under the null hypothesis and the denominator is the maximum likelihood under the full parameter space. A related statistic is the log-likelihood ratio, often transformed as -2 log Lambda(x).

In the framework of hypothesis testing, the likelihood ratio test (LRT) is used to decide whether to reject the null hypothesis. For simple hypotheses, the Neyman-Pearson lemma shows that the likelihood ratio test is the most powerful test for distinguishing the two hypotheses. For composite hypotheses, the generalized likelihood ratio test uses the maximum likelihood under the null divided by the maximum likelihood overall. Under regularity conditions and large samples, -2 log Lambda(x) often follows an approximately chi-square distribution, with degrees of freedom equal to the number of free parameters constrained under the null (Wilks’ theorem).

Interpretation and caveats: a small likelihood ratio indicates the data are much more likely under the alternative than under the null, providing evidence against the null hypothesis. The method assumes the model is correctly specified and relies on asymptotic approximations; results can be sensitive to model misspecification or nuisance parameters. In Bayesian contexts, likelihood ratios relate to Bayes factors, which compare marginal likelihoods over parameter spaces.