loglikelihoodratio

Log-likelihood ratio, often abbreviated as the log-likelihood ratio statistic, is a measure used in statistical inference to compare two competing statistical models or hypotheses based on their likelihoods given observed data. It typically compares a null model with an alternative model by evaluating how well each explains the data through their maximum likelihoods. If L(θ̂) denotes the maximum likelihood under the alternative (unrestricted) model and L(θ0) denotes the maximum likelihood under the null (restricted) model, the likelihood ratio is Λ = L(θ0) / L(θ̂). The common test statistic is D = -2 log Λ = -2 [log L(θ0) − log L(θ̂)], which is large when the null model is much less supported by the data than the alternative.

In large samples, under regularity conditions, the distribution of the statistic D under the null hypothesis approximates a chi-square distribution with degrees of freedom equal to the number of constrained parameters (the difference in dimension between the two models). This is known as Wilks’ theorem. The result provides a convenient way to perform hypothesis testing: compare the observed D to a chi-square critical value, or compute a p-value. The log-likelihood ratio test is widely used for nested models, where the null is a special case of the alternative, and can be extended to composite nulls and profile likelihoods to handle nuisance parameters.

Related concepts include the generalized likelihood ratio test (GLRT) for certain non-nested scenarios and information criteria such as AIC or BIC, which use the maximum log-likelihood but with different penalties for model complexity. Computing the statistic generally requires fitting both models via maximum likelihood estimation, often through numerical optimization. The log-likelihood ratio provides a flexible, data-driven framework for model comparison and hypothesis testing in many statistical settings.