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In statistics, a likelihood ratio test is a statistical test used to compare the goodness of fit of two models, one of which (the ''null model'') is a special case of the other (the ''alternative model''). The test is based on the likelihood ratio, which expresses how many times more likely the data are under one model than the other. This likelihood ratio, or equivalently its logarithm, can then be used to compute a ''p''-value, or compared to a critical value to decide whether to reject the null model in favour of the alternative model. When the logarithm of the likelihood ratio is used, the statistic is known as a log-likelihood ratio statistic, and the probability distribution of this test statistic, assuming that the null model is true, can be approximated using Wilks's theorem. In the case of distinguishing between two models, each of which has no unknown parameters, use of the likelihood ratio test can be justified by the Neyman–Pearson lemma, which demonstrates that such a test has the highest power among all competitors. ==Use== Each of the two competing models, the null model and the alternative model, is separately fitted to the data and the log-likelihood recorded. The test statistic (often denoted by ''D'') is twice the difference in these log-likelihoods: : The model with more parameters will always fit at least as well (have an equal or greater log-likelihood). Whether it fits significantly better and should thus be preferred is determined by deriving the probability or ''p''-value of the difference ''D''. Where the null hypothesis represents a special case of the alternative hypothesis, the probability distribution of the test statistic is approximately a chi-squared distribution with degrees of freedom equal to ''df''2 − ''df''1 . Symbols ''df''1 and ''df''2 represent the number of free parameters of models 1 and 2, the null model and the alternative model, respectively. Here is an example of use. If the null model has 1 parameter and a log-likelihood of −8024 and the alternative model has 3 parameters and a log-likelihood of −8012, then the probability of this difference is that of chi-squared value of +2·(8024 − 8012) = 24 with 3 − 1 = 2 degrees of freedom. Certain assumptions must be met for the statistic to follow a chi-squared distribution, and often empirical ''p''-values are computed. The likelihood-ratio test requires nested models, i.e. models in which the more complex one can be transformed into the simpler model by imposing a set of constraints on the parameters. If the models are not nested, then a generalization of the likelihood-ratio test can usually be used instead: the relative likelihood. 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Likelihood-ratio test」の詳細全文を読む スポンサード リンク
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