  in statistics, a likelihood ratio test is a statistical test used to compare the 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 neymanâpearson 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: : \begin{align} d & = -2\ln\left( \frac{\text{likelihood for null model}}{\text{likelihood for alternative model}} \right) \\ &= -2\ln(\text{likelihood for null model}) + 2\ln(\text{likelihood for alternative model}) \\ \end{align} 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&nbsp;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 df2&nbsp;&minus;&nbsp;df1&nbsp;. symbols df1 and df2 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 &minus;8024 and the alternative model has 3 parameters and a log-likelihood of &minus;8012, then the probability of this difference is that of chi-squar
Statistical_ratios Statistical_tests Statistical_theory