Likelihood ratio test

A hypothesis test that compares the goodness-of-fit of two models – an unconstrained model with all parameters free, and its corresponding model constrained by the null hypothesis to fewer parameters – to determine which offers a better fit for your sample data. For example, you can use this test to compare an unconstrained 3-parameter Weibull distribution to a constrained 2-parameter Weibull distribution to determine which is the better model for your data. The comparison is based on the ratio of the maximized likelihood function of the constrained model to the maximized likelihood function of the model without constraints. If the value of this ratio is sufficiently small, you conclude that the unconstrained model fits your sample data better than the simpler model constrained by the null hypothesis. Likelihood-ratio tests are useful methods for testing complicated models.

For example, Minitab's Individual Distribution Identification procedure in the Quality Tools menu uses a likelihood ratio test to compare the goodness-of-fit of a 1-parameter exponential distribution with the unconstrained 2-parameter exponential distribution. If the LRT p-value is less than your a-level (usually 0.05 or 0.10), you conclude that the unconstrained 2-parameter model offers significantly better goodness-of-fit than the 1-parameter model for your sample data.

If l is the value of the likelihood ratio, then for large samples (-2lnl) follows a chi-square distribution with degrees of freedom equal to the difference between the number of free parameters in the unconstrained and constrained models. Therefore, Minitab often provides the p-values associated with the likelihood ratio test from the chi-square distribution.