# A Likelihood Ratio Test

The likelihood ratio test is a test of the sufficiency of a smaller model versus a more complex model. The null hypothesis of the test states that the smaller model provides as good a fit for the data as the larger model. If the null hypothesis is rejected, then the alternative, larger model provides a significant improvement over the smaller model.

To use the likelihood ratio test, the null hypothesis model must be a model nested within, that is, a special case of, the alternative hypothesis model. For example, the scaled identity structure is a special case of the compound symmetry structure, and compound symmetry is a special case of the unstructured matrix. However, the autoregressive and compound symmetry structures are not special cases of each other.

The likelihood ratio test can be used to test repeated effect or random effect covariance structures, or both at the same time. For example, it is possible to test a model that has an identity structure for a random effect and an autoregressive structure for the repeated effect, versus a model that has a compound symmetry structure for the random effect and an unstructured matrix for the repeated effect. Simply make sure that the covariance structure for each effect in one model is nested within the covariance structures for the effects in the other model.