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### A1.7. Inference and Test Statistics

#### A1.7.1. Inference about the Covariance Parameters

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For inferences concerning the covariance parameters in your model, you can use likelihood-based statistics. One common such statistic is the Wald Z, which is computed as the parameter estimate divided by its estimated asymptotic standard error. The asymptotic standard errors are computed from the inverse of the second derivative matrix of the log likelihood with respect to the covariance parameters. The Wald Z test is valid for large samples, but it can be unreliable for small data sets and for parameters such as variance components that are known to have a skewed or bounded sampling distribution.

A better alternative is the likelihood ratio χ^{2}. This test compares two models that are nested with respect to the covariance parameters. The two models are often referred to as the full model and the reduced model. The reduced model is obtained from the full model by imposing one or more constraints on the covariance parameters—a process known as nesting models. The test statistic is the difference of the –2 log likelihoods between the reduced and the full model. Issues surrounding likelihood ratio testing in mixed models were addressed in A1.6.1—for example, the need for models to have the same X matrix when –2 Res Log Likelihoods are used to form test statistics and the nonstandard distribution theory when parameters are on the boundary of the parameter space. In order to perform likelihood ratio tests for covariance parameters, you need to run PROC MIXED twice to obtain the –2 (Res) Log Likelihoods under the full and reduced model.