The GLIMMIX Procedure

The Likelihood Ratio Test

The likelihood ratio test (LRT) compares the likelihoods of two models where parameter estimates are obtained in two parameter spaces, the space $\Omega $ and the restricted subspace $\Omega _0$. In the GLIMMIX procedure, the full model defines $\Omega $ and the test-specification in the COVTEST statement determines the null parameter space $\Omega _0$. The likelihood ratio procedure consists of the following steps (see, for example, Bickel and Doksum 1977, p. 210):

  1. Find the estimate $\widehat{\btheta }$ of $\btheta \in \Omega $. Compute the likelihood $L(\widehat{\btheta })$.

  2. Find the estimate $\widehat{\btheta }_0$ of $\btheta \in \Omega _0$. Compute the likelihood $L(\widehat{\btheta }_0)$.

  3. Form the likelihood ratio

    \[ \overline{\lambda } = \frac{L(\widehat{\btheta })}{L(\widehat{\btheta }_0)} \]

  4. Find a function $f\left(\overline{\lambda }\right)$ that has a known distribution. $f(\cdot )$ serves as the test statistic for the likelihood ratio test.

Please note the following regarding the implementation of these steps in the COVTEST statement of the GLIMMIX procedure.

  • The function $f(\cdot )$ in step 4 is always taken to be

    \[ \lambda = 2 \log \left\{ \overline{\lambda } \right\} \]

    which is twice the difference between the log likelihoods for the full model and the model under the COVTEST restriction.

  • For METHOD= RSPL and METHOD= RMPL, the test statistic is based on the restricted likelihood.

  • For GLMMs involving pseudo-data, the test statistics are based on the pseudo-likelihood or the restricted pseudo-likelihood and are based on the final pseudo-data.

  • The parameter space $\Omega $ for the full model is typically not an unrestricted space. The GLIMMIX procedure imposes boundary constraints for variance components and scale parameters, for example. The specification of the subspace $\Omega _0$ must be consistent with these full-model constraints; otherwise the test statistic $\lambda $ does not have the needed distribution. You can remove the boundary restrictions with the NOBOUND option in the PROC GLIMMIX statement or the NOBOUND option in the PARMS statement.