Dan Hall

University of Georgia

Variance Components Testing in Nonlinear and Generalized Linear Mixed Models

In recent years a great deal of statistical research has appeared that is concerned with broadening well-established classes of nonlinear regression models by incorporating random effects into the expectation function of the model. Both generalized linear models and normal error nonlinear regression models have been been broadened in this way. The resulting classes, generalized linear mixed models (GLMMs), and nonlinear mixed models (NLMMs), have been enthusiastically received by both researchers and practitioners for their flexibility and because they arise as natural extensions of existing models. However, substantial challenges remain related to the fitting of these models and to the properties of model parameter estimators. A natural question then, especially in light of these challenges, is whether the inclusion of random effects and the accompanying, often-cumbersome mixed model methodology is necessary for any particular data set. Lin (1997) has addressed this issue by proposing a score test for homogeneity in the GLMM. One of the nice features of her test is that it does not require fitting the mixed model, because its approximations to the score function and information matrix are derived under the null hypothesis of homogeneity. In this talk we propose an improvement to Lin's score test that exploits the fact that covariance parameters associated with random effects in the model are constrained to form a positive semidefinite covariance matrix. Therefore, an omnibus score test will have power against alternatives which are known never to occur. We improve on the power of Lin's test by concentrating the score test in directions given by the positive semidefiniteness constraint. Via simulation, we compare our test with Lin's in the context of the GLMM and with a score test analagous to her test in the NLMM.

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