Order-restricted score tests for homogeneity in generalised linear and nonlinear mixed models

Lin (1997), building on work by Solomon & Cox (1992) and Lin & Breslow (1996), introduced a score test for homogeneity in the generalised linear mixed model. In this paper we propose an improvement to Lin's test that exploits the fact that covariance parameters associated with random effects in the model are constrained so as to form a positive semidefinite covariance matrix. Therefore, an omnibus score test will have power against alternatives that are known never to occur. Our improvement works by concentrating the score test into directions given by the positive semidefiniteness constraint. We apply the same sort of modification to the bias-corrected version of her score test in the generalised linear mixed model context and, in a nonlinear mixed model context, to a score test analogous to hers. We show how to obtain the limiting distribution of our proposed test statistic. We also argue, via simulations and local alternative power calculations, that the restricted test has better power than the unrestricted one.