Generalized Linear mixed models (GLMMs) are widely used for regression analysis of data, continuous or discrete, that are

assumed to be clustered or correlated. Assessing model fit is

important for valid inference. We therefore propose a class of

chi-squared goodness-of-fit tests for GLMMs. Our test

statistic is a quadratic form in the differences between observed

values and the values expected under the estimated model in cells

defined by a partition of the covariate space. We show that this

test statistic has an asymptotic chi-squared distribution. We study

the power of the test through simulations for two special cases of

GLMMs, linear mixed models (LMMs) and logistic mixed models. For

LMMs, we further derive the analytical power of the test under

contiguous local alternatives and compare it with simulated

empirical power. Three examples are used to illustrate the proposed test.