  Many important problems in psychology and biomedical studies require testing
for overdispersion, correlation and heterogeneity in mixed effects and latent
variable models, and score tests are particularly useful for this purpose. But
the existing testing procedures depend on restrictive assumptions. In this
paper we propose a class of test statistics based on a general mixed effects
model to test the homogeneity hypothesis that all of the variance components
are zero. Under some mild conditions, not only do we derive asymptotic
distributions of the test statistics, but also propose a resampling procedure
for approximating their asymptotic distributions conditional on the observed
data. To overcome the technical challenge, we establish an invariance principle
for random quadratic forms indexed by a parameter. A simulation study is
conducted to investigate the empirical performance of the test statistics. A
real data set is analyzed to illustrate the application of our theoretical
results.
