  Cross-level interactions among fixed effects in linear mixed models (also
known as multilevel models) are often complicated by the variances stemming
from random effects and residuals. When these variances change across clusters,
tests of fixed effects (including cross-level interaction terms) are subject to
inflated Type I or Type II error. While the impact of variance
change/heterogeneity has been noticed in the literature, few methods have been
proposed to detect this heterogeneity in a simple, systematic way. In addition,
when heterogeneity among clusters is detected, researchers often wish to know
which clusters' variances differed from the others. In this study, we utilize a
recently-proposed family of score-based tests to distinguish between
cross-level interactions and heterogeneity in variance components, also
providing information about specific clusters that exhibit heterogeneity. These
score-based tests only require estimation of the null model (when variance
homogeneity is assumed to hold), and they have been previously applied to
psychometric models. We extend the tests to linear mixed models here, detailing
their implementation and performance when the data generating model is known.
We also include an empirical example illustrating the tests' use in practice.
