Score-based tests for detecting heterogeneity in linear mixed models

Cross-level interactions among fixed effects in linear mixed models (also known as multilevel models) can be complicated by heterogeneity stemming from random effects and residuals. When heterogeneity is present, 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 to detect measurement invariance. In this paper, we extend the tests to linear mixed models, allowing us to use the tests to differentiate between interaction and heterogeneity. We detail the tests' implementation and performance via simulation, provide an empirical example of the tests' use in practice, and provide code illustrating the tests' general application.