# Score-based Tests for Explaining Upper-Level Heterogeneity in Linear Mixed Models

## TL;DR

This paper introduces score-based tests to detect and explain heterogeneity in variance components within linear mixed models, helping researchers identify clusters with differing variances and improve fixed effect inference.

## Contribution

The study extends score-based tests to linear mixed models, providing a systematic method to detect variance heterogeneity and identify specific clusters with differing variances.

## Key findings

- Score-based tests effectively detect heterogeneity in variance components.
- The tests can identify specific clusters with differing variances.
- Application to empirical data demonstrates practical utility.

## Abstract

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.