Title: Detecting Random Effect Variance Heterogeneity in Linear Mixed Models
Through Recursive Partitioning 

Authors: Philip Buczak, Marie Beisemann, Markus Pauly, Philipp Doebler 

Abstract:
Hierarchical structures, where observations are nested in groups, are common
in psychological data, e.g., with students nested in classes or multiple
assessments nested within persons in longitudinal studies. Statistically, such
structures can be accounted for by linear mixed models (LMMs) which model
group-specific random effects (REs) for intercept and/or slope parameters. The
REs are often assumed to follow a multivariate normal distribution with a
common mean vector of fixed effects (FEs) and covariance matrix. Several
regression tree-based approaches have been proposed to detect subgroups
regarding the FEs, e.g., a treatment effect might be positive for younger
people, but negative for older people. In such approaches, the RE covariance
matrix is usually modelled globally, implying no differences between possible
subgroups in terms of effect heterogeneity. However, subgroups of people might
not only differ in terms of FEs, but also in terms of variability of the REs
around the FEs. For instance, in younger people, a treatment might work
similarly, implying a small RE variance, while in older people, a treatment
might work better for some and hardly at all for others, implying a larger RE
variance. In this work, we propose two methods to examine RE heterogeneity in
LMMs. First, a method based on Model-Based Recursive Partitioning that aims to
detect possible instability of RE (co-)variance parameters. Subgroups are
found in an unsupervised manner and data are split into the discovered
subgroups, yielding subgroup-specific LMM estimates. Second, a faster
alternative that uses a custom heuristic to identify RE heterogeneity. We
compare and assess both methods in a simulation study.