  Linear mixed models (LMMs), which typically assume normality for both the
random effects and error terms, are a popular class of methods for analyzing
longitudinal and clustered data. However, such models can be sensitive to
outliers, and this can lead to poor statistical results (e.g., biased inference
on model parameters and inaccurate prediction of random effects) if the data
are contaminated. We propose a new approach to robust estimation and inference
for LMMs using a hierarchical gamma divergence, which offers an automated,
data-driven approach to downweight the effects of outliers occurring in both
the error, and the random effects, using normalized powered density weights.
For estimation and inference, we develop a computationally scalable
minorization-maximization algorithm for the resulting objective function, along
with a clustered bootstrap method for uncertainty quantification and a
Hyvarinen score criterion for selecting a tuning parameter controlling the
degree of robustness. When the genuine and contamination mixed effects
distributions are sufficiently separated, then under suitable regularity
conditions assuming the number of clusters tends to infinity, we show the
resulting robust estimates can be asymptotically controlled even under a heavy
level of (covariate-dependent) contamination. Simulation studies demonstrate
hierarchical gamma divergence consistently outperforms several currently
available methods for robustifying LMMs, under a wide range of scenarios of
outlier generation at both the response and random effects levels. We
illustrate the proposed method using data from a multi-center AIDS cohort
study, where the use of a robust LMMs using hierarchical gamma divergence
approach produces noticeably different results compared to methods that do not
adequately adjust for potential outlier contamination.
