  Many scientific and engineering challenges -- ranging from pharmacokinetic
drug dosage allocation and personalized medicine to marketing mix (4Ps)
recommendations -- require an understanding of the unobserved heterogeneity in
order to develop the best decision making-processes. In this paper, we develop
a hypothesis test and the corresponding p-value for testing for the
significance of the homogeneous structure in linear mixed models. A robust
matching moment construction is used for creating a test that adapts to the
size of the model sparsity. When unobserved heterogeneity at a cluster level is
constant, we show that our test is both consistent and unbiased even when the
dimension of the model is extremely high. Our theoretical results rely on a new
family of adaptive sparse estimators of the fixed effects that do not require
consistent estimation of the random effects. Moreover, our inference results do
not require consistent model selection. We showcase that moment matching can be
extended to nonlinear mixed effects models and to generalized linear mixed
effects models. In numerical and real data experiments, we find that the
developed method is extremely accurate, that it adapts to the size of the
underlying model and is decidedly powerful in the presence of irrelevant
covariates.
