Fixed effects testing in high-dimensional linear mixed models

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.