  Hierarchical data analysis is crucial in various fields for making
discoveries. The linear mixed model is often used for training hierarchical
data, but its parameter estimation is computationally expensive, especially
with big data. Subsampling techniques have been developed to address this
challenge. However, most existing subsampling methods assume homogeneous data
and do not consider the possible heterogeneity in hierarchical data. To address
this limitation, we develop a new approach called group-orthogonal subsampling
(GOSS) for selecting informative subsets of hierarchical data that may exhibit
heterogeneity. GOSS selects subdata with balanced data size among groups and
combinatorial orthogonality within each group, resulting in subdata that are
$D$- and $A$-optimal for building linear mixed models. Estimators of parameters
trained on GOSS subdata are consistent and asymptotically normal. GOSS is shown
to be numerically appealing via simulations and a real data application.
Theoretical proofs, R codes, and supplementary numerical results are accessible
online as Supplementary Materials.
