  Penalized variable selection for high dimensional longitudinal data has
received much attention as accounting for the correlation among repeated
measurements and providing additional and essential information for improved
identification and prediction performance. Despite the success, in longitudinal
studies the potential of penalization methods is far from fully understood for
accommodating structured sparsity. In this article, we develop a sparse group
penalization method to conduct the bi-level gene-environment (G$\times$E)
interaction study under the repeatedly measured phenotype. Within the quadratic
inference function (QIF) framework, the proposed method can achieve
simultaneous identification of main and interaction effects on both the group
and individual level. Simulation studies have shown that the proposed method
outperforms major competitors. In the case study of asthma data from the
Childhood Asthma Management Program (CAMP), we conduct G$\times$E study by
using high dimensional SNP data as the Genetic factor and the longitudinal
trait, forced expiratory volume in one second (FEV1), as phenotype. Our method
leads to improved prediction and identification of main and interaction effects
with important implications.
