  The explosion of biobank data offers immediate opportunities for
gene-environment (GxE) interaction studies of complex diseases because of the
large sample sizes and the rich collection in genetic and non-genetic
information. However, the extremely large sample size also introduces new
computational challenges in GxE assessment, especially for set-based GxE
variance component (VC) tests, which are a widely used strategy to boost
overall GxE signals and to evaluate the joint GxE effect of multiple variants
from a biologically meaningful unit (e.g., gene). In this work, we focus on
continuous traits and present SEAGLE, a Scalable Exact AlGorithm for
Large-scale set-based GxE tests, to permit GxE VC tests for biobank-scale data.
SEAGLE employs modern matrix computations to achieve the same "exact" results
as the original GxE VC tests without imposing additional assumptions or relying
on approximations. SEAGLE can easily accommodate sample sizes in the order of
$10^5$, is implementable on standard laptops, and does not require specialized
computing equipment. We demonstrate SEAGLE's performance through extensive
simulations. We illustrate its utility by conducting genome-wide gene-based GxE
analysis on the Taiwan Biobank data to explore the interaction of gene and
physical activity status on body mass index.
