  Integrative analysis of data from multiple sources is critical to making
generalizable discoveries. Associations that are consistently observed across
multiple source populations are more likely to be generalized to target
populations with possible distributional shifts. In this paper, we model the
heterogeneous multi-source data with multiple high-dimensional regressions and
make inferences for the maximin effect (Meinshausen, B{\"u}hlmann, AoS, 43(4),
1801--1830). The maximin effect provides a measure of stable associations
across multi-source data. A significant maximin effect indicates that a
variable has commonly shared effects across multiple source populations, and
these shared effects may be generalized to a broader set of target populations.
There are challenges associated with inferring maximin effects because its
point estimator can have a non-standard limiting distribution. We devise a
novel sampling method to construct valid confidence intervals for maximin
effects. The proposed confidence interval attains a parametric length. This
sampling procedure and the related theoretical analysis are of independent
interest for solving other non-standard inference problems. Using genetic data
on yeast growth in multiple environments, we demonstrate that the genetic
variants with significant maximin effects have generalizable effects under new
environments.
