  The practice of pooling several individual test statistics to form aggregate
tests is common in many statistical application where individual tests may be
underpowered. While selection by aggregate tests can serve to increase power,
the selection process invalidates the individual test-statistics, making it
difficult to identify the ones that drive the signal in follow-up inference.
Here, we develop a general approach for valid inference following selection by
aggregate testing. We present novel powerful post-selection tests for the
individual null hypotheses which are exact for the normal model and
asymptotically justified otherwise. Our approach relies on the ability to
characterize the distribution of the individual test statistics after
conditioning on the event of selection. We provide efficient algorithms for
estimation of the post-selection maximum-likelihood estimates and suggest
confidence intervals which rely on a novel switching regime for good coverage
guarantees. We validate our methods via comprehensive simulation studies and
apply them to data from the Dallas Heart Study, demonstrating that single
variant association discovery following selection by an aggregated test is
indeed possible in practice.
