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Summary  The practice of pooling several individual test statistics to form aggregate tests is common in many statistical applications where individual tests may be underpowered. Although selection by aggregate tests can serve to increase power, the selection process invalidates inference based on the individual test statistics, making it difficult to identify those 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 computation 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 aggregate test is indeed possible in practice.

Post‐selection estimation and testing following aggregate association tests