It is frequently of interest to jointly analyze two paired sequences of multiple tests. This paper studies the problem of detecting whether there are more pairs of tests that are significant in both sequences than would be expected by chance. The asymptotic detection boundary is derived in terms of parameters such as the sparsity of non-null cases in each sequence, the effect sizes of the signals, and the magnitude of the dependence between the two sequences. A new test for detecting weak dependence is also proposed, shown to be asymptotically adaptively optimal, studied in simulations, and applied to study genetic pleiotropy in 10 pediatric autoimmune diseases.
Keywords: Detection boundary; Higher criticism; Independence testing; Optimal adaptivity; Sparsity.