Simultaneous Testing of Grouped Hypotheses: Finding Needles in Multiple Haystacks
Tony Cai and Wenguang Sun
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Abstract:
In large-scale multiple testing problems, data are often collected
from heterogeneous sources and hypotheses form into groups that
exhibit different characteristics. Conventional approaches, including
the pooled and separate analyses, fail to efficiently utilize the
external grouping information. We develop a compound decision
theoretic framework for testing grouped hypotheses and introduce an
oracle procedure that minimizes the false non-discovery rate subject
to a constraint on the false discovery rate. It is shown that both the
pooled and separate analyses can be uniformly improved by the oracle
procedure. We then propose a data-driven procedure that is shown to be
asymptotically optimal. Simulation studies show that our procedures
enjoy superior performance and yield the most accurate results in
comparison with both the pooled and separate procedures. A real data
example with grouped hypotheses is studied in detail using different
methods. Both theoretical and numerical results demonstrate that
exploiting external information of the sample can greatly improve the
efficiency of a multiple testing procedure. The results also provide
insights on how the grouping information is incorporated for optimal
simultaneous inference.
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