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We consider the problem of sequential multiple hypothesis testing with nontrivial data collection cost. This problem appears, for example, when conducting biological experiments to identify differentially expressed genes in a disease process. This work builds on the generalized α-investing framework that enables control of the false discovery rate in a sequential testing setting. We make a theoretical analysis of the long term asymptotic behavior of α-wealth which motivates a consideration of sample size in the α-investing decision rule. Using the game theoretic principle of indifference, we construct a decision rule that optimizes the expected return (ERO) of α-wealth and provides an optimal sample size for the test. We show empirical results that a cost-aware ERO decision rule correctly rejects more false null hypotheses than other methods. We extend cost-aware ERO investing to finite-horizon testing which enables the decision rule to hedge against the risk of unproductive tests. Finally, empirical tests on a real data set from a biological experiment show that cost-aware ERO produces actionable decisions as to which tests to conduct and if so at what sample size.
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