  We consider the problem of sequential multiple hypothesis testing with
nontrivial data collection costs. This problem appears, for example, when
conducting biological experiments to identify differentially expressed genes of
a disease process. This work builds on the generalized $\alpha$-investing
framework which enables control of the false discovery rate in a sequential
testing setting. We make a theoretical analysis of the long term asymptotic
behavior of $\alpha$-wealth which motivates a consideration of sample size in
the $\alpha$-investing decision rule. Posing the testing process as a game with
nature, we construct a decision rule that optimizes the expected
$\alpha$-wealth reward (ERO) and provides an optimal sample size for each test.
Empirical results show that a cost-aware ERO decision rule correctly rejects
more false null hypotheses than other methods for $n=1$ where $n$ is the sample
size. When the sample size is not fixed cost-aware ERO uses a prior on the null
hypothesis to adaptively allocate of the sample budget to each test. We extend
cost-aware ERO investing to finite-horizon testing which enables the decision
rule to allocate samples in a non-myopic manner. Finally, empirical tests on
real data sets from biological experiments show that cost-aware ERO balances
the allocation of samples to an individual test against the allocation of
samples across multiple tests.
