α-investing: a procedure for sequential control of expected false discoveries
α-investing is an adaptive sequential methodology that encompasses a large family of procedures for testing multiple hypotheses. All control mFDR, which is the ratio of the expected number of false rejections to the expected number of rejections. mFDR is a weaker criterion than the false discovery rate, which is the expected value of the ratio. We compensate for this weakness by showing that α-investing controls mFDR at every rejected hypothesis. α-investing resembles α-spending that is used in sequential trials, but it has a key difference. When a test rejects a null hypothesis, α-investing earns additional probability towards subsequent tests. α-investing hence allows us to incorporate domain knowledge into the testing procedure and to improve the power of the tests. In this way, α-investing enables the statistician to design a testing procedure for a specific problem while guaranteeing control of mFDR.