Online control of the familywise error rate

Specifically, without knowing the future p -values, the analyst must irrevocably decide at each step whether to reject the null, such that with probability at least 1 α, there are no false rejections in the entire sequence. This paper unifies algorithm design concepts developed for offline FWER control and for online false discovery rate (FDR) control. Though Bonferroni, fallback procedures and Sidak's method can trivially be extended to the online setting, our main contribution is the design of new, adaptive online algorithms that control the FWER and per-family error rate (PFER) when the p -values are independent or locally dependent in time. Our experiments demonstrate substantial gains in power, also formally proved in an idealized Gaussian model. 1 Introduction Online multiple testing refers to the setting in which a potentially infinite stream of hypotheses H 1,H 2,... (respectively p -values P 1,P 2,...) is tested sequentially one at a time. At each step t N, one must decide whether to reject the current null hypothesis H t or not, without knowing the outcomes of all the future tests. Typically, we reject the null hypothesis when P t is smaller than some threshold α t. Let R represent the set of rejected null hypotheses, and H 0 be the unknown set of true null hypotheses; then, V R H 0 is the set of incorrectly rejected null hypotheses, also known as false discoveries. Denoting V V, some common error metrics are the false discovery rate (FDR), family wise error rate (FWER), per-family error rate (PFER) and power which are defined as FDR E null V R 1 null, FWER Pr{ V 1}, PFER E [V ], power E null H c 0 R H c 0 null .