Online control of the familywise error rate

Biological research often involves testing a growing number of null hypotheses as new data are accumulated over time. We study the problem of online control of the familywise error rate, that is testing an a priori unbounded sequence of hypotheses (p-values) one by one over time without knowing the future, such that with high probability there are no false discoveries in the entire sequence. This paper unifies algorithmic concepts developed for offline (single batch) familywise error rate control and online false discovery rate control to develop novel online familywise error rate control methods. Though many offline familywise error rate methods (e.g., Bonferroni, fallback procedures and Sidak's method) can trivially be extended to the online setting, our main contribution is the design of new, powerful, adaptive online algorithms that control the familywise error rate when the p-values are independent or locally dependent in time. Our numerical experiments demonstrate substantial gains in power, that are also formally proved in an idealized Gaussian sequence model. A promising application to the International Mouse Phenotyping Consortium is described.