Beyond Neyman-Pearson: e-values enable hypothesis testing with a data-driven alpha

Abstract

A standard practice in statistical hypothesis testing is to mention the p-value alongside the accept/reject decision. We show the advantages of mentioning an e-value instead. With p-values, it is not clear how to use an extreme observation (e.g. p $\ll \alpha$) for getting better frequentist decisions. With e-values it is straightforward, since they provide Type-I risk control in a generalized Neyman-Pearson setting with the decision task (a general loss function) determined post-hoc, after observation of the data -- thereby providing a handle on `roving $\alpha$'s'. When Type-II risks are taken into consideration, the only admissible decision rules in the post-hoc setting turn out to be e-value-based. Similarly, if the loss incurred when specifying a faulty confidence interval is not fixed in advance, standard confidence intervals and distributions may fail whereas e-confidence sets and e-posteriors still provide valid risk guarantees. Sufficiently powerful e-values have by now been developed for a range of classical testing problems. We discuss the main challenges for wider development and deployment.

arXiv commentThird, once again thoroughly revised version. Part of the material in the first version has moved to another paper, "The E-Posterior", to appear in Phil. Trans. Royal Soc. of London Series A. Compared to the second version, the technical treatment in this version has been considerably simplified