We develop the theory of hypothesis testing based on the e-value, a notion of evidence that, unlike the p-value, allows for effortlessly combining results from several studies in the common scenario where the decision to perform a new study may depend on previous outcomes. Tests based on e-values are safe, i.e. they preserve Type-I error guarantees, under such optional continuation. We define growth-rate optimality (GRO) as an analogue of power in an optional continuation context, and we show how to construct GRO e-variables for general testing problems with composite null and alternative, emphasizing models with nuisance parameters. GRO e-values take the form of Bayes factors with special priors. We illustrate the theory using several classic examples including a one-sample safe t-test and the 2 x 2 contingency table. Sharing Fisherian, Neymanian and Jeffreys-Bayesian interpretations, e-values may provide a methodology acceptable to adherents of all three schools.

翻译：我们基于e值发展了假设检验理论。与p值不同，e值作为一种证据度量，在决定开展新研究可能取决于先前结果的常见场景中，能够轻松整合多项研究的结果。基于e值的检验是安全的，即在可选继续检验的情况下，仍能保持I类错误控制。我们定义了增长率最优性（GRO）作为可选继续检验背景下检验效力的对应概念，并展示了如何为具有复合原假设和备择假设的一般检验问题构造GRO e变量，重点关注含 nuisance 参数的模型。GRO e值采用具有特殊先验的贝叶斯因子形式。我们通过几个经典实例（包括单样本安全t检验和2×2列联表）阐述了该理论。由于兼收Fisher学派、Neyman学派和Jeffreys-Bayes学派的解释，e值可能提供一种能被三个学派拥护者共同接受的方法论。