  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.
