Authors:Peter Grünwald, Rianne de Heide, Wouter Koolen (Submitted on 18 Jun 2019) Abstract: We present a new theory of hypothesis testing. The main concept is the S-value, a notion of evidence which, unlike p-values, allows for effortlessly combining evidence from several tests, even in the common scenario where the decision to perform a new test depends on the previous test outcome: safe tests based on S-values generally preserve Type-I error guarantees under such "optional continuation".