A Continuous Generalization of Hypothesis Testing

Testing has developed into the fundamental statistical framework for falsifying hypotheses. Unfortunately, tests are binary in nature: a test either rejects a hypothesis or not. Such binary decisions do not reflect the reality of many scientific studies, which often aim to present the evidence against a hypothesis and do not necessarily intend to establish a definitive conclusion. To solve this, we propose the continuous generalization of a test, which we use to measure the evidence against a hypothesis. Such a continuous test can be interpreted as a non-randomized interpretation of the classical 'randomized test'. This offers the benefits of a randomized test, without the downsides of external randomization. Another interpretation is as a literal measure, which measures the amount of binary tests that reject the hypothesis. Our work also offers a new perspective on the $e$-value: the $e$-value is recovered as a continuous test with $\alpha \to 0$, or as an unbounded measure of the amount of rejections.