Two-Sample Tests that are Safe under Optional Stopping, with an Application to Contingency Tables

We develop E variables for testing whether two data streams come from the same source or not, and more generally, whether the difference between the sources is larger than some minimal effect size. These E variables lead to tests that remain safe, i.e. keep their Type-I error guarantees, under flexible sampling scenarios such as optional stopping and continuation. In special cases our E variables also have an optimal `growth' property under the alternative. We illustrate the generic construction through the special case of 2x2 contingency tables, where we also allow for the incorporation of different restrictions on a composite alternative. Comparison to p-value analysis in simulations and a real-world example show that E variables, through their flexibility, often allow for early stopping of data collection, thereby retaining similar power as classical methods.