A multiple test correction for streams and cascades of statistical hypothesis tests

Statistical hypothesis testing is a popular and powerful tool for inferring knowledge from data. For every such test performed, there is always a non-zero probability of making a false discovery, i.e.~rejecting a null hypothesis in error. Familywise error rate (FWER) is the probability of making at least one false discovery during an inference process. The expected FWER grows exponentially with the number of hypothesis tests that are performed, almost guaranteeing that an error will be committed if the number of tests is big enough and the risk is not managed; a problem known as the multiple testing problem. State-of-the-art methods for controlling FWER in multiple comparison settings require that the set of hypotheses be predetermined. This greatly hinders statistical testing for many modern applications of statistical inference, such as model selection, because neither the set of hypotheses that will be tested, nor even the number of hypotheses, can be known in advance. This paper introduces Subfamilywise Multiple Testing, a multiple-testing correction that can be used in applications for which there are repeated pools of null hypotheses from each of which a single null hypothesis is to be rejected and neither the specific hypotheses nor their number are known until the final rejection decision is completed. To demonstrate the importance and relevance of this work to current machine learning problems, we further refine the theory to the problem of model selection and show how to use Subfamilywise Multiple Testing for learning graphical models. We assess its ability to discover graphical models on more than 7,000 datasets, studying the ability of Subfamilywise Multiple Testing to outperform the state of the art on data with varying size and dimensionality, as well as with varying density and power of the present correlations. Subfamilywise Multiple Testing provides a significant improvement in statistical efficiency, often requiring only half as much data to discover the same model, while strictly controlling FWER.