Multiple hypothesis testing with persistent homology

Multiple hypothesis testing requires a control procedure: the error probabilities in statistical testing compound when several tests are performed for the same conclusion. A common type of multiple hypothesis testing error rates is the FamilyWise Error Rate (FWER) which measures the probability that any one of the performed tests rejects its null hypothesis erroneously. These are often controlled using Bonferroni’s method or later more sophisticated approaches all of which involve replacing the test level α with α/k, reducing it by a factor of the number of simultaneous tests performed. Common paradigms for hypothesis testing in persistent homology are often based on permutation testing, however increasing the number of permutations to meet a Bonferroni-style threshold can be prohibitively expensive. In this paper we propose a null model based approach to testing for acyclicity (ie trivial homology), coupled with a Family-Wise Error Rate (FWER) control method that does not suffer from these computational costs.