Optimal Transport Based Testing in Factorial Designs

Abstract

We introduce a general framework for testing statistical hypotheses in factorial designs for probability measures supported on finite spaces. The suggested methodology is based on the pairwise comparison of measures using optimal transport (OT). The formulation of hypotheses is intuitive: It is a direct extension of those underlying the analysis of variance (ANOVA) and its nonparametric counterparts to test for linear relationships between (discrete) probability measures in factorial designs. To this end, means or cumulative distribution functions simply will be replaced by measures. We derive under the null hypotheses and under (local) alternatives the asymptotic distribution of the corresponding empirical OT test statistic, which is the optimal value of a linear program with random objective function. It turns out that this requires to extend existing techniques from probability measures to signed measures, and we show directional Hadamard differentiability and the validity of the functional delta method. We discuss computational issues, permutation and bootstrap tests, and back up our findings with simulations. We illustrate our methodology on datasets from cellular biophysics and from biometric identification.

48 pages