The $k$-sample testing problem tests whether or not $k$ groups of data points are sampled from the same distribution. Multivariate analysis of variance (MANOVA) is currently the gold standard for $k$-sample testing but makes strong, often inappropriate, parametric assumptions. Moreover, independence testing and $k$-sample testing are tightly related, and there are many nonparametric multivariate independence tests with strong theoretical and empirical properties, including distance correlation (Dcorr) and Hilbert-Schmidt-Independence-Criterion (Hsic). We prove that universally consistent independence tests achieve universally consistent $k$-sample testing and that $k$-sample statistics like Energy and Maximum Mean Discrepancy (MMD) are exactly equivalent to Dcorr. Empirically evaluating these tests for $k$-sample scenarios demonstrates that these nonparametric independence tests typically outperform MANOVA, even for Gaussian distributed settings. Finally, we extend these non-parametric $k$-sample testing procedures to perform multiway and multilevel tests. Thus, we illustrate the existence of many theoretically motivated and empirically performant $k$-sample tests. A Python package with all independence and k-sample tests called hyppo is available from https://hyppo.neurodata.io/.
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