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In this article, we propose a nonparametric graphical test based on optimal matching, for assessing the equality of multiple unknown multivariate probability distributions. Our procedure pools the data from the different classes to create a graph based on the minimum non-bipartite matching, and then utilizes the number of edges connecting data points from different classes to examine the closeness between the distributions. The proposed test is exactly distribution-free (the null distribution does not depend on the distribution of the data) and can be efficiently applied to multivariate as well as non-Euclidean data, whenever the inter-point distances are well-defined. We show that the test is universally consistent, and prove a distributional limit theorem for the test statistic under general alternatives. Through simulation studies, we demonstrate its superior performance against other common and well-known multisample tests. The method is applied to single cell transcriptomics data obtained from the peripheral blood, cancer tissue, and tumor-adjacent normal tissue of human subjects with hepatocellular carcinoma and non-small-cell lung cancer. Our method unveils patterns in how biochemical metabolic pathways are altered across immune cells in a cancer setting, depending on the tissue location. All of the methods described herein are implemented in the R package multicross.  Supplementary materials  for this article are available online.

Distribution-Free Multisample Tests Based on Optimal Matchings With Applications to Single Cell Genomics