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

Abstract

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.

KEYWORDS:

• Biological pathways
• Distribution-free tests
• Graph-based methods
• Optimal matching

Acknowledgments

The authors thank the editor, the associate editor, and the anonymous referees for their detailed and thoughtful comments, which greatly improved the quality and the presentation of the article.