A New Graph-Based Two-Sample Test for Multivariate and Object Data

ABSTRACT

Two-sample tests for multivariate data and especially for non-Euclidean data are not well explored. This article presents a novel test statistic based on a similarity graph constructed on the pooled observations from the two samples. It can be applied to multivariate data and non-Euclidean data as long as a dissimilarity measure on the sample space can be defined, which can usually be provided by domain experts. Existing tests based on a similarity graph lack power either for location or for scale alternatives. The new test uses a common pattern that was overlooked previously, and works for both types of alternatives. The test exhibits substantial power gains in simulation studies. Its asymptotic permutation null distribution is derived and shown to work well under finite samples, facilitating its application to large datasets. The new test is illustrated on two applications: The assessment of covariate balance in a matched observational study, and the comparison of network data under different conditions.

KEYWORDS:

• General alternatives
• Nonparametrics
• Permutation null distribution
• Similarity graph

Acknowledgment

The authors thank Dylan Small for very helpful discussions and for kindly providing the data for the analysis of college students matchings. The authors also thank two anonymous referees for very helpful comments.

Funding

Hao Chen is supported in part by NSF award DMS-1513653.