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An exact distribution‐free test comparing two multivariate distributions based on adjacency

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  • Paul R. Rosenbaum

Abstract

Summary. A new test is proposed comparing two multivariate distributions by using distances between observations. Unlike earlier tests using interpoint distances, the new test statistic has a known exact distribution and is exactly distribution free. The interpoint distances are used to construct an optimal non‐bipartite matching, i.e. a matching of the observations into disjoint pairs to minimize the total distance within pairs. The cross‐match statistic is the number of pairs containing one observation from the first distribution and one from the second. Distributions that are very different will exhibit few cross‐matches. When comparing two discrete distributions with finite support, the test is consistent against all alternatives. The test is applied to a study of brain activation measured by functional magnetic resonance imaging during two linguistic tasks, comparing brains that are impaired by arteriovenous abnormalities with normal controls. A second exact distribution‐free test is also discussed: it ranks the pairs and sums the ranks of the cross‐matched pairs.

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  • Paul R. Rosenbaum, 2005. "An exact distribution‐free test comparing two multivariate distributions based on adjacency," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 67(4), pages 515-530, September.
    • Handle: RePEc:bla:jorssb:v:67:y:2005:i:4:p:515-530
    DOI: 10.1111/j.1467-9868.2005.00513.x
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    Cited by:

    1. Jorge Luis Garcia & James J. Heckman & Victor Ronda, 2021. "The Lasting Effects of Early Childhood Education on Promoting the Skills and Social Mobility of Disadvantaged African Americans," Working Papers 2021-037, Human Capital and Economic Opportunity Working Group.
    2. Geng, Sen & Peng, Yujia & Shachat, Jason & Zhong, Huizhen, 2015. "Adolescents, cognitive ability, and minimax play," Economics Letters, Elsevier, vol. 128(C), pages 54-58.
    3. Hao Chen & Dylan S. Small, 2022. "New multivariate tests for assessing covariate balance in matched observational studies," Biometrics, The International Biometric Society, vol. 78(1), pages 202-213, March.
    4. Shin-ichi Tsukada, 2019. "High dimensional two-sample test based on the inter-point distance," Computational Statistics, Springer, vol. 34(2), pages 599-615, June.
    5. W. Lok & Stephen Lee, 2011. "A new statistical depth function with applications to multimodal data," Journal of Nonparametric Statistics, Taylor & Francis Journals, vol. 23(3), pages 617-631.
    6. Zhi Peng Ong & Aixiang Andy Chen & Tianming Zhu & Jin-Ting Zhang, 2023. "Testing Equality of Several Distributions at High Dimensions: A Maximum-Mean-Discrepancy-Based Approach," Mathematics, MDPI, vol. 11(20), pages 1-21, October.
    7. Denys Pommeret & Laurence Reboul & Anne-francoise Yao, 2023. "Testing the equality of the laws of two strictly stationary processes," Statistical Inference for Stochastic Processes, Springer, vol. 26(1), pages 193-214, April.
    8. Martin Boďa & Mariana Považanová, 2020. "Productivity patterns in Europe: adaptation of the Malmquist index to measuring group performance and productivity change over time," Empirica, Springer;Austrian Institute for Economic Research;Austrian Economic Association, vol. 47(4), pages 949-989, November.
    9. Albert Vexler & Young Min Kim & Jihnhee Yu & Nicole A. Lazar & Alan D. Hutson, 2014. "Computing Critical Values of Exact Tests by Incorporating Monte Carlo Simulations Combined with Statistical Tables," Scandinavian Journal of Statistics, Danish Society for Theoretical Statistics;Finnish Statistical Society;Norwegian Statistical Association;Swedish Statistical Association, vol. 41(4), pages 1013-1030, December.
    10. Biswas, Munmun & Ghosh, Anil K., 2014. "A nonparametric two-sample test applicable to high dimensional data," Journal of Multivariate Analysis, Elsevier, vol. 123(C), pages 160-171.
    11. Anil K. Ghosh & Munmun Biswas, 2016. "Distribution-free high-dimensional two-sample tests based on discriminating hyperplanes," TEST: An Official Journal of the Spanish Society of Statistics and Operations Research, Springer;Sociedad de Estadística e Investigación Operativa, vol. 25(3), pages 525-547, September.
    12. Gertler,Paul J.,Heckman,James J.,Pinto,Rodrigo Ribeiro Antunes,Chang-Lopez,Susan M.,Grantham-Mcgregor,Sally,Vermeersch,Christel M. J.,Walker,Susan,Wright,Amika S., 2021. "Effect of the Jamaica Early Childhood Stimulation Intervention on Labor Market Outcomes at Age 31," Policy Research Working Paper Series 9787, The World Bank.
    13. Petrie, Adam, 2016. "Graph-theoretic multisample tests of equality in distribution for high dimensional data," Computational Statistics & Data Analysis, Elsevier, vol. 96(C), pages 145-158.
    14. Ruoqi Yu, 2021. "Evaluating and improving a matched comparison of antidepressants and bone density," Biometrics, The International Biometric Society, vol. 77(4), pages 1276-1288, December.
    15. Modarres, Reza, 2014. "On the interpoint distances of Bernoulli vectors," Statistics & Probability Letters, Elsevier, vol. 84(C), pages 215-222.
    16. Ruiyi Zhang & R. Todd Ogden & Martin Picard & Anuj Srivastava, 2022. "Nonparametric k‐sample test on shape spaces with applications to mitochondrial shape analysis," Journal of the Royal Statistical Society Series C, Royal Statistical Society, vol. 71(1), pages 51-69, January.
    17. Ye Dong & Stephen Lee, 2014. "Depth functions as measures of representativeness," Statistical Papers, Springer, vol. 55(4), pages 1079-1105, November.
    18. Mondal, Pronoy K. & Biswas, Munmun & Ghosh, Anil K., 2015. "On high dimensional two-sample tests based on nearest neighbors," Journal of Multivariate Analysis, Elsevier, vol. 141(C), pages 168-178.
    19. García, Jorge Luis & Heckman, James J. & Ziff, Anna L., 2018. "Gender differences in the benefits of an influential early childhood program," European Economic Review, Elsevier, vol. 109(C), pages 9-22.
    20. Lovato, Ilenia & Pini, Alessia & Stamm, Aymeric & Vantini, Simone, 2020. "Model-free two-sample test for network-valued data," Computational Statistics & Data Analysis, Elsevier, vol. 144(C).
    21. Luai Al-Labadi & Forough Fazeli Asl & Zahra Saberi, 2022. "A Bayesian nonparametric multi-sample test in any dimension," AStA Advances in Statistical Analysis, Springer;German Statistical Society, vol. 106(2), pages 217-242, June.
    22. Nicolas Städler & Sach Mukherjee, 2017. "Two-sample testing in high dimensions," Journal of the Royal Statistical Society Series B, Royal Statistical Society, vol. 79(1), pages 225-246, January.
    23. Reza Modarres, 2020. "Graphical Comparison of High‐Dimensional Distributions," International Statistical Review, International Statistical Institute, vol. 88(3), pages 698-714, December.
    24. Matias D. Cattaneo & Fang Han & Zhexiao Lin, 2023. "On Rosenbaum's Rank-based Matching Estimator," Papers 2312.07683, arXiv.org, revised Jan 2024.
    25. Paul, Biplab & De, Shyamal K. & Ghosh, Anil K., 2022. "Some clustering-based exact distribution-free k-sample tests applicable to high dimension, low sample size data," Journal of Multivariate Analysis, Elsevier, vol. 190(C).

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