Multi-Sample Problems

  • Mayer Alvo
  • Philip L. H. Yu
Part of the Springer Series in the Data Sciences book series (SSDS)


In this chapter, we present a unified theory of hypothesis testing based on ranks. The theory consists of defining two sets of ranks, one consistent with the alternative and the other consistent with the data itself in a notion to be described. The test statistic is then constructed by measuring the distance between the two sets. Critchlow (19861992) utilized a different definition for measuring the distance between sets. The problem can embedded into a smooth parametric alternative framework which then leads to a test statistic. It is seen that the locally most powerful tests can be obtained from this construction. We illustrate the approach in the cases of testing for ordered as well as unordered multi-sample location problems. In addition, we also consider dispersion alternatives. The tests are derived in the case of the Spearman and the Hamming distance functions. The latter were chosen to exemplify that different approaches may be needed to obtain the asymptotic distributions under the hypotheses.


  1. Alvo, M. (2008). Nonparametric tests of hypotheses for umbrella alternatives. Canadian Journal of Statistics, 36:143–156.MathSciNetCrossRefGoogle Scholar
  2. Alvo, M. (2016). Bridging the gap: a likelihood: function approach for the analysis of ranking data. Communications in Statistics - Theory and Methods, Series A, 45:5835–5847.MathSciNetCrossRefGoogle Scholar
  3. Alvo, M. and Berthelot, M.-P. (2012). Nonparametric tests of trend for proportions. International Journal of Statistics and Probability, 1:92–104.CrossRefGoogle Scholar
  4. Alvo, M. and Cabilio, P. (1995). Testing ordered alternatives in the presence of incomplete data. Journal of the American Statistical Association, 90:1015–1024.MathSciNetCrossRefGoogle Scholar
  5. Alvo, M. and Pan, J. (1997). A general theory of hypothesis testing based on rankings. Journal of Statistical Planning and Inference, 61:219–248.MathSciNetCrossRefGoogle Scholar
  6. Alvo, M. and Yu, P. L. H. (2014). Statistical Methods for Ranking Data. Springer.Google Scholar
  7. Box, G. and Cox, D. (1964). An analysis of transformations. Journal of the American Statistical Association, 26:211–252.zbMATHGoogle Scholar
  8. Cabilio, P. and Peng, J. (2008). Multiple rank-based testing for ordered alternatives with incomplete data. Statistics and Probability Letters, 78:2609–2613.MathSciNetCrossRefGoogle Scholar
  9. Critchlow, D. (1986). A unified approach to constructing nonparametric rank tests. Technical Report 86–15, Department of Statistics, Purdue University.Google Scholar
  10. Critchlow, D. (1992). On rank statistics: An approach via metrics on the permutation group. Journal of Statistical Planning and Inference, 32(325–346).MathSciNetCrossRefGoogle Scholar
  11. Gao, X. and Alvo, M. (2005a). A nonparametric test for interaction in two-way layouts. The Canadian Journal of Statistics, 33:1–15.MathSciNetCrossRefGoogle Scholar
  12. Gao, X. and Alvo, M. (2005b). A unified nonparametric approach for unbalanced factorial designs. Journal of the American Statistical Association, 100:926–941.MathSciNetCrossRefGoogle Scholar
  13. Gao, X. and Alvo, M. (2008). Nonparametric multiple comparison procedures for unbalanced two-way layouts. Journal of Statistical Planning and Inference, 138:3674–3686.MathSciNetCrossRefGoogle Scholar
  14. Gao, X., Alvo, M., Chen, J., and Li, G. (2008). Nonparametric multiple comparison procedures for unbalanced one-way factorial designs. Journal of Statistical Planning and Inference, 138:2574–2591.MathSciNetCrossRefGoogle Scholar
  15. Hajek, J. (1968). Asymptotic normality of simple linear rank statistics under alternatives. Ann. Math. Statist., 39:325–346.MathSciNetCrossRefGoogle Scholar
  16. Hájek, J. and Sidak, Z. (1967). Theory of Rank Tests. Academic Press, New York.zbMATHGoogle Scholar
  17. Jin, W. R., Riley, R. M., Wolfinger, R. D., White, K. P., Passador-Gundel, G., and Gibson, G. (2001). The contribution of sex, genotype and age to transcriptional variance in drosophila melanogaster. Nature Genetics, 29:389–395.CrossRefGoogle Scholar
  18. Kruskal, W. H. (1952). A nonparametric test for the several sample problem. Annals of Mathematical Statistics, 23:525–540.MathSciNetCrossRefGoogle Scholar
  19. Kruskal, W. H. and Wallis, W. A. (1952). Use of ranks in one-criterion variance analysis. Journal of the American Statistical Association, 47(260):583–621.CrossRefGoogle Scholar
  20. Page, E. (1963). Ordered hypotheses for multiple treatments: a significance test for linear ranks. Journal of the American Statistical Association, 58:216–230.MathSciNetCrossRefGoogle Scholar
  21. Schach, S. (1979). An alternative to the Friedman test with certain optimality properties. Ann. Statist., 7(3):537–550.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Mayer Alvo
    • 1
  • Philip L. H. Yu
    • 2
  1. 1.Department of Mathematics and StatisticsUniversity of OttawaOttawaCanada
  2. 2.Department of Statistics and Actuarial ScienceUniversity of Hong KongHong KongChina

Personalised recommendations