Rank Methods for Combination of Independent Experiments in Analysis of Variance

  • J. L. HodgesJr.
  • E. L. Lehmann
Open Access
Part of the Selected Works in Probability and Statistics book series (SWPS)

Introduction and summary

It is now coming to be generally agreed that in testing for shift in the two-sample problem, certain tests based on ranks have considerable advantage over the classical t-test. From the beginning, rank tests were recognized to have one important advantage: their significance levels are exact under the sole assumption that the samples are randomly drawn (or that the assignment of treatments to subjects is performed at random), whereas the t-test in effect is exact only when we are dealing with random samples from normal distributions. On the other hand, it was felt that this advantage had to be balanced against the various optimum properties possessed by the t-test under the assumption of normality. It is now being recognized that these optimum properties are somewhat illusory and that, under realistic assumptions about extreme observations or gross errors, the t-test in practice may well be less efficient than such rank tests as the Wilcoxon or normal scores test [6], [7].


Null Distribution Balance Incomplete Block Design Asymptotic Efficiency Null Variance Restricted Randomization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


  1. [1]
    Benard, A. and Elteren, Ph. van (1953). A generalization of the method of m-rank-ings. Indag. Math. 15 358–369.Google Scholar
  2. [2]
    Durbin, J. (1951).Incomplete blocks in ranking experiments. British J. Psych. 4 85–90.Google Scholar
  3. [3]
    Elteren, Ph. van (1960); On the combination of independent two sample tests of Wil-coxon. Bull. Inst. Internat. Statist. 37 351–361.MathSciNetMATHGoogle Scholar
  4. [4]
    Elteren, Ph. van and Noether, G. E. (1959). The asymptotic efficiency of the κx2 n-test for a balanced incomplete block design. Biometrika 46 475–477.MATHCrossRefGoogle Scholar
  5. [5]
    Friedman, M. (1937). The use of ranks to avoid the assumption of normality implicit in the analysis of variance. J. Amer. Statist. Assoc. 32 675–698.CrossRefGoogle Scholar
  6. [6]
    Hodges, J. L., Jr. and Lehmann, E. L. (1956). The efficiency of some non-parametric competitors of the t-test. Ann. Math. Statist. 27 324–335.MathSciNetMATHCrossRefGoogle Scholar
  7. [7]
    Hodges, J. L., Jr. and Lehmann, E. L. (1961). Comparison of the normal scores and Wilcoxon tests. Proc. Fourth Berkeley Symp. Math. Statist. Prob. 1 307–317. Univ. of California Press.MathSciNetGoogle Scholar
  8. [8]
    Loève, M. (I960). Probability Theory, 2nd ed. Van Nostrand, Princeton.Google Scholar
  9. [9]
    Walsh, John E. (1959). Exact nonparametric tests for randomized blocks. Ann. Math. Statist. 30 1034–1040.MathSciNetMATHCrossRefGoogle Scholar
  10. [10]
    Wilcoxon, F. (1946). Individual comparisons of grouped data by ranking methods. J. of Entomology 39 269–270.Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2012

Authors and Affiliations

  • J. L. HodgesJr.
    • 1
  • E. L. Lehmann
    • 1
  1. 1.University of CaliforniaBerkeleyUSA

Personalised recommendations