Robustness of Multiple Comparisons Against Variance Heterogeneity

  • Jan B. Dijkstra
Part of the Theory and Decision Library book series (TDLB, volume 1)


If H01 = … = μk is rejected for normal populations with classical one way analysis of variance, it is usually of interest to know where the differences may be. If the population variances are equal there are several approaches one might consider:
  1. 1.

    Least Significant Difference test (Fisher, 1935)

  2. 2.

    Multiple Range test for equal sample sizes (Newman, 1939)

  3. 3.

    An adaptation for unequal sample sizes (Kramer, 1956)

  4. 4.

    Multiple F-test (Duncan, 1951)

  5. 5.

    Multiple Comparisons test (Duncan, 1952).


For all these methods (including the one way analysis of variance) alternatives exist that are robust against variance heterogeneity. A modification of (3) has some unattractive properties if the variances and the sample size differ greatly. The adaptations for unequal variances of (4) and (5) seem better than (1) for cases with many samples. Test (2) is rather robust in itself if the variances are not too much different. Modifications exist that allow slight unequalities in the sample sizes.


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  1. 1.
    Dijkstra, Jan B. and Werter, Paul S. P. J. Testing the equality of several means when the population variances are unequal. Communications in Statistics (1981) B 10, 6.Google Scholar
  2. 2.
    Brown, M. B. and Forsythe, A. B. The small sample behavior of some statistics which test the equality of several means. Technometrics 16 (1974), 129–132.MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Welch, B.L.: The comparison of several mean values: an alternative approach. Biometrika 38 (1951), 330–336.MathSciNetzbMATHGoogle Scholar
  4. 4.
    James, G. S. The comparison of several groups of observations when the ratios of the population variances are unknown. Biometrika 38 (1951), 324–329.MathSciNetzbMATHGoogle Scholar
  5. 5.
    Fisher, R. A. The Design of Experiments. Oliver & Boyd (1935) Edinburgh and London.Google Scholar
  6. 6.
    Miller, R. G. Simultaneous Statistical Inference. McGraw Hill Book Company, New York (1966).zbMATHGoogle Scholar
  7. 7.
    Behrens, W. V.: Ein Beitrag zur Fehlerberechnung bei wenigen Beobach-tungen. Landwirtschaftliche Jahrbucher 68 (1929), 807–837.Google Scholar
  8. 8.
    Welch, B.L. Further note on Mrs. Aspin’s tables and on certain approximations to the tabled function. Biometrika 36 (1949), 293–296.MathSciNetGoogle Scholar
  9. 9.
    Wang, Y. Y. Probabilities of type I errors of the Welch tests for the Behrens-Fisher problem. Journal of the American Statistical Association 66 (1971).Google Scholar
  10. 10.
    Ury, H. K. and Wiggins, A. D. Large sample and other multiple comparisons among means. British Journal of Mathematical and Statistical Psychology 24 (1971), 174–194.CrossRefGoogle Scholar
  11. 11.
    Hochberg, Y. A modification of the T-method of multiple comparisons for a one-way lay-out with unequal variances. Journal of the American Association 71 (1976), 200–203.MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Tamhane, A. C. Multiple Comparisons in model-1 one way ANOVA with unequal variances. Comunications in Statistics A 6(1) (1977), 15–32.MathSciNetCrossRefGoogle Scholar
  13. 13.
    Banerjee, S. K. On confidence intervals for two-means problem based on separate estimates of variances and tabulated values of t-variable. Sankhya, A 23 (1961).Google Scholar
  14. 14.
    Newman, D. The distribution of the range in samples from a normal population, expressed in terms of an independent estimate of standard deviation. Biometrika 31 (1939), 20–30.MathSciNetzbMATHGoogle Scholar
  15. 15.
    Keuls, M. The use of the “studentized range” in connection with an analysis of variance. Euphytica 1 (1952), 112–122.CrossRefGoogle Scholar
  16. 16.
    Winer, B. J. Statistical principles in experimental design. New York, McGraw-Hill (1962).CrossRefGoogle Scholar
  17. 17.
    Kramer, C. Y. Extension of multiple range tests to group means with unequal numbers of replications. Biometrics 12 (1956), 307–310.MathSciNetCrossRefGoogle Scholar
  18. 18.
    Ramseyer, G. C. and Tcheng, T. The robustness of the studentized range statistic to violations of the normality and homogeneity of variance assumptions. American Educational Research Journal 10 (1973).Google Scholar
  19. 19.
    Games, P. A. and Howell, J. F. Pairwise multiple comparison procedures with unequal N’s and/or Variances: a Monte Carlo Study. Journal of Educational Statistics 1 (1976), 113–125.CrossRefGoogle Scholar
  20. 20.
    Duncan, D. B. A significance test for differences between ranked treatments in an analysis of variances. Virginia Journal of Science 2 (1951).Google Scholar
  21. 21.
    Duncan, D. B. Multiple range and Multiple F-tests. Biometrics 11 (1955), 1–42.MathSciNetCrossRefGoogle Scholar
  22. 22.
    Duncan, D. B. On the properties of the multiple comparisons test. Virginia Journal of Science 3 (1952).Google Scholar
  23. 23.
    Ekbohm, G. On testing the equality of several means with small samples. The Agricultural College of Sweden, Uppsala (1976).Google Scholar
  24. 24.
    Ryan, T. A. Significance Tests for multiple comparison of proportions, variances and other statistics. Psychological Bulletin 57 (1960), 318–328.CrossRefGoogle Scholar
  25. 25.
    Einot, I. and Gabriel, K. R. A study of the powers of several methods of multiple comparisons. Journal of the American Statistical Association 70 (1975), 574–583.zbMATHCrossRefGoogle Scholar

Copyright information

© Academy of Agricultural Sciences of the GDR, Research Centre of Animal Production, Dummerstorf-Rostock, DDR 2551 Dummerstorf. 1984

Authors and Affiliations

  • Jan B. Dijkstra
    • 1
  1. 1.Computing CentreEindhoven University of TechnologyNetherlands

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