Statistics and Computing

, Volume 5, Issue 4, pp 289–296 | Cite as

Two permutation tests of equality of variances

  • Rose D. Baker


The F-ratio test for equality of dispersion in two samples is by no means robust, while non-parametric tests either assume a common median, or are not very powerful. Two new permutation tests are presented, which do not suffer from either of these problems. Algorithms for Monte Carlo calculation of P values and confidence intervals are given, and the performance of the tests are studied and compared using Monte Carlo simulations for a range of distributional types. The methods used to speed up Monte Carlo calculations, e.g. stratification, are of wider applicability.


Permutation test F-ratio test scale problem Monte Carlo stratification 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Bartlett, M. S. (1937) Properties of sufficiency and statistical tests. Proceedings of the Royal Society Series A, 160, 268–82.Google Scholar
  2. Box, G. E. P. (1953) Non-normality and tests on variances. Biometrika, 40, 318–35.Google Scholar
  3. Diciccio, T. J. and Romano, J. P. (1988) A review of bootstrap confidence intervals (with discussion). Journal of the Royal Statistical Society, Series B, 50, 338–70.Google Scholar
  4. Duran, B. S. (1976) A survey of nonparametric tests for scale. Communications in Statistics— J.Theory and Methods, A5, 1287–312.Google Scholar
  5. Fisher, R. A. (1924) On a distribution yielding the error functions of several well known statistics. Proc. Int. Congress Math., Toronto, 2, 805–13.Google Scholar
  6. Gabriel, K. R. and Hall, W. J. (1983) Rerandomisation inference on regression and shift effects: computationally feasible methods. Journal of the American Statistical Association, 78, 827–36.Google Scholar
  7. Gibbons, J. D. and Chakraborti, S. (1992) Nonparametric Statistical Inference, 3rd edn. Marcel Dekker, New York.Google Scholar
  8. Good, P. (1994) Permutation Tests. Springer-Verlag, New York.Google Scholar
  9. Hinkley, D. V. (1988) Bootstrap methods. Journal of the Royal Statistical Society Series B, 50, 321–37.Google Scholar
  10. Mehta, C. R., Patel, N. R., Senchaudhuri, P. (1988) Importance sampling for estimating exact probabilities in permutational inference. Journal of the American Statistical Association, 83, 999–1005.Google Scholar
  11. Moses, L. E. (1963) Rank tests for dispersion. Annals of Mathematical Statistics, 34, 973–83.Google Scholar
  12. Oden, N. L. (1991) Allocation of effort in Monte-Carlo simulation for power of permutation tests. Journal of the American Statistical Association, 86, 1074–6.Google Scholar
  13. Press, W. H., Teukolsky, S. A., Vetterling, W. T. and Flannery, B. P. (1992) Numerical Recipes in C: the Art of Scientific Computing, 2nd edn. Cambridge University Press, New York.Google Scholar
  14. Rivest, L. P. (1986) Bartlett's, Cochran's and Hartley's tests on variances are liberal when the underlying distribution is long-tailed. Journal of the American Statistical Association, 81, 124–8.Google Scholar
  15. Tritchler, D. (1984) On inverting permutation tests. Journal of the American Statistical Association, 79, 200–7.Google Scholar

Copyright information

© Chapman & Hall 1995

Authors and Affiliations

  • Rose D. Baker
    • 1
  1. 1.Centre for OR and Applied Statistics, Department of Mathematics and Computer ScienceUniversity of SalfordSalfordUK

Personalised recommendations