Statistical Science

Comparing Variances and Other Measures of Dispersion

Dennis D. Boos and Cavell Brownie

Full-text: Open access

Abstract

Testing hypotheses about variance parameters arises in contexts where uniformity is important and also in relation to checking assumptions as a preliminary to analysis of variance (ANOVA), dose-response modeling, discriminant analysis and so forth. In contrast to procedures for tests on means, tests for variances derived assuming normality of the parent populations are highly nonrobust to nonnormality. Procedures that aim to achieve robustness follow three types of strategies: (1) adjusting a normal-theory test procedure using an estimate of kurtosis, (2) carrying out an ANOVA on a spread variable computed for each observation and (3) using resampling of residuals to determine p values for a given statistic. We review these three approaches, comparing properties of procedures both in terms of the theoretical basis and by presenting examples. Equality of variances is first considered in the two-sample problem followed by the k-sample problem (one-way design).

Article information

Source
Statist. Sci., Volume 19, Number 4 (2004), 571-578.

Dates
First available in Project Euclid: 18 April 2005

Permanent link to this document
https://projecteuclid.org/euclid.ss/1113832721

Digital Object Identifier
doi:10.1214/088342304000000503

Mathematical Reviews number (MathSciNet)
MR2185578

Zentralblatt MATH identifier
1100.62586

Keywords
Comparing variances measures of spread permutation method resampling resamples variability

Citation

Boos, Dennis D.; Brownie, Cavell. Comparing Variances and Other Measures of Dispersion. Statist. Sci. 19 (2004), no. 4, 571--578. doi:10.1214/088342304000000503. https://projecteuclid.org/euclid.ss/1113832721


Export citation

References

  • Boos, D. D. (1986). Comparing $k$ populations with linear rank statistics. J. Amer. Statist. Assoc. 81 1018--1025.
  • Boos, D. D. and Brownie, C. (1989). Bootstrap methods for testing homogeneity of variances. Technometrics 31 69--82.
  • Boos, D. D., Janssen, P. and Veraverbeke, N. (1989). Resampling from centered data in the two-sample problem. J. Statist. Plann. Inference 21 327--345.
  • Box, G. E. P. (1953). Non-normality and tests on variances. Biometrika 40 318--335.
  • Box, G. E. P. and Andersen, S. L. (1955). Permutation theory in the derivation of robust criteria and the study of departures from assumption (with discussion). J. Roy. Statist. Soc. Ser. B 17 1--34.
  • Brown, M. B. and Forsythe, A. B. (1974). Robust tests for the equality of variances. J. Amer. Statist. Assoc. 69 364--367.
  • Carroll, R. J. (2003). Variances are not always nuisance parameters. Biometrics 59 211--220.
  • Conover, W. J., Johnson, M. E. and Johnson, M. M. (1981). A comparative study of tests for homogeneity of variances, with applications to the outer continental shelf bidding data. Technometrics 23 351--361.
  • Fairfull, R. W., Crober, D. C. and Gowe, R. S. (1985). Effects of comb dubbing on the performance of laying stocks. Poultry Science 64 434--439.
  • Games, P. A., Winkler, H. B. and Probert, D. A. (1972). Robust tests for homogeneity of variance. Educational and Psychological Measurement 32 887--909.
  • Groot, A., Fan, Y., Brownie, C., Jurenka, R. A., Gould, F. and Schal, C. (2005). Effect of PBAN on pheromone production by mated Heliothis virescens and Heliothis subflexa females. J. Chemical Ecology 31 15--28.
  • Hall, P. (1992). The Bootstrap and Edgeworth Expansion. Springer, New York.
  • Klotz, J. (1962). Nonparametric tests for scale. Ann. Math. Statist. 33 498--512.
  • Layard, M. W. J. (1973). Robust large-sample tests for homogeneity of variances. J. Amer. Statist. Assoc. 68 195--198.
  • Levene, H. (1960). Robust tests for equality of variances. In Contributions to Probability and Statistics (I. Olkin, ed.) 278--292. Stanford Univ. Press, Stanford, CA.
  • Lim, T.-S. and Loh, W.-Y. (1996). A comparison of tests of equality of variances. Comput. Statist. Data Anal. 22 287--301.
  • Miller, R. G. (1968). Jackknifing variances. Ann. Math. Statist. 39 567--582.
  • Nair, V. and Pregibon, D. (1988). Analyzing dispersion effects from replicated factorial experiments. Technometrics 30 247--257.
  • O'Brien, P. C. (1992). Robust procedures for testing equality of covariance matrices. Biometrics 48 819--827.
  • O'Brien, R. G. (1978). Robust techniques for testing heterogeneity of variance effects in factorial designs. Psychometrika 43 327--342.
  • O'Brien, R. G. (1979). A general ANOVA method for robust tests of additive models for variances. J. Amer. Statist. Assoc. 74 877--880.
  • Rousseeuw, P. J. and Croux, C. (1993). Alternatives to the median absolute deviation. J. Amer. Statist. Assoc. 88 1273--1283.
  • SAS Institute Inc. (1999). SAS online doc, version 8. SAS Institute Inc., Cary, NC.
  • Shoemaker, L. H. (2003). Fixing the $F$ test for equal variances. Amer. Statist. 57 105--114.