Journal of Statistical Theory and Practice

, Volume 6, Issue 3, pp 524–535 | Cite as

Testing Equality of Variances for Multiple Univariate Normal Populations

  • David AllinghamEmail author
  • J. C. W. Rayner


To test for equality of variances in independent random samples from multiple univariate normal populations, the test of first choice would usually be the likelihood ratio test, the Bartlett test. This test is known to be powerful when normality can be assumed. Here two Wald tests of equality of variances are derived. The first test compares every variance with every other variance and was announced in Mather and Rayner (2002), but no proof was given there. The second test is derived from a quite different model using orthogonal contrasts, but is identical to the first. This second test statistic is similar to one given in Rippon and Rayner (2010), for which no empirical assessment has been given. These tests are compared with the Bartlett test in size and power. The Bartlett test is known to be nonrobust to the normality assumption, as is the orthogonal contrasts test. To deal with this difficulty an analogue of the new test is given. An indicative empirical assessment shows that it is more robust than the Bartlett test and competitive with the Levene test in its robustness to fat-tailed distributions. Moreover, it is a Wald test and has good power properties in large samples. Advice is given on how to implement the new test.

AMS Subject Classification

62Gxx 62Fxx 


Bartlett’s test Levene test Orthogonal and nonorthogonal contrasts Wald tests 


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  1. Allingham, D., and J. C. W. Rayner. 2011. A nonparametric two-sample Wald test of equality of variances. Adv. Decision Sci., article ID 748580. 8 doi:10.1155/2011/748580Google Scholar
  2. Mather, K. J., and J. C. W. Rayner. 2002. Testing equality of corresponding variances from multiple covariance matrices. In Advances in statistics, combinatorics and related areas, ed. C. Gulati, Y.-X. Lin, S. Mishra, and J. Rayner. 157–166. Singapore, World Scientific Press.CrossRefGoogle Scholar
  3. NIST/SEMATECH. 2006. Data used for chi-square test for the standard deviation, (accessed November 2011).Google Scholar
  4. Rayner, J. C. W. 1997. The asymptotically optimal tests. J. R. Stat. Soc., Ser. D (The Statistician), 46(3), 337–346.CrossRefGoogle Scholar
  5. Rippon, P., and J. C. W. Rayner. 2010. Generalised score and Wald tests. Adv. Decision Sci., article ID 292013. 8 doi:10.1155/2010/292013Google Scholar
  6. Stuart, A., and J. K. Ord. 1994. Kendall’s advanced theory of statistics, Vol. 1: Distribution theory (6th ed.), London, Hodder Arnold.Google Scholar

Copyright information

© Grace Scientific Publishing 2012

Authors and Affiliations

  1. 1.Centre for Computer-Assisted Research Mathematics and its Applications, School of Mathematical and Physical SciencesUniversity of NewcastleNewcastleAustralia
  2. 2.Centre for Statistical and Survey Methodology, School of Mathematics and Applied StatisticsUniversity of WollongongWollongongAustralia
  3. 3.School of Mathematical and Physical SciencesUniversity of NewcastleNewcastleAustralia

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