Abstract
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.
Similar content being viewed by others
References
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:10.1155/2011/748580
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.
NIST/SEMATECH. 2006. Data used for chi-square test for the standard deviation, https://doi.org/www.itl.nist.gov/div898/handbook/eda/section3/eda3581.htm (accessed November 2011).
Rayner, J. C. W. 1997. The asymptotically optimal tests. J. R. Stat. Soc., Ser. D (The Statistician), 46(3), 337–346.
Rippon, P., and J. C. W. Rayner. 2010. Generalised score and Wald tests. Adv. Decision Sci., article ID 292013. 8 doi:10.1155/2010/292013
Stuart, A., and J. K. Ord. 1994. Kendall’s advanced theory of statistics, Vol. 1: Distribution theory (6th ed.), London, Hodder Arnold.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Allingham, D., Rayner, J.C.W. Testing Equality of Variances for Multiple Univariate Normal Populations. J Stat Theory Pract 6, 524–535 (2012). https://doi.org/10.1080/15598608.2012.695703
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1080/15598608.2012.695703