## Centre for Statistical & Survey Methodology Working Paper Series

#### Publication Date

2012

#### Recommended Citation

Allingham, David and Rayner, J C., Testing equality of variances for multiple univariate normal populations, Centre for Statistical and Survey Methodology, University of Wollongong, Working Paper 4-12, 2012, 12.

http://ro.uow.edu.au/cssmwp/91

#### 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 non-robust 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 that 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.