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Testing homogeneity of variances with unequal sample sizes

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Abstract

When sample sizes are unequal, problems of heteroscedasticity of the variables given by the absolute deviation from the median arise. This paper studies how the best known heteroscedastic alternatives to the ANOVA F test perform when they are applied to these variables. This procedure leads to testing homoscedasticity in a similar manner to Levene’s (1960) test. The difference is that the ANOVA method used by Levene’s test is non-robust against unequal variances of the parent populations and Levene’s variables may be heteroscedastic. The adjustment proposed by O’Neil and Mathews (Aust Nz J Stat 42:81–100, 2000) is approximated by the Keyes and Levy (J Educ Behav Stat 22:227–236, 1997) adjustment and used to ensure the correct null hypothesis of homoscedasticity. Structural zeros, as defined by Hines and O’Hara Hines (Biometrics 56:451–454, 2000), are eliminated. To reduce the error introduced by the approximate distribution of test statistics, estimated critical values are used. Simulation results show that after applying the Keyes–Levy adjustment, including estimated critical values and removing structural zeros the heteroscedastic tests perform better than Levene’s test. In particular, Brown–Forsythe’s test controls the Type I error rate in all situations considered, although it is slightly less powerful than Welch’s, James’s, and Alexander and Govern’s tests, which perform well, except in highly asymmetric distributions where they are moderately liberal.

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Acknowledgments

The author is sincerely grateful to two anonymous referees and the Associate Editor for their time and effort in providing very constructive, helpful and valuable comments and suggestions that have led to a substantial improvement in the quality of the paper.

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Authors and Affiliations

  1. Department of Quantitative Methods for Economics and Business, Economics and Business School, University of Murcia, Murcia, Spain

    I. Parra-Frutos

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  1. I. Parra-Frutos
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Correspondence to I. Parra-Frutos.

Appendix

Appendix

See Tables 1 and 2.

Table 1 Type I error rates in small and unequal sample sizes
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Table 2 Estimated power
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Parra-Frutos, I. Testing homogeneity of variances with unequal sample sizes. Comput Stat 28, 1269–1297 (2013). https://doi.org/10.1007/s00180-012-0353-x

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