Tests of homogeneity

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Tests of homogeneity

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Many statistical tests and models assume that the variances of different samples that are being compared, or data drawn from different locations in space or time, have approximately the same variances. The assumption of homogeneity of variances or homoscedasticity, can be tested in a number of ways. Pairwise tests could be carried out, but this would require that every pairwise combination was checked. Multiple comparisons can be made using tests such as those due to Bartlett , Levene, and Fligner and Killeen. These are described in the subsections that follow. Of course, it is quite common to find that variances of samples are not homoskedastic, and are instead heteroskedastic, i.e. they vary significantly between samples, times or locations. In such cases a variance stabilizing transform may be tried, if appropriate, or an alternative model used that does not assume homoscedasticity. Conover at al. (1981, [CON1]) used Monte Carlo methods to evaluate the performance of a very large number of alternative tests for homogeneity, and found that relatively few were both powerful and robust. They concluded that:

"Many of the tests for variances that receive widespread usage have uncontrolled risk of Type I errors when the populations are asymmetric and heavy-tailed [e.g. Bartlett's test]. Even the more popular nonparametric tests show unstable error rates when they are modified for the case in which the population means are unknown. Thus, it is important to find some tests for variances, when the population means are unknown, that show stable error rates and reasonable power. After extensive simulation involving different distributions, sample sizes, means, and variances, three tests appear to be superior selections in terms of robustness and power [Levene and two Fligner-Killeen non-parametric tests based on ranked data using the absolute value of the difference between data items and the median for that sample] "

References

[CON1] Conover W J, Johnson M E, 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