Interval Estimation for the Difference of Two Independent Variances

Abstract

Analytical methods for interval estimation of differences between variances have not been described. A simple analytical method is given for interval estimation of the difference between variances of two independent samples. It is shown, using simulations, that confidence intervals generated with this method have close to nominal coverage even when sample sizes are small and unequal and observations are highly skewed and leptokurtic, provided the difference in variances is not very large. The method is also adapted for testing the hypothesis of no difference between variances. The test is robust but slightly less powerful than Bonett's test with small samples.

Keywords:

• Difference in variances
• F-test
• Homoscedasticity
• Heterogeneity of variances
• Kurtosis
• Levene's test

Mathematics Subject Classification:

• 62E17
• 62F25

Acknowledgments

We thank the LIPID trial management committee and the NHMRC Clinical Trials Centre for providing access to the LIPID trial data. Les Irwig and Adrienne Kirby provided helpful comments. Funding was received from NHMRC program grant 402764. Rob Herbert is supported by a research fellowship provided by the Australian NHMRC. We are grateful to two anonymous reviewers made who constructive suggestions that improved the article.

Notes

*Code used in this table and in figures to denote combinations of distributions examined.

**See column headed “Code” for codes. The first letter indicates the distribution of observations in population C, and the second letter indicates the distribution of the differences between populations C and T.

^#Pilot simulations only.

^§Only for the following 6 combinations of parameters: n _C  = n _T  = 10, 100, 100; LS distribution; δ = 0, 1.

Mean and worst coverage probabilities are calculated across all sample sizes for equal and unequal sample sizes. *lowest coverage of any test. # all combinations of N, l, and r distributions and all combinations of N, L, R, and S distributions except the NN distribution (total of 28 combinations of distributions; see Table 1 for explanation of codes).

*Highest rejection probability of any test.

^#Lowest rejection probability of any test.

^§Mean and worst coverage probabilities calculated across simulations with equal and unequal sample sizes.

^¶Only for equal sample sizes (n _C  = n _T ) with the LS distribution.