Comparing Variances and Other Measures of Dispersion

Testing hypotheses about variance parameters arises in contexts where uniformity is important and also in relation to checking assumptions as a preliminary to analysis of variance (ANOVA), dose-response modeling, discriminant analysis and so forth. In contrast to procedures for tests on means, tests for variances derived assuming normality of the parent populations are highly nonrobust to nonnormality. Procedures that aim to achieve robustness follow three types of strategies: (1) adjusting a normal-theory test procedure using an estimate of kurtosis, (2) carrying out an ANOVA on a spread variable computed for each observation and (3) using resampling of residuals to determine p values for a given statistic. We review these three approaches, comparing properties of procedures both in terms of the theoretical basis and by presenting examples. Equality of variances is first considered in the two-sample problem followed by the k-sample problem (one-way design).