Abstract
A directional Bayesian pure significance test for the equality of variances is developed. The approach is based on the assessment of observed departure conditioned on the direction of departure in multivariate models. The resulting one-dimensional directional distribution is easily interpreted. Normality is not required. Robustness of prior selection is discussed focusing on directional properties of the multivariate prior. Several examples are considered.
Highlights
Statistical significance tests for the equality of variation across treatment groups are often used in experimental settings where ANOVA based analysis is required
Robustness of prior selection is discussed focusing on directional properties of the multivariate prior
Approximate 75, 90 and 95 per cent Highest Posterior Density (H.P.D.) regions are given in Box and Tiao (1973), p. 138 with the null hypothesis ψ0 = (0, 0) lying within the 90% region, providing limited support for the null hypothesis of homogeneity
Summary
Statistical significance tests for the equality of variation across treatment groups are often used in experimental settings where ANOVA based analysis is required. This is an issue for example in toxicological and genetic applications (Gastwirth et al 2009). As shown in Box (1954), the robustness of the one-way ANOVA overall F-test to non-normality is dependent on the degree of inequality among the group variances. In settings where homogeneity of variation is to be formally tested, the commonly applied Bartlett test (Bartlett, 1937) which is a slight modification of a likelihood ratio test is often used, but is sensitive to the assumption of normality. The non-parametric Levene test can provide a more robust test with the trade-off of lower power.
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