Subgroup Inconsistency and the Dilution Effect in Levene's Test for Homoscedasticity
Abstract
Levene's test for homoscedasticity is a standard procedure used to evaluate whether multiple groups of independent observations share a common variance. Because Levene's test relies on an Analysis of Variance (ANOVA) applied to transformed absolute or squared deviations, it structurally mirrors the statistical properties of ANOVA itself. Let μj and σ2j denote the expectation and variance of the observations in group j. It was previously established that for a given significance level α, ANOVA can result in a logical contradiction: failing to reject the global null hypothesis H0: μ1 = μ2 = μ3 while simultaneously rejecting the localized hypothesis H0': μ1 = μ2 with the same or higher confidence. In this paper, we show that Levene's test directly inherits this same ``paradox'' regarding group variances σ2j. We provide theoretical reasoning and a numerical illustration of this inconsistency, demonstrating how the addition of a well-behaved third group can dilute the test statistic and mask a significant localized variance discrepancy.
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BibTeX
@misc{gurvich2026subgroup,
title = {{Subgroup Inconsistency and the Dilution Effect in Levene's Test for Homoscedasticity}},
author = {Vladimir Gurvich and Mariya Naumova},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.17746},
eprint = {2609.17746},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.17746},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Subgroup Inconsistency and the Dilution Effect in Levene's Test for Homoscedasticity
AU - Vladimir Gurvich
AU - Mariya Naumova
PY - 2026
DA - 2026/09/15
PB - arXiv
DO - 10.48550/arxiv.2609.17746
UR - https://arxiv.org/abs/2609.17746
AB - Levene's test for homoscedasticity is a standard procedure used to evaluate whether multiple groups of independent observations share a common variance. Because Levene's test relies on an Analysis of Variance (ANOVA) applied to transformed absolute or squared deviations, it structurally mirrors the statistical properties of ANOVA itself. Let μ_j and σ^2_j denote the expectation and variance of the observations in group j. It was previously established that for a given significance level α, ANOVA can result in a logical contradiction: failing to reject the global null hypothesis H_0: μ_1 = μ_2 = μ_3 while simultaneously rejecting the localized hypothesis H_0': μ_1 = μ_2 with the same or higher confidence. In this paper, we show that Levene's test directly inherits this same ``paradox'' regarding group variances σ^2_j. We provide theoretical reasoning and a numerical illustration of this inconsistency, demonstrating how the addition of a well-behaved third group can dilute the test statistic and mask a significant localized variance discrepancy.
ER -