Levene 方差齐性检验中的子组不一致性与稀释效应

中文摘要 AI 辅助

Levene 方差齐性检验是一种标准程序，用于评估多组独立观测是否具有共同方差。由于 Levene 检验依赖于对变换后的绝对偏差或平方偏差进行方差分析（ANOVA），它在结构上镜像了 ANOVA 本身的统计性质。设 $\mu_j$ 和 $\sigma^2_j$ 分别表示第 $j$ 组观测的期望和方差。先前已证明，对于给定的显著性水平 $\alpha$，ANOVA 可能导致逻辑矛盾：在未能拒绝全局零假设 $H_0: \mu_1 = \mu_2 = \mu_3$ 的同时，却以相同或更高的置信度拒绝了局部假设 $H_0': \mu_1 = \mu_2$。在本文中，我们证明 Levene 检验直接继承了关于组方差 $\sigma^2_j$ 的同一“悖论”。我们提供了理论推理和数值示例来说明这种不一致性，展示了添加一个行为良好的第三组如何稀释检验统计量并掩盖显著的局部方差差异。

英文摘要

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.

发表机构

• National Research University Higher School of Economics（国立研究高等经济学院）
• Rutgers University（罗格斯大学）

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