Distribution of the Sample Version of the Measure of Association, Gamma
关联度量 Gamma 的样本统计量分布AI 解读 ↓
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关联度量 Gamma 的样本统计量分布
- 研究问题
- Goodman–Kruskal Gamma 的渐近正态理论在小样本中是否适用，方差估计、关联强度与边际集中程度如何影响尾部近似？
- 方法与思路
- 作者对多组五乘五总体列联表开展随机抽样实验，比较不同标准化方式下的尾部分布，并考察零关联检验功效。
- 材料陈述的贡献
- 摘要称，使用理论渐近方差时部分小样本尾部近似较好；代入估计方差后强关联会加重尾部偏离，类别高度集中也影响近似。
- 阅读关注点
- 应核对总体列联表设计、各种方差公式与估计方式、不同关联强度下的尾部偏离，以及零关联与独立性的设定关系。
- 阅读边界
- 来源为 OpenAlex 索引摘要，稿件版本未注明；结果来自特定五乘五列联表实验，不能推广至任意维度、边际分布或关联结构。
原文摘要
Random sampling experiments were performed to determine whether Goodman and Kruskal's asymptotic theory (2) for the sample version (G) of the measure of association Gamma (γ) is applicable to small samples. First, the asymptotic theory was checked against the results obtained with a series of 5 × 5 population cross classifications with values of γ covering the full range. For samples of size fifty, the tails of the distribution of (G — γ)/σ corresponded closely to the tails of the unit normal distribution when σ2 was the appropriate asymptotic variance derived in (2). Asymptotic normality still worked fairly well in the tails when σ2 was replaced by its maximum likelihood estimator, providing that |γ| <.50. There was, however, a clear tendency to have more observations in the tails than predicted from the asymptotic theory and this became very marked for large γ's. When σ was replaced by an upper bound derived in (2), (G — γ)/σ was of course more tightly concentrated. For the same series of population cross classifications, the power of a test that γ = 0, using a test statistic based on G and a specially applicable variance formula, was investigated. For sample sizes twenty-five and fifty, the power of this test was high except when γ approached closely to zero. For the case γ = 0 (independence of the two classifications), and 5 × 5 population cross classifications, an investigation was made of the effect of varying the marginal totals. The tails of the obtained distribution corresponded fairly closely to the tails of the unit normal distribution for samples of size twenty-five and fifty unless the population cross classifications had a high percentage of the population concentrated in one category.