Distribution of the Sample Version of the Measure of Association, Gamma
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关联度量 Gamma 样本版本的分布
- 研究问题
- Gamma 样本关联度量的渐近正态近似在小样本中是否可靠，方差估计、关联强度与边际分布如何影响尾部？
- 方法与思路
- 作者针对一系列五乘五总体列联表开展随机抽样实验，比较不同方差标准化方式下的尾部分布及零关联检验功效。
- 材料陈述的贡献
- 摘要称，采用理论渐近方差时近似较好；估计方差下强关联更易出现尾部偏差，边际高度集中也会影响近似。
- 阅读关注点
- 应核对列联表的构造、方差替代方式与尾部比较，并区分 Gamma 为零和独立性，查看边际集中程度的影响。
- 阅读边界
- 材料为版本未注明的 OpenAlex 索引摘要；结果限于所述表格与样本规模，不能外推到任意列联表或关联结构。
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.