A Warning on the Use of Chi-Squared Statistics with Frequency Tables with Small Expected Cell Counts
小期望格频数频数表使用卡方统计量的警示AI 解读 ↓
← 返回论文列表与筛选
阅读说明
本页整理公开书目信息，不生成学术质量评分。AI 解读根据题名及所注明的摘要或原文片段生成，未完成全文核验，可能有误；具体假设、证明与结论请核对原文。摘要保留来源文本，未经模型改写。
DOI 首次登记不等于接收或发表日期。部分更正、评论等条目没有独立摘要；缺少可靠材料时保留待补状态。
引用量采用 OpenAlex 单条关联记录，不合计多个记录；关联记录可能包含正式发表版本，不等于此版本单独引用。缺失不视为零，引用指标不代表论文质量。
关注作者按书目中的完整姓名匹配，同名和姓名变体需人工辨别。相关论文先按共享细分方向和具体方法排列，再参考通用方法与发表日期。
分类依据与纠错
AI 根据所提供的论文材料归类：摘要研究一般多项频数表的检验性质，没有对应所列经济计量细分方向的直接证据。
反馈分类问题正在核对关联版本…
代码、数据与补充材料
正在核对对应版本的资源…
关注这篇论文的作者
小期望格频数频数表使用卡方统计量的警示
- 研究问题
- 多格且期望格频数不均的频数表中，Pearson 卡方检验为何可能在原假设近似良好时仍出现低功效甚至不一致？
- 方法与思路
- 作者分析多项频数表下统计量的期望与方差，结合正态近似、离散 Edgeworth 展开和精确计算考察检验水平与功效。
- 材料陈述的贡献
- 摘要称即使原假设下各格期望数至少为五，功效仍可能低于检验水平；适当扩大格数和选择概率可使功效接近零。
- 阅读关注点
- 应核对格数增长的渐近设定、异常功效所需的概率配置，以及替代统计量能够缓解和仍无法消除的问题范围。
- 阅读边界
- 依据 OpenAlex 索引摘要，稿件版本未注明，不能推断刊本变化；所述反例不能解释为所有小格频数表必然发生同样失效。
原文摘要
When applied to frequency tables with small expected cell counts, Pearson chi-squared test statistics may be asymptotically inconsistent even in cases in which a satisfactory chi-squared approximation exists for the distribution under the null hypothesis. This problem is particularly important in cases in which the number of cells is large and the expected cell counts are quite variable. To illustrate this bias of the chi-squared test, this article considers the Pearson chi-squared test of the hypothesis that the cell probabilities for a multinomial frequency table have specified values. In this case, the expected value and variance of the Pearson chi-square may be evaluated under both the null and alternative hypotheses. When the number of cells is large, normal approximations and discrete Edgeworth expansions may also be used to assess the size and power of the Pearson chi-squared test. These analyses show that unless all cell probabilities are equal, it is possible to select a significance level and cell probabilities under the alternative hypothesis such that the power is less than the size of the test. As shown by exact calculations, the difference may be substantial even in cases in which all expected cell sizes are at least 5 under the null hypothesis. The use of moments shows that given any minimum expected cell size under the null hypothesis and given any significance level, it is possible to make the power arbitrarily close to 0 by the selection of a large enough number of cells in the table and suitable cell probabilities for the null and alternative hypotheses. The normal approximations for the distribution of the Pearson chi-squared statistic permit the size of this bias to be assessed in less-extreme cases involving tables with many cells. These results imply that caution must be exercised in the application of Pearson chi-squared statistics to sparse contingency tables with many cells. An alternative to the Pearson chi-square, proposed by Zelterman (1986), avoids some of the problems. Exact calculation, however, shows that the alternative statistic does not eliminate all problems of bias. The problems described in this article clearly extend to more general applications of the Pearson chi-squared statistic.