Cramér's V 的精确条件置信区间：在保证区间过宽且软件区间不覆盖时，接近名义水平且更紧致
Exact Conditional Confidence Intervals for Cramér's V: Near-Nominal and Tight Where the Guaranteed Interval Is Wide and the Software Interval Does Not Cover
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中文总结 AI 辅助
针对Cramér's V置信区间难以构造的问题，本文通过条件于观测边际计算精确分布并求逆，得到接近名义水平且宽度仅为投影区间三分之一到一半的区间，以估计目标改变为代价。
中文摘要 AI 辅助
Cramér's V 是几乎每个卡方检验旁都会报告的效果量，但几乎从不伴随置信区间，因为该区间是一个困难的 nuisance 参数问题：无数个列联表共享同一个效果量值。分析者可以获取的两种区间都不令人满意。联合置信区域在效果量上的投影对每个表格都有保证，但区间过宽，在较大的表格上覆盖率基本为 1.000；软件输出的非中心反转区间以及 bootstrap 区间虽然窄，但不覆盖（在本研究的网格上中位覆盖率为 0.31）。本文提供了一个既有效又信息丰富的区间。以观测到的边际为条件，使得非零关联下 Pearson 统计量的精确条件分布无需渐近近似和 Monte Carlo 方法即可计算；对其求逆可得到 Cramér's V 的接近名义水平的置信区间，其宽度仅为投影区间的三分之一到一半，且优势随表格尺寸增大而增加。一个明确的代价是估计目标变为在观测边际处的效果量；mid-p 默认方法对小效应覆盖不足，此时保守变体或投影区间仍可作为备选。
英文摘要
Cramér's V, the effect size reported beside almost every chi-square test, is almost never accompanied by a confidence interval, because the interval is a hard nuisance-parameter problem: infinitely many tables share one effect-size value. The two intervals an analyst can reach for are unsatisfactory. The projection of a joint confidence region onto the effect size is guaranteed for every table but wide, over-covering at essentially 1.000 on larger tables; the noncentral inversion the software prints, and the bootstrap, are narrow but do not cover (median coverage 0.31 on this study's grid). This paper supplies an interval that is both valid and informative. Conditioning on the observed margins makes the exact conditional distribution of the Pearson statistic under a non-null association computable with no asymptotics and no Monte Carlo; inverting it yields a near-nominal confidence interval for Cramér's V that is a third to a half the width of the projection, the advantage growing with table size. The one price, stated plainly, is a change of estimand to the effect size at the observed margins; the mid-p default undercovers small effects, where a conservative variant or the projection remains the fallback.
发表机构
- University of Massachusetts Lowell（马萨诸塞大学洛厄尔分校）
机构由 AI 辅助整理，请以论文原文为准。
