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
Abstract
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.
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BibTeX
@misc{dwyer2026exact,
title = {{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}},
author = {William J. Dwyer},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.26163},
eprint = {2609.26163},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.26163},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - 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
AU - William J. Dwyer
PY - 2026
DA - 2026/09/01
PB - arXiv
DO - 10.48550/arxiv.2609.26163
UR - https://arxiv.org/abs/2609.26163
AB - 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.
ER -