{"title":"A theory of contrasts for modified Freeman–Tukey statistics and its applications to Tukey’s post-hoc tests for contingency tables","authors":"Yoshio Takane, Eric J. Beh, Rosaria Lombardo","doi":"10.1007/s00180-024-01537-7","DOIUrl":null,"url":null,"abstract":"<p>This paper presents a theory of contrasts designed for modified Freeman–Tukey (FT) statistics which are derived through square-root transformations of observed frequencies (proportions) in contingency tables. Some modifications of the original FT statistic are necessary to allow for ANOVA-like exact decompositions of the global goodness of fit (GOF) measures. The square-root transformations have an important effect of stabilizing (equalizing) variances. The theory is then used to derive Tukey’s post-hoc pairwise comparison tests for contingency tables. Tukey’s tests are more restrictive, but are more powerful, than Scheffè’s post-hoc tests developed earlier for the analysis of contingency tables. Throughout this paper, numerical examples are given to illustrate the theory. Modified FT statistics, like other similar statistics for contingency tables, are based on a large-sample rationale. Small Monte-Carlo studies are conducted to investigate asymptotic (and non-asymptotic) behaviors of the proposed statistics.\n</p>","PeriodicalId":55223,"journal":{"name":"Computational Statistics","volume":"32 1","pages":""},"PeriodicalIF":1.0000,"publicationDate":"2024-08-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Computational Statistics","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1007/s00180-024-01537-7","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
Abstract
This paper presents a theory of contrasts designed for modified Freeman–Tukey (FT) statistics which are derived through square-root transformations of observed frequencies (proportions) in contingency tables. Some modifications of the original FT statistic are necessary to allow for ANOVA-like exact decompositions of the global goodness of fit (GOF) measures. The square-root transformations have an important effect of stabilizing (equalizing) variances. The theory is then used to derive Tukey’s post-hoc pairwise comparison tests for contingency tables. Tukey’s tests are more restrictive, but are more powerful, than Scheffè’s post-hoc tests developed earlier for the analysis of contingency tables. Throughout this paper, numerical examples are given to illustrate the theory. Modified FT statistics, like other similar statistics for contingency tables, are based on a large-sample rationale. Small Monte-Carlo studies are conducted to investigate asymptotic (and non-asymptotic) behaviors of the proposed statistics.
本文介绍了一种对比理论，该理论是为修正的弗里曼-图基（FT）统计量而设计的，该统计量是通过对或然率表中的观察频率（比例）进行平方根变换而得出的。为了对全局拟合优度（GOF）进行类似方差分析的精确分解，有必要对原始 FT 统计量进行一些修改。平方根变换具有稳定（均衡）方差的重要作用。然后，利用该理论推导出针对或然表的 Tukey 事后配对比较检验。Tukey 检验比 Scheffè 早先为分析或然率表而开发的事后检验更具限制性，但更强大。本文通篇以数字示例来说明理论。修正的 FT 统计法与其他类似的或然率统计法一样，都是基于大样本的原理。本文进行了小规模的蒙特卡洛研究，以研究拟议统计量的渐近（和非渐近）行为。
期刊介绍：
Computational Statistics (CompStat) is an international journal which promotes the publication of applications and methodological research in the field of Computational Statistics. The focus of papers in CompStat is on the contribution to and influence of computing on statistics and vice versa. The journal provides a forum for computer scientists, mathematicians, and statisticians in a variety of fields of statistics such as biometrics, econometrics, data analysis, graphics, simulation, algorithms, knowledge based systems, and Bayesian computing. CompStat publishes hardware, software plus package reports.
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