{"title":"列联表的f散度广义cram<s:1>系数","authors":"Wataru Urasaki, Tomoyuki Nakagawa, Tomotaka Momozaki, Sadao Tomizawa","doi":"10.1007/s11634-023-00560-8","DOIUrl":null,"url":null,"abstract":"Abstract Various measures in two-way contingency table analysis have been proposed to express the strength of association between row and column variables in contingency tables. Tomizawa et al. (2004) proposed more general measures, including Cramér’s coefficient, using the power-divergence. In this paper, we propose measures using the f -divergence that has a wider class than the power-divergence. Unlike statistical hypothesis tests, these measures provide quantification of the association structure in contingency tables. The contribution of our study is proving that a measure applying a function that satisfies the condition of the f -divergence has desirable properties for measuring the strength of association in contingency tables. With this contribution, we can easily construct a new measure using a divergence that has essential properties for the analyst. For example, we conducted numerical experiments with a measure applying the $$\\theta$$ <mml:math xmlns:mml=\"http://www.w3.org/1998/Math/MathML\"> <mml:mi>θ</mml:mi> </mml:math> -divergence. Furthermore, we can give further interpretation of the association between the row and column variables in the contingency table, which could not be obtained with the conventional one. We also show a relationship between our proposed measures and the correlation coefficient in a bivariate normal distribution of latent variables in the contingency tables.","PeriodicalId":49270,"journal":{"name":"Advances in Data Analysis and Classification","volume":null,"pages":null},"PeriodicalIF":1.4000,"publicationDate":"2023-10-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Generalized Cramér’s coefficient via f-divergence for contingency tables\",\"authors\":\"Wataru Urasaki, Tomoyuki Nakagawa, Tomotaka Momozaki, Sadao Tomizawa\",\"doi\":\"10.1007/s11634-023-00560-8\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract Various measures in two-way contingency table analysis have been proposed to express the strength of association between row and column variables in contingency tables. Tomizawa et al. (2004) proposed more general measures, including Cramér’s coefficient, using the power-divergence. In this paper, we propose measures using the f -divergence that has a wider class than the power-divergence. Unlike statistical hypothesis tests, these measures provide quantification of the association structure in contingency tables. The contribution of our study is proving that a measure applying a function that satisfies the condition of the f -divergence has desirable properties for measuring the strength of association in contingency tables. With this contribution, we can easily construct a new measure using a divergence that has essential properties for the analyst. For example, we conducted numerical experiments with a measure applying the $$\\\\theta$$ <mml:math xmlns:mml=\\\"http://www.w3.org/1998/Math/MathML\\\"> <mml:mi>θ</mml:mi> </mml:math> -divergence. Furthermore, we can give further interpretation of the association between the row and column variables in the contingency table, which could not be obtained with the conventional one. We also show a relationship between our proposed measures and the correlation coefficient in a bivariate normal distribution of latent variables in the contingency tables.\",\"PeriodicalId\":49270,\"journal\":{\"name\":\"Advances in Data Analysis and Classification\",\"volume\":null,\"pages\":null},\"PeriodicalIF\":1.4000,\"publicationDate\":\"2023-10-05\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Advances in Data Analysis and Classification\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s11634-023-00560-8\",\"RegionNum\":4,\"RegionCategory\":\"计算机科学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q2\",\"JCRName\":\"STATISTICS & PROBABILITY\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Advances in Data Analysis and Classification","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s11634-023-00560-8","RegionNum":4,"RegionCategory":"计算机科学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q2","JCRName":"STATISTICS & PROBABILITY","Score":null,"Total":0}
引用次数: 0
摘要
摘要:在双向列联表分析中,人们提出了各种度量来表达列联表中行变量和列变量之间的关联强度。Tomizawa et al.(2004)利用幂散度提出了更一般的度量方法,包括cramsamrs系数。在本文中,我们提出了使用具有比幂散度更宽的类的f散度测度。与统计假设检验不同,这些措施在列联表中提供了关联结构的量化。我们的研究的贡献是证明了应用满足f -散度条件的函数的度量具有测量列联表中关联强度的理想性质。有了这个贡献,我们可以很容易地使用对分析人员具有基本属性的散度构造一个新的度量。例如,我们对应用$$\theta$$ θ -散度的测量进行了数值实验。此外,我们可以进一步解释列联表中行变量和列变量之间的关联,这是传统列联表所不能得到的。我们还在列联表中显示了我们提出的措施与潜在变量的二元正态分布中的相关系数之间的关系。
Generalized Cramér’s coefficient via f-divergence for contingency tables
Abstract Various measures in two-way contingency table analysis have been proposed to express the strength of association between row and column variables in contingency tables. Tomizawa et al. (2004) proposed more general measures, including Cramér’s coefficient, using the power-divergence. In this paper, we propose measures using the f -divergence that has a wider class than the power-divergence. Unlike statistical hypothesis tests, these measures provide quantification of the association structure in contingency tables. The contribution of our study is proving that a measure applying a function that satisfies the condition of the f -divergence has desirable properties for measuring the strength of association in contingency tables. With this contribution, we can easily construct a new measure using a divergence that has essential properties for the analyst. For example, we conducted numerical experiments with a measure applying the $$\theta$$ θ -divergence. Furthermore, we can give further interpretation of the association between the row and column variables in the contingency table, which could not be obtained with the conventional one. We also show a relationship between our proposed measures and the correlation coefficient in a bivariate normal distribution of latent variables in the contingency tables.
期刊介绍:
The international journal Advances in Data Analysis and Classification (ADAC) is designed as a forum for high standard publications on research and applications concerning the extraction of knowable aspects from many types of data. It publishes articles on such topics as structural, quantitative, or statistical approaches for the analysis of data; advances in classification, clustering, and pattern recognition methods; strategies for modeling complex data and mining large data sets; methods for the extraction of knowledge from data, and applications of advanced methods in specific domains of practice. Articles illustrate how new domain-specific knowledge can be made available from data by skillful use of data analysis methods. The journal also publishes survey papers that outline, and illuminate the basic ideas and techniques of special approaches.