Mathematical Sciences: Multivariate Analysis, Rank Data andMultivariate Ranks

数学科学：多元分析、排名数据和多元排名

• 批准号: 9504525
• 负责人: John Marden
• 金额: $ 14.7万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 1995
• 资助国家: 美国
• 起止时间: 1995-07-15 至 1999-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=9504525&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Multivariate; Analysis; Rank

Proposal: DMS 9504525 PI: John Marden Institution: University of Illinois Title: Multivariate Analysis, Rank Data and Multivariate Ranks ABSTRACT This research explores several areas of multivariate statistical analysis and the modeling and analysis of rank data. In rank data, several judges rank a set of objects from best to worse. Research in the analysis and modeling of such data includes evaluating distances between rank vectors by looking at the set of distances as a convex cone; extending unidimensional unfolding models to several dimensions by using orthogonal ``contrasts" of objects, each contrast being a collection of groups of objects; and using quotient groups of the group of permutations with respect to particular subgroups to represent contrasts and/or patterns of ties in rank data. Related to rank data is rank-based nonparametric analysis of variance, wherein the the cells in the design are the objects. Main and interaction effects are defined nonparametrically in multiway layouts, rank-based procedures for hypothesis testing of these models are developed, and invariance and likelihood principles are used to find universally efficient rank-based test procedures. Finally, an approach to extending univariate rank procedures to multivariate procedures, including multivariate analogs of popular univariate statistics (Wilcoxon/Mann-Whitney, Kruskal-Wallis, Jonckheere-Terpstra, Kendall tau, Spearman rho, etc.) and tests of symmetry hypotheses on covariance matrices, is detailed relying on a particular definition of multivariate rank. Iteration of the rank function may provide an approach to defining multivariate probability functions and inverse functions, with applications to generating random vectors and multivariate QQ-plots. The main focus of this research is to analyze and extend a number of popular statistical procedures. Rank data, in which a number of judges ranks a number of objects from best to worse, arise in many are as, including political science, sociology, education, psychology, and consumer preferences. Such data typically show a high degree of complexity, often due to the presence of distinct camps among the judges, natural groupings of the objects, or a continuum of possible preferences of the judges (along, e.g., a liberal/conservative continuum). New statistical models based on groupings of objects are able to capture such complexities in many cases. This research extends the scope of these models by considering multiple groupings or continuums. (There may be separate liberal/conservative continuums for social issues and for economic issues.) Related research is applicable to many fields, including agriculture, biology, economics, and environmental sciences as well as the social sciences. Most statistical studies are directed towards comparing individuals or treatments (e.g., fertilizers, drugs, educational methods) based on one or more measurement, or finding relationships between different measurements (e.g., diet and health). The basic methods depend on fairly restrictive assumptions, assumptions which often do not apply in real scientific situations. Consequently, a great amount of work has been dedicated to developing procedures which work well even when the assumptions are violated to a certain extent. It is particularly important that the procedures not fail in the presence of a few unusual values. When comparisons are based on only one attribute, rank-based procedures are easily applied and understood, and valid under reasonably mild conditions. These methods are applied by taking the data (on, e.g., cholesterol level) and replacing them with their ranks (i.e., the lowest level becomes ``1", the next ``2", etc.) It is very common to wish to make comparisons based on more than one attribute, e.g., cholesterol level and blood pressure. In such cases, ranking individuals is more problematic since their order with respect to cholesterol may not be the same as that with respect to blood pressure. A major component of this research uses a particular approach to ranking individuals on several attributes simultaneously. Procedures are developed based on these rankings that parallel those for single attributes. These methods are easy to use and widely valid, being especially resistant to unusual observations.

提案：DMS 9504525 PI：John马尔登机构：伊利诺伊大学标题：多元分析、秩数据和多元秩 摘要 本研究探讨了多元统计分析和秩数据的建模与分析的几个领域。 在排名数据中， 评判员将一组物品从好到坏进行排序。 这些数据的分析和建模研究包括 通过将所述距离集合看作凸锥来评估等级向量之间的距离;通过使用对象的正交"对比”将一维展开模型扩展到若干维度，每个对比是对象组的集合;以及使用所述置换组相对于特定子群的商组来表示等级数据中的对比和/或关系模式。与秩数据相关的是基于秩的非参数方差分析，其中设计中的单元格是对象。 主效应和相互作用的非参数定义在多向布局，基于秩的程序，这些模型的假设检验的开发，和不变性和似然原理被用来找到普遍有效的秩为基础的测试程序。 最后，将单变量秩过程扩展到多变量过程的方法，包括流行的单变量统计的多变量类似物（Wilcoxon/Mann-Whitney，Kruskal-Wallis，Jonckheere-Terpstra，Kendall tau，斯皮尔曼rho等）。 以及协方差矩阵对称性假设的检验 关于多元秩的一个特殊定义 秩函数的迭代可以提供定义多变量 概率函数和反函数及其在随机生成中的应用 向量和多变量QQ图。 本研究的主要重点是分析和扩展一些流行的 统计程序。排名数据，其中一些法官从最好到最差的一些对象排名，出现在许多领域，包括政治学，社会学，教育学，心理学和消费者偏好。 这些数据通常显示出高度的复杂性，这通常是由于在裁判员中存在不同的阵营、对象的自然分组或裁判员的可能偏好的连续体（沿着，例如，自由/保守连续体）。基于对象分组的新统计模型能够在许多情况下捕捉这种复杂性。 本研究通过考虑多个分组或连续体扩展了这些模型的范围。 （在社会问题和经济问题上，可能存在独立的自由/保守连续体。相关研究适用于许多领域，包括农业，生物学，经济学，环境科学以及社会科学。大多数统计学研究都是针对比较个体或治疗（例如，肥料、药物、教育方法）基于一个或多个测量或发现 不同测量之间的关系（例如，饮食与健康）。 这些基本方法依赖于相当严格的假设，而这些假设在真实的科学情况中往往并不适用。因此，大量的工作一直致力于开发程序，即使在一定程度上违反假设时也能很好地工作。 特别重要的是，程序在出现一些异常值时不会失败。当比较仅基于一个属性时，基于等级的程序易于应用和理解，并且在合理温和的条件下有效。这些方法是通过获取数据（例如，胆固醇水平）并用它们的等级（即，最低级别变为“1”， 下一个“2”，等等）很常见的是希望 基于一个以上的属性进行比较，例如， 胆固醇水平和血压。在这种情况下，对个体进行排名更成问题，因为他们关于胆固醇的顺序可能与关于血压的顺序不同。 这项研究的一个主要组成部分是使用一种特殊的方法来同时对几个属性进行排名。 程序是根据这些排名，平行那些为 单一属性。这些方法简单易行，适用范围广， 特别是对不寻常的观察有抵抗力。