Mathematical Sciences: Multivariate Analysis, Rank Data andMultivariate Ranks

题目：多变量分析、排名数据和多变量排名摘要本研究探讨了多变量统计分析和排名数据建模与分析的几个领域。在排名数据中，几位法官对一组物体从最好到最差进行排名。这些数据的分析和建模研究包括通过将距离集视为凸锥来评估秩向量之间的距离；通过使用对象的正交“对比”，将一维展开模型扩展到多个维度，每个对比是对象组的集合；并使用排列群相对于特定子群的商群来表示秩数据中的对比和/或联系模式。与排名数据相关的是基于排名的非参数方差分析，其中设计中的单元格是对象。本文对多路布局中的主效应和交互效应进行了非参数化定义，建立了基于秩的模型假设检验程序，并利用不变性和似然原理找到了普遍有效的基于秩的检验程序。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（Wilcoxon/Mann-Whitney, Kruskal-Wallis, Jonckheere-Terpstra, Kendall tau， Spearman rho等）的多变量类似物和协方差矩阵上对称假设的检验，详细介绍了依赖于多变量秩的特定定义。秩函数的迭代可以提供一种定义多元概率函数和反函数的方法，并应用于生成随机向量和多元q -图。本研究的主要重点是分析和扩展一些流行的统计程序。排名数据是指一些法官对一些事物从好到坏进行排名，这种数据出现在许多学科中，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，通常是由于法官之间存在不同的阵营，对象的自然分组，或法官可能偏好的连续体（例如，自由/保守连续体）。在许多情况下，基于对象分组的新统计模型能够捕捉到这种复杂性。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主义/保守主义统一体。）相关研究适用于许多领域，包括农业、生物、经济、环境科学以及社会科学。大多数统计研究的目的是根据一种或多种测量方法比较个人或治疗方法（例如，肥料、药物、教育方法），或发现不同测量方法之间的关系（例如，饮食和健康）。基本方法依赖于相当严格的假设，这些假设通常不适用于实际的科学情况。因此，已经投入了大量的工作来制定即使在一定程度上违反假设也能正常工作的程序。特别重要的是，在存在一些不寻常的值时，程序不会失败。当比较仅基于一个属性时，基于等级的过程很容易应用和理解，并且在相当温和的条件下有效。这些方法是通过获取数据（例如，胆固醇水平）并用它们的等级替换它们（即，最低的水平变成“1”，下一个变成“2”，等等）来应用的。希望基于多个属性进行比较是很常见的，例如，胆固醇水平和血压。在这种情况下，对个体进行排名就更有问题了，因为他们在胆固醇方面的排名可能与在血压方面的排名不同。这项研究的一个主要组成部分使用了一种特殊的方法来同时根据几个属性对个体进行排名。程序是根据这些排序开发的，这些排序与单个属性的排序相似。这些方法易于使用且广泛有效，尤其对不寻常的观察结果具有抵抗力。

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.