Mathematical Sciences: Multivariate Analysis, Rank Data andMultivariate Ranks

建议：DMS 9504525 PI：约翰·马登研究所：伊利诺伊大学标题：多变量分析，排名数据和多变量排名摘要本研究探索了多变量统计分析的几个领域，以及排名数据的建模和分析。在排名数据中，几位评委对一组对象进行从好到差的排名。对这种数据的分析和建模的研究包括：通过将距离集合视为凸锥来评估排序向量之间的距离；通过使用对象的正交的“对比”将一维展开模型扩展到几个维度，每个对比是对象组的集合；以及使用相对于特定子组的排列组的商组来表示排序数据中的对比和/或平局模式。与等级数据相关的是基于等级的非参数方差分析，其中设计中的单元格是对象。在多向布局中非参数地定义了主效应和交互效应，给出了这些模型的秩基假设检验方法，并利用不变性和似然原理来寻找普遍有效的秩基检验方法。最后，将单变量排序过程扩展到多变量过程，包括流行的单变量统计的多变量模拟(Wilcoxon/Mann-Whitney，Kruskal-Wallis，Jonckheere-Terpstra，Kendall tau，Spearman Rho等)。以及对协方差矩阵的对称性假设的检验，依赖于多变量等级的特定定义。秩函数的迭代可以提供一种定义多变量概率函数和反函数的方法，以及生成随机向量和多变量QQ图的应用。这项研究的主要重点是分析和推广一些流行的统计程序。在排名数据中，许多评委对许多对象进行了从好到差的排序，这些数据出现在许多领域，包括政治学、社会学、教育学、心理学和消费者偏好。这类数据通常表现出高度的复杂性，这往往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)。基于对象分组的新统计模型能够在许多情况下捕捉到这种复杂性。本研究通过考虑多个分组或连续体来扩展这些模型的范围。(对于社会问题和经济问题，可能存在不同的自由/保守连续体。)相关研究适用于农业、生物、经济、环境科学和社会科学等多个领域。大多数统计研究旨在根据一项或多项衡量标准比较个人或治疗方法(如化肥、药物、教育方法)，或找出不同衡量标准(如饮食和健康)之间的关系。基本方法依赖于相当严格的假设，这些假设往往不适用于真实的科学情况。因此，一直致力于开发即使在一定程度上违反假设也能很好地工作的程序。尤其重要的是，在存在一些不寻常的值的情况下，这些程序不会失败。当比较只基于一个属性时，基于排名的程序很容易应用和理解，并且在相当温和的条件下有效。应用这些方法的方法是，获取数据(例如，关于胆固醇水平)，并用它们的等级取代它们(即，最低水平变成“1”，下一个变成“2”，等等)。希望基于一个以上的属性进行比较是非常常见的，例如，胆固醇水平和血压。在这种情况下，对个人进行排名更成问题，因为他们在胆固醇方面的顺序可能与在血压方面的顺序不同。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在几个属性上对个人进行排名。程序是基于这些排名开发的，这些排名与单一属性的排名相当。这些方法易于使用，而且普遍有效，特别是对不寻常的观察具有抵抗力。

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.