Reduced Rank Models for Contingency Tables
Reduced Rank Models for Contingency Tables
复制标题
列联表的降阶模型
DOI：
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发表时间：
1991
期刊：
影响因子：
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通讯作者：
P. Heijden
中科院分区：
文献类型：
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作者：
J. Leeuw;P. Heijden
Biomtlrika（1991），78，1，pp. 229-32英国印刷JAN DE LEEUW心理学和数学系，加州大学洛杉矶分校，加州90024-1563，美国和PETER G. M.货车DER HEIJDEN Department of Empirical and Theoretical Sociology，University of Utrecht，3508 TC Utrecht，The Netherlands概要一些关键词：典型分析;对应分析;潜在类分析;降秩模型。1.引言近年来，人们一直非常关注的双向列联表，可以制定一个矩阵的概率降低秩方面的模型。一个著名的降秩模型是独立模型，其中秩为1。对于秩高于1的不同类别的降秩模型是可能的。每一个都有独立模型作为秩1的特例。第一类这样的模型与所谓的规范分析或对应分析密切相关。Goodman（1985，1986，1987）和Gilula & Haberman（1986，1988）对这些模型的极大似然估计进行了研究。第二类可以用降秩公式表示的模型是双向表的潜在类分析（LCA）。Lazersfeld（1950 a，B）提出了隐类分析。参见Clogg（1981）的最新评论。在本文中，我们将这些类别的模型相互关联。Gilula（1979，1983，1984），Gilula & Haberman（1986），Goodman（1987），以及货车der Heijden，Mooijaart & de Leeuw（1989）都讨论过这个关系。我们总结现有的结果，在一个简单的方式使用新的证明。Gilula（1979）提供了秩-2对应分析必须成立的条件，以隐含秩-2潜在类分析。我们在这里表明，秩-2对应分析总是意味着秩-2潜在类分析。这意味着Gilula（1979）给出的定理和例子是不正确的。2.一般降秩模型本文研究的基本模型假定n × m概率矩阵II的秩为p，其中p = min（n，m）。我们称这个模型为R p。概率矩阵II的所有元素都是非负的，而ny的和等于1。除非另有说明，我们假设II是满的，因为它的行和irl+和列和v +J都是正的。因此，没有行或列等于零。我们比较这个模型与典型的模型CP，其中最多p - 1的典型相关表的行和列变量之间的非零。这些典型相关性是积矩相关系数的固定值，被视为行得分和列得分的函数。2011年5月18日从加州大学洛杉矶分校的biomet.oxfordjournals.org下载介绍了用于分析双向列联表的降秩模型。通过典型分析和潜在类分析，将降秩模型分为两类。这两个类之间的关系进行了讨论。在文献中提到的主题的结果是多余的或不准确的。
Biomtlrika (1991), 78, 1, pp. 229-32 Printed in Great Britain Reduced rank models for contingency tables BY JAN DE LEEUW Departments of Psychology and Mathematics, University of California, Los Angeles, California 90024-1563, U.S.A. AND PETER G. M. VAN DER HEIJDEN Department of Empirical and Theoretical Sociology, University of Utrecht, 3508 TC Utrecht, The Netherlands SUMMARY Some key words: Canonical analysis; Correspondence analysis; Latent class analysis; Reduced rank models. 1. INTRODUCTION In recent years much attention has been given to models for two-way contingency tables that can be formulated in terms of reduced rank of a matrix with probabilities. A well-known reduced rank model is the independence model, where the rank is one. For rank higher than one distinct classes of reduced rank models are possible. Each has the independence model as the special case for rank one. A first class of such models is closely related to what is known under names as canonical analysis or correspondence analysis. Recently much attention has been given to the maximum likelihood estimation of versions of these models by Goodman (1985, 1986, 1987) and Gilula & Haberman (1986, 1988). A second class of models that can be formulated in terms of reduced rank is latent class analysis, LCA, for two-way tables. Latent class analysis was proposed by Lazersfeld (1950a, b). See Clogg (1981) for a more recent review. In this paper we relate these classes of models to each other. The relation has been discussed earlier by Gilula (1979, 1983, 1984), Gilula & Haberman (1986), Goodman (1987), and van der Heijden, Mooijaart & de Leeuw (1989). We summarize existing results in a simple way using new proofs. Gilula (1979) provided conditions that had to hold for rank-2 correspondence analysis to imply rank-2 latent class analysis. We show here that rank-2 correspondence analysis always implies rank-2 latent class analysis. This implies that the theorem and the example given by Gilula (1979) are incorrect. 2. GENERAL REDUCED RANK MODELS The basic model studied in this paper assumes that a n n x m probability matrix II has rank p, where p = min (n,m). We call this model R p . The probability matrix II has all elements nonnegative, while the sum of the ny is equal to one. We suppose, unless indicated otherwise, that II is full, in the sense that its row sums ir l+ and its column sums v +J are all positive. Thus no row or column is equal to zero. We compare this model with the canonical model C p , in which at most p - 1 of the canonical correlations between the row and the column variables of the table are nonzero. These canonical correlations are the stationary values of the product moment correlation coefficient, seen as a function of scores for rows and scores for columns. Downloaded from biomet.oxfordjournals.org at University of California, Los Angeles on May 18, 2011 Reduced rank models for the analysis of two-way contingency tables are introduced. Two classes of reduced rank models are discerned, with well-known exponents canonical analysis and latent class analysis. The relation between these two classes is discussed. Results on the subject mentioned earlier in the literature are shown to be either redundant or inaccurate.