Correspondence analysis of incomplete contingency tables

Correspondence analysis can be described as a technique which decomposes the departure from independence in a two-way contingency table. In this paper a form of correspondence analysis is proposed which decomposes the departure from the quasi-independence model. This form seems to be a good alternative to ordinary correspondence analysis in cases where the use of the latter is either impossible or not recommended, for example, in case of missing data or structural zeros. It is shown that Nora's reconstitution of order zero, a procedure well-known in the French literature, is formally identical to our correspondence analysis of incomplete tables. Therefore, reconstitution of order zero can also be interpreted as providing a decomposition of the residuals from the quasi-independence model. Furthermore, correspondence analysis of incomplete tables can be performed using existing programs for ordinary correspondence analysis.     

Exploratory analysis of contingency tables by loglinear formulation and generalizations of correspondence analysis

Goodman's (1979, 1981, 1985) loglinear formulation for bi-way contingency tables is extended to tables with or without missing cells and is used for exploratory purposes. A similar formulation is done for three-way tables and generalizations of correspondence analysis are deduced. A generalized version of Goodman's algorithm, based on Newton's elementary unidimensional method is used to estimate the scores in all cases.This research was partially supported by National Science and Engineering Research Council of Canada, Grant No. A8724. The author is grateful to the reviewers and the editor for helpful comments.     

Correspondence analysis used complementary to loglinear analysis

Loglinear analysis and correspondence analysis provide us with two different methods for the decomposition of contingency tables. In this paper we will show that there are cases in which these two techniques can be used complementary to each other. More specifically, we will show that often correspondence analysis can be viewed as providing a decomposition of the difference between two matrices, each following a specific loglinear model. Therefore, in these cases the correspondence analysis solution can be interpreted in terms of the difference between these loglinear models. A generalization of correspondence analysis, recently proposed by Escofier, will also be discussed. With this decomposition, which includes classical correspondence analysis as a special case, it is possible to use correspondence analysis complementary to loglinear analysis in more instances than those described for classical correspondence analysis. In this context correspondence analysis is used for the decomposition of the residuals of specific restricted loglinear models.     

Analysis of individual differences in multidimensional scaling via an n-way generalization of “Eckart-Young” decomposition

An individual differences model for multidimensional scaling is outlined in which individuals are assumed differentially to weight the several dimensions of a common “psychological space”. A corresponding method of analyzing similarities data is proposed, involving a generalization of “Eckart-Young analysis” to decomposition of three-way (or higher-way) tables. In the present case this decomposition is applied to a derived three-way table of scalar products between stimuli for individuals. This analysis yields a stimulus by dimensions coordinate matrix and a subjects by dimensions matrix of weights. This method is illustrated with data on auditory stimuli and on perception of nations.     

Modelling Trends in Ordered Correspondence Analysis Using Orthogonal Polynomials

The core of the paper consists of the treatment of two special decompositions for correspondence analysis of two-way ordered contingency tables: the bivariate moment decomposition and the hybrid decomposition, both using orthogonal polynomials rather than the commonly used singular vectors. To this end, we will detail and explain the basic characteristics of a particular set of orthogonal polynomials, called Emerson polynomials. It is shown that such polynomials, when used as bases for the row and/or column spaces, can enhance the interpretations via linear, quadratic and higher-order moments of the ordered categories. To aid such interpretations, we propose a new type of graphical display—the polynomial biplot.     

Resampling programs for multiway contingency tables with fixed marginal frequency totals

A resampling algorithm is presented for analyzing multiway contingency tables with fixed marginal frequency totals. Applications are illustrated with extensions of Fisher's exact, Pearson's chi-squared, and likelihood-ratio tests to three-way contingency tables.     

A Generic Disjunctive/Conjunctive Decomposition Model for n-ary Relations

This paper discusses a generic decomposition model that represents an arbitrary n-ary relation as a disjunctive or conjunctive combination of a number of n-ary component relations of a prespecified type. An important subclass of order-preserving decompositions is defined and its properties are derived. The generic model is shown to subsume various known models as special cases, including the models of Boolean factor analysis, hierarchical classes analysis, and disjunctive/conjunctive nonmetric factor analysis. Moreover, it also subsumes a broad range of new models as exemplified with a novel model for multidimensional parallelogram analysis and novel three-way extensions of nonmetric factor analysis. Copyright 1999 Academic Press.     

Dual scaling for the analysis of categorical data

Dual scaling is a set of related techniques for the analysis of a wide assortment of categorical data types including contingency tables and multiple-choice, rank order, and paired comparison data. When applied to a contingency table, dual scaling also goes by the name "correspondence analysis," and when applied to multiple-choice data in which there are more than 2 items, "optimal scaling" and "multiple correspondence analysis. " Our aim of this article was to explain in nontechnical terms what dual scaling offers to an analysis of contingency table and multiple-choice data.     

Simplicity and Typical Rank Results for Three-Way Arrays

Matrices can be diagonalized by singular vectors or, when they are symmetric, by eigenvectors. Pairs of square matrices often admit simultaneous diagonalization, and always admit block wise simultaneous diagonalization. Generalizing these possibilities to more than two (non-square) matrices leads to methods of simplifying three-way arrays by nonsingular transformations. Such transformations have direct applications in Tucker PCA for three-way arrays, where transforming the core array to simplicity is allowed without loss of fit. Simplifying arrays also facilitates the study of array rank. The typical rank of a three-way array is the smallest number of rank-one arrays that have the array as their sum, when the array is generated by random sampling from a continuous distribution. In some applications, the core array of Tucker PCA is constrained to have a vast majority of zero elements. Both simplicity and typical rank results can be applied to distinguish constrained Tucker PCA models from tautologies. An update of typical rank results over the real number field is given in the form of two tables.     

The typical rank of tall three-way arrays

The rank of a three-way array refers to the smallest number of rank-one arrays (outer products of three vectors) that generate the array as their sum. It is also the number of components required for a full decomposition of a three-way array by CANDECOMP/PARAFAC. The typical rank of a three-way array refers to the rank a three-way array has almost surely. The present paper deals with typical rank, and generalizes existing results on the typical rank ofI × J × K arrays withK = 2 to a particular class of arrays withK ≥ 2. It is shown that the typical rank isI when the array is tall in the sense thatJK − J < I < JK. In addition, typical rank results are given for the case whereI equalsJK − J. The author is obliged to Henk Kiers, Tom Snijders, and Philip Thijsse for helpful comments.     

Relationships among several methods of linearly constrained correspondence analysis

This paper shows essential equivalences among several methods of linearly constrained correspondence analysis. They include Fisher's method of additive scoring, Hayashi's second type of quantification method, ter Braak's canonical correspondence analysis, Nishisato's type of quantification method, ter Braak's canonical correspondence analysis, Nishisato's ANOVA of categorical data, correspondence analysis of manipulated contingency tables, Böckenholt and Böckenholt's least squares canonical analysis with linear constraints, and van der Heijden and Meijerink's zero average restrictions. These methods fall into one of two classes of methods corresponding to two alternative ways of imposing linear constraints, the reparametrization method and the null space method. A connection between the two is established through Khatri's lemma.The work reported in this paper has been supported by grant A6394 from the Natural Sciences and Engineering Research Council of Canada to the first author. We wish to thank Carolyn Anderson, Ulf Böckenholt, Henk Kiers, Shizuhiko Nishisato, Jim Ramsay, Tadashi Shibayama, Cajo ter Braak, and Peter van der Heijden for their helpful comments on earlier drafts of this paper.     

Taxicab Correspondence Analysis

Taxicab correspondence analysis is based on the taxicab singular value decomposition of a contingency table, and it shares some similar properties with correspondence analysis. It is more robust than the ordinary correspondence analysis, because it gives uniform weights to all the points. The visual map constructed by taxicab correspondence analysis has a larger sweep and clearer perspective than the map obtained by correspondence analysis. Two examples are provided. This research was financed by the Natural Sciences and Engineering Research Council of Canada. The author thanks Serge Vienneau for his help regarding the graphical displays, and also thanks the editor, associate editor, and two reviewers for their constructive comments.     

Analysis of contingency tables by ideal point discriminant analysis

Cross-classified data are frequently encountered in behavioral and social science research. The loglinear model and dual scaling (correspondence analysis) are two representative methods of analyzing such data. An alternative method, based on ideal point discriminant analysis (DA), is proposed for analysis of contingency tables, which in a certain sense encompasses the two existing methods. A variety of interesting structures can be imposed on rows and columns of the tables through manipulations of predictor variables and/or as direct constraints on model parameters. This, along with maximum likelihood estimation of the model parameters, allows interesting model comparisons. This is illustrated by the analysis of several data sets.Presented as the Presidential Address to the Psychometric Society's Annual and European Meetings, June, 1987. Preparation of this paper was supported by grant A6394 from the Natural Sciences and Engineering Research Council of Canada. Thanks are due to Chikio Hayashi of University of the Air in Japan for providing the ISM data, and to Jim Ramsay and Ivo Molenaar for their helpful comments on an earlier draft of this paper.     

Taxicab Correspondence Analysis of Contingency Tables with One Heavyweight Column

The aim of this paper is to study the analysis of contingency tables with one heavyweight column or one heavyweight entry by taxicab correspondence analysis (TCA). Given that the mathematics of TCA is simpler than the mathematics of correspondence analysis (CA), the influence of one heavyweight column on the outputs of TCA is studied explicitly without recourse to asymptotics as done by Benzécri (Les Cahiers de L’Analyse des Données 4:413–16, 1979). A reweighting of the heavyweight column is proposed, which can also be applied to CA. A real data set is analyzed. This research is financed by NSERC of Canada. The author is grateful to the Editor, Professor B. Junker, to an Associate Editor and to three referees for their constructive comments.     

Sufficient conditions for uniqueness in Candecomp/Parafac and Indscal with random component matrices

A key feature of the analysis of three-way arrays by Candecomp/Parafac is the essential uniqueness of the trilinear decomposition. We examine the uniqueness of the Candecomp/Parafac and Indscal decompositions. In the latter, the array to be decomposed has symmetric slices. We consider the case where two component matrices are randomly sampled from a continuous distribution, and the third component matrix has full column rank. In this context, we obtain almost sure sufficient uniqueness conditions for the Candecomp/Parafac and Indscal models separately, involving only the order of the three-way array and the number of components in the decomposition. Both uniqueness conditions are closer to necessity than the classical uniqueness condition by Kruskal. Part of this research was supported by (1) the Flemish Government: (a) Research Council K.U. Leuven: GOA-MEFISTO-666, GOA-Ambiorics, (b) F.W.O. project G.0240.99, (c) F.W.O. Research Communities ICCoS and ANMMM, (d) Tournesol project T2004.13; and (2) the Belgian Federal Science Policy Office: IUAP P5/22. Lieven De Lathauwer holds a permanent research position with the French Centre National de la Recherche Scientifique (C.N.R.S.). He also holds an honorary research position with the K.U. Leuven, Leuven, Belgium.     

A noniterative method of joint correspondence analysis

Correspondence analysis leads to a graphical representation of the associations between categories of the row and column variables of a contingency table. Greenacre's (1988) formulation of joint correspondence analysis is a multivariate extension which finds the optimal joint display of contingency tables between all pairs of variables in a set. Greenacre presented a discrepancy function and an alternating least squares algorithm for its minimization. Boik (1996) presented an alternative algorithm, also of the alternating least squares type, for minimizing the same discrepancy function. In this paper, a noniterative procedure, not based on the minimization of any discrepancy function, is described.     

Correspondence analysis and association models constrained by a conditional independence graph

The manner in which the conditional independence graph of a multiway contingency table effects the fitting and interpretation of the Goodman association model (RC) and of correspondence analysis (CA) is considered.Estimation of the row and column scores is presented in this context by developing a unified framework that includes both models. Incorporation of the conditional independence constraints inherent in the graph may lead to equal or additive scores for the corresponding marginal tables, depending on the topology of the graph. An example of doubly additive scores in the analysis of a Burt subtable is given.Thanks are due to anonymous referees who substantially improved the original draft of this paper.     

The LMPCA program: A graphical user interface for fitting the linked-mode PARAFAC-PCA model to coupled real-valued data

In behavioral research, PARAFAC analysis, a three-mode generalization of standard principal component analysis (PCA), is often used to disclose the structure of three-way three-mode data. To get insight into the underlying mechanisms, one often wants to relate the component matrices resulting from such a PARAFAC analysis to external (two-way two-mode) information, regarding one of the modes of the three-way data. To this end, linked-mode PARAFAC-PCA analysis can be used, in which the three-way and the two-way data set, which have one mode in common, are simultaneously analyzed. More specifically, a PARAFAC and a PCA model are fitted to the three-way and the two-way data, respectively, restricting the component matrix for the common mode to be equal in both models. Until now, however, no software program has been publicly available to perform such an analysis. Therefore, in this article, the LMPCA program, a free and easy-to-use MATLAB graphical user interface, is presented to perform a linked-mode PARAFAC-PCA analysis. The LMPCA software can be obtained from the authors at http://ppw.kuleuven.be/okp/software/LMPCA. For users who do not have access to MATLAB, a stand-alone version is provided.     

Categorical independence tests for large sparse r-way contingency tables

A nonasymptotic chi-squared technique is shown to have very useful properties for the analysis of large sparse r-way contingency tables. Examples of analyses of 4 x 5, 5 x 6, 6 x 7, and two 2 x 2 x 2 sparse contingency tables provide comparisons of the nonasymptotic chi-squared technique with asymptotic chi-squared and exact chi-squared techniques. The asymptotic chi-squared analyses yield inflated probability values for the five tables. The nonasymptotic chi-squared technique yields probability values much closer to the exact probability values than the asymptotic chi-squared technique for the five tables.