Rosaria Lombardo

Nonsymmetric Correspondence Analysis: A Tool for Analysing Contingency TablesWith a Dependence Structure

In this article a case is made for using nonsymmetric correspondence analysis to evaluate contingency tables with a dependence structure. A nontechnical overview of the method is presented in the article itself, while the relevant formulae are given in the Appendix. The technique is illustrated with three examples from such diverse areas as the intergenerational transfer of attachment, division of household tasks between recently married couples illustrating the handling of longitudinal data via supplementary variables, and attractiveness of products for shoplifting illustrating categorical multiple regression.

Non-symmetric correspondence analysis with ordinal variables using orthogonal polynomials

Non-symmetrical correspondence analysis (NSCA) is a useful tool for graphically detecting the asymmetric relationship between two categorical variables. Most of the theory associated with NSCA does not distinguish between a two-way contingency table of ordinal variables and a two-way one of nominal variables. Typically, singular value decomposition (SVD) is used in classical NSCA for dimension reduction. A bivariate moment decomposition (BMD) for ordinal variables in contingency tables using orthogonal polynomials and generalized correlations is proposed. This method not only takes into account the ordinal nature of the two categorical variables, but also permits for the detection of significant association in terms of location, dispersion and higher order components.

Simple and multiple correspondence analysis for ordinal-scale variables using orthogonal polynomials

Correspondence analysis (CA) has gained a reputation for being a very useful statistical technique for determining the nature of association between two or more categorical variables. For simple and multiple CA, the singular value decomposition (SVD) is the primary tool used and allows the user to construct a low-dimensional space to visualize this association. As an alternative to SVD, one may consider the bivariate moment decomposition (BMD), a method of decomposition that involves using orthogonal polynomials to reflect the structure of ordered categorical responses. When the features of BMD are combined with SVD, a hybrid decomposition (HD) is formed. The aim of this paper is to show the applicability of HD when performing simple and multiple CA.

Multiple Correspondence Analysis via Polynomial Transformations of Ordered Categorical Variables

We present an alternative approach to Multiple Correspondence Analysis (MCA) that is appropriate when the data consist of ordered categorical variables. MCA displays objects (individuals, units) and variables as individual points and sets of category points in a low-dimensional space. We propose a hybrid decomposition on the basis of the classical indicator super-matrix, using the singular value decomposition, and the bivariate moment decomposition by orthogonal polynomials. When compared to standard MCA, the hybrid decomposition will give the same representation of the categories of the variables, but additionally, we obtain a clear association interpretation among the categories in terms of linear, quadratic and higher order components. Moreover, the graphical display of the individual units will show an automatic clustering.

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.

Studying the dependence between ordinal-nominal categorical variables via orthogonal polynomials

In situations where the structure of one of the variables of a contingency table is ordered recent theory involving the augmentation of singular vectors and orthogonal polynomials has shown to be applicable for performing symmetric and non-symmetric correspondence analysis. Such an approach has the advantage of allowing the user to identify the source of variation between the categories in terms of components that reflect linear, quadratic and higher-order trends. The purpose of this paper is to focus on the study of two asymmetrically related variables cross-classified to form a two-way contingency table where only one of the variables has an ordinal structure.

Non-symmetric correspondence analysis and biplot representation: Comparing differences in market share distribution

In this paper Non-Symmetric Correspondence Analysis (NSCA, Lauro and D’Ambra, 1984; D’Ambra and Lauro, 1989, 1992) is proposed as a useful technique for evaluating contingency table with a dependence structure, in particular within the context of comparing market share differences. Technical aspects of the method are discussed with a view towards application, giving special attention to the biplot representation ofNSCA as compared to the symmetric graphical display. Two examples dealing with canned food and the car market, respectively, are used to illustrate the usefulness of the technique and its kind of representation.

CATANOVA for two-way cross classified categorical data

In this article we develop an extension of categorical analysis of variance for one response and two factors, based on a partitioning of a measure of predictability for three-way contingency tables, known as Gray and Williamss index. At the first instance moment the decomposition of this multiple measure of association in partial association measures is shown. Finally, for ordinal-scale variables, we propose an extension of this decomposition using a particular set of orthogonal polynomials.

Non Parametric Control Chart by Multivariate Additive Partial Least Squares via Spline

Statistical process control (SPC) chart is aimed at monitoring a process over time in order to detect any special event that may occur and find assignable causes for it. Controlling both product quality variables and process variables is a complex problem. Multivariate methods permit to treat all the data simultaneously extracting information on the “directionality” of the process variation. Highlighting the dependence relationships between process variables and product quality variables, we propose the construction of a non-parametric chart, based on Multivariate Additive Partial Least Squares Splines; proper control limits are built by applying the Bootstrap approach.

Multivariate Additive PLS Spline Boosting in Agro-Chemistry studies

Routinely, the multi-response Partial Least-Squares (PLS) is used in regression and classification problems showing good performances in many applied studies. In this paper, we aim to present PLS via spline functions focusing on supervised classification studies and showing how PLS methods historically belongs to L2 boosting family. The theory of the PLS boost models is presented and used in classification studies. As a natural enrichment of linear PLS boost, we present its multi-response non-linear version by univariate and bivariate spline functions to transform the predictors. Three case studies of different complexities concerning soils and its products will be discussed, showing the gain in diagnostic provided by the non-linear additive PLS boost discriminant analysis compared to the linear one.