Dimensionality Reduction Methods for Contingency Tables with Ordinal Variables (original) (raw)
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A dimensional reduction method for ordinal three-way contingency table
Compstat 2006 - Proceedings in Computational Statistics, 2006
For the study of association in three-way, and more generally multiway, contingency tables the literature offers a large number of techniques that can be considered. When there is an asymmetric dependence structure between the variables the Marcotorchino index [Mar84] (as apposed to the Pearson chi-squared statistic) can be used to measure the strength of their association. When the variables have an ordinal structure, this information is often not take into account. In this paper we introduce a partition of the Marcotorchino index for three ordered categorical variables using a special class of orthogonal polynomials. A graphical procedure is also considered to obtain a visual summary of the asymmetrical relationship between the variables.
Non-symmetric correspondence analysis with ordinal variables using orthogonal polynomials
Computational Statistics & Data Analysis, 2007
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.
Symmetrical and Non-symmetrical Variants of Three-Way Correspondence Analysis for Ordered Variables
Statistical Science, 2021
In the framework of multi-way data analysis, this paper presents symmetrical and non-symmetrical variants of three-way correspondence analysis that are suitable when a three-way contingency table is constructed from ordinal variables. In particular, such variables may be modelled using general recurrence formulae to generate orthogonal polynomial vectors instead of singular vectors coming from one of the possible three-way extensions of the singular value decomposition. As we shall see, these polynomials, that until now have been used to decompose two-way contingency tables with ordered variables, also constitute an alternative orthogonal basis for modelling symmetrical, nonsymmetrical associations and predictabilities in three-way contingency tables. Consequences with respect to modelling and graphing will be highlighted.
A Unified Approach for the Multivariate Analysis of Contingency Tables
Open Journal of Statistics, 2015
We present a unified approach to describing and linking several methods for representing categorical data in a contingency table. These methods include: correspondence analysis, Hellinger distance analysis, the log-ratio alternative, which is appropriate for compositional data, and the non-symmetrical correspondence analysis. We also present two solutions working with cummulative frequencies.
Journal of Physics: Conference Series
In this paper, we confined our attention to compare two methods to obtain a graphical depiction of the association (dependency) between three categorical variables. We shall first describe how to recode a three-way contingency table by discussing the Burt matrix form of the data. This method is known as multiple correspondence analysis (MCA). Another method is to preserve a three-way contingency table form using Tucker3, it's known as a three-way correspondence analysis (CA3). As a case study, we pay attention to analyze the association between race and gender in occupation field that may have contributes to differences in employment opportunity and the continuing increases in women's educational attainment. The results show that CA3 is more simple in computation and provide the graphical depiction of three-way association simultaneously, while MCA's plot can't. Consider to the cumulative inertia on the two-dimensional plot, the percentage inertia of CA3's plot is better than MCA's plot.
Journal of Applied Statistics, 2010
For many questionnaires and surveys in the marketing, business, and health disciplines, items often involve ordinal scales (such as the Likert scale and rating scale) that are associated in sometimes complex ways. Techniques such as classical correspondence analysis provide a simple graphical means of describing the nature of the association. However, the procedure does not allow the researcher to specify how one item may be associated with another, nor does the analysis allow for the ordinal structure of the scales to be reflected. This article presents a graphical approach that can help the researcher to study in depth the complex association of the items and reflect the structure of the items. We will demonstrate the applicability of this approach using data collected from a study that involves identifying major factors that influence the level of patient satisfaction in a Neapolitan hospital.
Studying the dependence between ordinal-nominal categorical variables via orthogonal polynomials
Journal of Applied Statistics, 2011
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.
2009
Correspondence analysis is an exploratory technique for analyzing the interaction in a contingency table. Tables with meaningful orders of the rows and columns may be analyzed using a model-based correspondence analysis that incorporates order constraints. However, if there exists a permutation of the rows and columns of the contingency table so that the rows are regression dependent on the columns and, vice versa, the columns are regression dependent on the rows, then both implied orders are reflected in the first dimension of the unconstrained correspondence analysis [Schriever, B.F., 1983. Scaling of order dependent categorical variables with correspondence analysis. International Statistical Review 51, 225-238]. Thus, using unconstrained correspondence analysis, we may still find that the data fit an ordinal stochastic model. Fit measures are formulated that may be used to verify whether the re-ordered contingency table is regression dependent in either the rows or columns. Using several data examples, it is shown that the fit indices may complement the usual geometric interpretation of the unconstrained correspondence analysis solution in low-dimensional space.
Correspondence analysis of incomplete contingency tables
Psychometrika, 1988
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.