Dimensionality reduction methods for contingency tables with ordinal variables

Correspondence analysis is a widely used tool for obtaining a graphical representation of the interdependence between the rows and columns of a contingency table, by using a dimensionality reduction of the spaces. The maximum information regarding the association between the two categorical variables is then visualized allowing to understand its nature. Several extensions of this method take directly into account the possible ordinal structure of the variables by using different dimensionality reduction tools. Aim of this paper is to present an unified theoretical framework of several methods of correspondence analysis with ordinal variables

Decomposition of the Gray-Williams "tau" in main and interaction effects by ANOVA in three-way contingency table

The identification of meaningful relationships between two or more categorical variables is an important, and ongoing, element to the analysis of contingency tables. It involves detecting categories that are similar and/or different to other categories. Correspondence analysis can be used to detect such relationships by providing a graphical interpretation of the association between the variables, and it is especially useful when it is known that this association is of a symmetric nature. (Greenacre 1984), (Lebart et al. 1984). In this paper, we will explore the Gray-Williams index when used as the measure of association in non-symmetrical correspondence analysis (NSCA). It will be shown that, by concatenating a predictor variable of a three-way contingency table, the two measures are equivalent. The paper will analyse the sum of squares for nominal data partitioning the Sum of squares for main effects and the interaction in the sense of analysis of variance giving an orthogonal decomposition of Gray Williams index

Cumulative Correspondence Analysis using Orthogonal Polynomials

Taguchi's statistic has long been known to be a more appropriate measure of association the dependence for ordinal variables than the Pearson chi-squared statistic. Therefore, there is some advantage in using Taguchi's statistic in the correspondence analysis context when a two-way contingency table consists at least an ordinal categorical variable. The aim of this paper, considering contingency table with two ordinal categorical variables, is to show a decomposition of Taguchi's index into linear, quadratic and higher order components. This decomposition has been developed using Emerson's orthogonal polynomials. Moreover two cases study to explain the methodology has been analyzed

Visualizing main effects and interaction in multiple non-symmetric correspondence analysis

Non-symmetric correspondence analysis (NSCA) is a useful technique for analysing a two-way contin- gency table. Frequently, the predictor variables are more than one; in this paper, we consider two categorical variables as predictor variables and one response variable. Interaction represents the joint effects of pre- dictor variables on the response variable. When interaction is present, the interpretation of the main effects is incomplete or misleading. To separate the main effects and the interaction term, we introduce a method that, starting from the coordinates of multiple NSCA and using a two-way analysis of variance without interaction, allows a better interpretation of the impact of the predictor variable on the response variable. The proposed method has been applied on a well-known three-way contingency table proposed by Bocken- holt and Bockenholt in which they cross-classify subjects by person’ s attitude towards abortion, number of years of education and religion. We analyse the case where the variables education and religion influence a person’s attitude towards abortion

Cumulative correspondence analysis to improve the public train transport

To a company, improving customer satisfaction (CS) is a strategic element to increase market share, for example in public transport a growth of users implies a positive effect as private transport reduced.Consequently, the use of statistical methods directed to the analysis of CS is very important to get information necessary to support strategic decisions. In this paper, through Taguchi method for design of experiment, we set nine different hypothetic public train transport (called scenarios) and we asked to potential users to evaluate them by an ordinal scale. In order to analyse this data we use an integrated approach based on Cumulative Correspondence Analysis and the Taguchi's index, it allows to find a scenario that maximizes the satisfaction of potential users

Evaluation of Passenger Satisfaction using three-way contingence table with ordinal variables

The aim of this paper is to evaluate the Passenger Satisfaction (PS) starting from quality factors (punctuality, safeness, staff aspect and conduct, modal integration, etc.). Carrying out two or more ways contingence tables, crossing the overall satisfaction (PS) and the quality factors we can study the dependency between the overall satisfaction and quality factors. In particular, the partition of Marcotorchino index for a three-way contingency table with one, two and three ordered categorical variables (Beh E.J., Simonetti B., D'Ambra L., 2007) will allow us to analyze the asymmetric and ordinal structure of the data and to pick up the nonlinear relationship within the data. To complement the survey Ordered Non-Symmetric Correspondence Analysis (ONSCA) will be carried out