Explorative Data Analysis and CATANOVA for Ordinal Variables: An Integrated Approach

Categorical analysis of variance (CATANOVA) is a statistical method designed to analyse variability between treatments of interest to the researcher. There are well-established links between CATANOVA and the τ statistic of Goodman and Kruskal which, for the purpose of the graphical identification of this variation, is partitioned using singular value decomposition for Non-Symmetrical Correspondence Analysis (NSCA) (D'Ambra & Lauro, 1989). The aim of this paper is to show a decomposition of the Between Sum of Squares (BSS), measured both in CATANOVA framework and in the statistic τ, into location, dispersion and higher order components. This decomposition has been developed using Emerson's orthogonal polynomials. Starting from this decomposition, a statistical test and a confidence circle have been calculated for each component and for each modality in which the BSS was decomposed, respectively. A Customer Satisfaction study has been considered to explain the methodology.     

Partitioning Pearson's Chi-squared Statistic for Singly Ordered Two-way Contingency Tables

This paper presents a partition of Pearson's chi-squared statistic for singly ordered two-way contingency tables. The partition involves using orthogonal polynomials for the ordinal variable while generalized basic vectors are used for the non-ordinal variable. The benefit of this partition is that important information about the structure of the ordered variable can be identified in terms of locations, dispersion and higher order components. For the non-ordinal variable, it is shown that the squared singular values from the singular value decomposition of the transformed dataset can be partitioned into location, dispersion and higher order components. The paper also uses the chi-squared partition to present an alternative to the maximum likelihood technique of parameter estimation for the log-linear analysis of the contingency table.     

Partitioning Pearson's Chi-Squared Statistic for a Completely Ordered Three-Way Contingency Table

The paper presents a partition of the Pearson chi-squared statistic for triply ordered three-way contingency tables. The partition invokes orthogonal polynomials and identifies three-way association terms as well as each combination of two-way associations. This partition provides information about the structure of each variable by identifying important bivariate and trivariate associations in terms of location (linear), dispersion (quadratic) and higher order components. The significance of each term in the partition, and each association within each term can also be determined.
The paper compares the chi-squared partition with the log-linear models of Agresti (1994) for multi-way contingency tables with ordinal categories, by generalizing the model proposed by Haberman (1974).