Eric J. Beh

A Comparative Study of Scores for Correspondence Analysis with Ordered Categories

Ordered categorical data can be analysed using correspondence analysis with the ordered categories taken into consideration. Such an analysis was proposed by Beh (1997) and uses orthogonal polynomials which require the input of a scoring scheme to reflect the ordered structure of the categories. This method of correspondence analysis visualises the relationship between the categories, in terms of the location, dispersion and higher order components. The impact of the scoring method on the orthogonal polynomials, and hence upon the correspondence plot and other output of the analysis should therefore be considered. This paper aims at identifying this impact by considering four scoring schemes: integer valued (natural) scores, midrank scores, Nishisato scores and singular vectors from the classical correspondence analysis of the data. It is shown that while the latter two maximise the location component, generally there is little difference when comparing them with the output of the former two scoring schemes. A simple comparative study of profile co-ordinates using different scoring schemes is also discussed.

The information in aggregate data

Ecological analysis involves using aggregate data for a set of groups to make inferences concerning individual level relationships. Typically the data available for analysis consists of the means or totals of variables of interest for geographical areas, although the groups can be organisations such as schools or hospitals. Attention has focused on developing methods of estimating the parameters characterising the individual level relationships across the whole population, but also in some cases the relationships for each of the groups. Applying standard methods used to analyse individual level data, such as linear or logistic regression or contingency table analysis, to aggregate data will usually produce biased estimates of individual level relationships. Thus much of the effort in ecological analysis has concentrated on developing methods of analysing aggregate data that can produce unbiased, or less biased, parameter estimates. There has been less work done on inference procedures, such as constructing confidence intervals and hypothesis testing. Fundamental to these inferential issues is the question of how much information is contained in aggregate data and what evidence such data can provide concerning important assumptions and hypotheses.

Catanova for two-way contingency tables with ordinal variables using orthogonal polynomials

ABSTRACT The analysis of variance of cross-classified (categorical) data (CATANOVA) is a technique designed to identify the variation between treatments of interest to the researcher. There are well-established links between CATANOVA and the Goodman and Kruskal tau statistic as well as the Light and Margolin R 2 for the purposes of the graphical identification of this variation. The aim of this article is to present a partition of the numerator of the tau statistic, or equivalently, the BSS measure in the CATANOVA framework, into location, dispersion, and higher order components. Even if a CATANOVA identifies an overall lack of variation, by considering this partition and calculations derived from them, it is possible to identify hidden, but statistically significant, sources of variation.

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.

Partitioning Pearson's chi-squared statistic for singly ordered two-way contingency tables

This paper presents a partition of Pearsons 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.

Theory & Methods: 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).

The aggregate association index

Recently (Beh, 2008, JSPI) presented an index that helps to identify how likely two dichotomous categorical variables may be associated given only the aggregate (or marginal) information. Such an index was referred to as the aggregate association index. This paper will further consider some of the issues concerned with that index. These include variations of the original index as well as adaptations for quantifying the possibility that there exists a statistically significant positive or negative association between the two dichotomous variables.

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.

Correspondence analysis of ranked data

In the past correspondence analysis has been generally applied to two-way and multi-way contingency tables. However, data is sometimes presented m the form of rankings, and the analysis of treatments and their rankings often needs to be made. To apply correspondence analysis to ranked data, Andersons chi-square statistic is considered instead of the classical Pearson statistic. The alternative approach to correspondence analysis discussed by Beh (1997) is also applicable to rank data if the researcher is interested in how the ranks and treatments compare in terms of location, dispersion and higher order components A brief discussion on the interpretation of the transition formulae of Beh (1997) is also made by considering the 3x3 bean example of Anderson (1959).

Correspondence Analysis of Cumulative Frequencies Using a Decomposition of Taguchi's Statistic

Taguchis statistic has long been known to be a more appropriate measure of association for ordinal variables than the Pearson chi-squared statistic. Therefore, there is some advantage in using Taguchis statistic for performing correspondence analysis when a two-way contingency table consists of one ordinal categorical variable. This article will explore the development of correspondence analysis using a decomposition of Taguchis statistic.