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The Aggregate Prediction Index and Non-Symmetrical Correspondence Analysis of Aggregate Data: The 2 X2 Table
No full textFeatures of the polynomial biplots for ordered contingency tables
No full textFor more than 20 years variants of correspondence analysis have arisen that accommodate for the structure of ordered categorical variables using orthogonal polynomials. When the visual display from this analysis is the biplot, projections linking the origin to the standard coordinate of each category is a common feature.
In the case when a column variable, say, consists of ordered categories, the biplot can be constructed so that their standard coordinate is determined using orthogonal polynomials which require a set of a priori scores that reflect the ordered structure of the categories. When the first two polynomials are used to construct the biplot they produce a configuration of standard coordinates that appear to be parabolic in shape.
This paper verifies the exact nature of this parabolic relationship and examines the various features of this configuration of points. Particular emphasis is given to the focus, vertex, intercepts and directrix of this relationship and we also briefly examine the impact of choosing different a priori scores on these features. The R function, parabola.exe(), used to perform these calculates is included as supplementary material to this paper
The prediction index of aggregate data
No full textThe analysis of the association between the two dichotomous variables of a 2 × 2 table arises as an important statistical issue in a number of diverse settings, such as in biomedical, medical, epidemiological, pharmaceutical or environmental research. When only the aggregate (or marginal) information is available, the analyst may determine the likely strength of the association between the variables. In this paper, we propose a new measure, called aggregate prediction index, that assesses the likely statistical significance of the association between the rows and columns of a 2 × 2 table where one variable is treated as a predictor variable and the other is treated as a response variable. Further insight into the predictor’s potential strength can be visually obtained by performing an asymmetric version of correspondence analysis and considering a biplot display of the two variables - this issue shall also be explored in light of the new index
Visualizing departures from symmetry. A study on cardiovascular risk among patients with diabetes.
No full textSingly and doubly ordered cumulative correspondence analysis.
No full textThe classical approach to correspondence analysis (CA) is designed to allow its user to a graphically summarize
the association between two or more categorical variables that form a contingency table. Despite its
popularity and utility, the classical approach does not take in consideration the structure of ordered variables.
One way to performing CA when the variables have an ordered structure is to consider the Taguchi’s statistic
(Taguchi, 1974). Beh, D’Ambra, Simonetti (2010) demonstrated the applicability of considering this statistic
which takes into account the ordered structure by considering the cumulative sum of cell frequencies across
the variable. Thus, the statistic is defined by summing the chi-squared statistic for each I × 2 contingency
table obtained by aggregating the column categories 1 to j and aggregating the column categories (j+1)
to J. For this reason, the Taguchi’s statistic is also referred to as cumulative chi-squared statistic (Nair; 1987).
Cuadras (2002) proposes an approach to correspondence analysis based on double cumulative frequencies.
However, it does not decompose any known index. In this paper we explore a generalization of Taguchi’s
statistic which takes into account the presence of two ordinal categorical variables by considering their cumulative
sum of cell frequencies. This generalization is analogous to the doubly cumulative chi-squared statistic
which is constructed by summing the chi-squared statistic for each 2×2 sub-table formed by pooling adjacent
rows and columns of the original contingency table; see Hirotsu (1986).
We illustrate this approach to CA using a partition of the statistic proposed by Hirotsu. Its application
presents some interesting properties and allows the analyst to represent the variations of row and column
categories rather than the categories on the space generated by cumulative frequencies
CATANOVA for two-way contingency tables with ordinal variables using orthogonal polynomials
No full textThe 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
A European perception of food using two methods of correspondence analysis
No full textIn a recent issue of this journal, Guerrero et al. (2010) studied an interesting data set involving the analysis
of consumer-driven associations to the word ‘‘Traditional’’, from a food perspective, in six European
countries. As part of their analysis, they demonstrated the sources of association between the words
studied and the country of origin of those interviewed using correspondence analysis. In this paper,
we focus on this association by assuming that the country of origin is a predictor of the words associated
with ‘‘Traditional’’. This analysis is performed using another member of the correspondence analysis
family – non-symmetric correspondence analysis. This paper will also explore the use of both these
correspondence analysis techniques on their data and consider the dendrogram and the semantic differential
plot as alternative approaches to visually summarising the association
Three-way ordinal non symmetrical correspondence analysis for the evaluation of the patient satisfaction
No full textIn some recent articles, emphasis has been given to the partition of the Goodman-Kruskal’s tau index using orthogonal polynomials for the study of the non symmetrical relations in three-way contingency tables. New graphical techniques that consider such a partition and allow for the analysis of asymmetric relationships have been proposed, including three-way ordinal non symmetrical correspondence analysis (Simonetti, 2003). Such a procedure takes into account the presence of an ordinal predictor and response variables. In this paper we demonstrate the applicability of such a technique for the patient satisfaction evaluation
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