Advanced Studies in Behaviormetrics and Data Science by Unknown

Author:Unknown
Language: eng
Format: epub
ISBN: 9789811527005
Publisher: Springer Singapore

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Nishisato [13] discussed the condensed response-pattern table, corresponding to the contingency table, as shown in Table 2, as equivalent to the incidence table, consisting of 1s and 0s. The equivalence of the condensed response-pattern table and the (1, 0) incidence table can be demonstrated by Benzécri’s principle of distributional equivalence (Benzécri [3]) and Nishisato’s principle of equivalent partitioning of forced classification (Nishisato [14]).

Chapter 4 of his 1980 book contains a description of a number of mathematical relations between the contingency table and the response-pattern table formats. The most relevant to our discussion is that the condensed response-pattern table contains both rows and columns of the contingency table in columns. From the graphical point of view, this means that quantification of the response-pattern table yields coordinates for both rows and columns of the contingency table in common space (see Young & Householder [30]). Since our interest lies in finding coordinates of these variables in common space, we now look at only the quantification outcome of the response-pattern table. For detailed comparisons of the two response formats, please refer to Nishisato [20] and Nishisato et al. [22].

First, let us note that multidimensional data structure of rows and columns of the contingency table can be represented by principal coordinates, obtained from the response-pattern representation of the contingency table. Second, notice that those principal coordinates must be identified from a set of components in terms of some mathematical constraints associated with multidimensional decomposition. Table 3 (Principal Coordinates in Different Types of Space) shows the decompositions of the response-pattern structure of the contingency table, as reported in Nishisato [20] and Nishisato et al. [22].Table 2Response-pattern table

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