Article

Measures of fit in multiple correspondence analysis of crisp and fuzzy coded data

Department of Economics and Business, Universitat Pompeu Fabra, Economics Working Papers 01/2008; DOI: 10.2139/ssrn.1107815
Source: RePEc

ABSTRACT When continuous data are coded to categorical variables, two types of coding are possible: crisp coding in the form of indicator, or dummy, variables with values either 0 or 1; or fuzzy coding where each observation is transformed to a set of “degrees of membership” between 0 and 1, using co-called membership functions. It is well known that the correspondence analysis of crisp coded data, namely multiple correspondence analysis, yields principal inertias (eigenvalues) that considerably underestimate the quality of the solution in a low-dimensional space. Since the crisp data only code the categories to which each individual case belongs, an alternative measure of fit is simply to count how well these categories are predicted by the solution. Another approach is to consider multiple correspondence analysis equivalently as the analysis of the Burt matrix (i.e., the matrix of all two-way cross-tabulations of the categorical variables), and then perform a joint correspondence analysis to fit just the off-diagonal tables of the Burt matrix – the measure of fit is then computed as the quality of explaining these tables only. The correspondence analysis of fuzzy coded data, called “fuzzy multiple correspondence analysis”, suffers from the same problem, albeit attenuated. Again, one can count how many correct predictions are made of the categories which have highest degree of membership. But here one can also defuzzify the results of the analysis to obtain estimated values of the original data, and then calculate a measure of fit in the familiar percentage form, thanks to the resultant orthogonal decomposition of variance. Furthermore, if one thinks of fuzzy multiple correspondence analysis as explaining the two-way associations between variables, a fuzzy Burt matrix can be computed and the same strategy as in the crisp case can be applied to analyse the off-diagonal part of this matrix. In this paper these alternative measures of fit are defined and applied to a

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    ABSTRACT: This article provides a largely nontechnical discussion of the acquisition of membership values in fuzzy set analyses. First the basic properties of a membership are discussed. Then the three common strategies of membership assignment—direct subjective assign- ment, indirect subjective assignment, and transformation—are critically examined in turn. Examples are used to illustrate the techniques. The connection with existing psy- chometric and statistical methods is particularly emphasized, focusing on the notion of a membership value as a random variable as a means to assess uncertainty in assignment.
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