A Geometrical Framework for Covariance Matrices of Continuous and Categorical Variables

Autor/inn/en	Vernizzi, Graziano; Nakai, Miki
Titel	A Geometrical Framework for Covariance Matrices of Continuous and Categorical Variables
Quelle	In: Sociological Methods & Research, 44 (2015) 1, S.48-79 (32 Seiten) PDF als Volltext Verfügbarkeit
Sprache	englisch
Dokumenttyp	gedruckt; online; Zeitschriftenaufsatz
ISSN	0049-1241
DOI	10.1177/0049124114543243
Schlagwörter	Matrices; Geometry; Statistical Analysis; Correlation; Surveys; Social Science Research + Suchen Sie Ihr Suchwort? Matrizenrechnung; Geometrie; Statistische Analyse; Korrelation; Survey; Umfrage; Befragung; Social scientific research; Sozialwissenschaftliche Forschung
Abstract	It is well known that a categorical random variable can be represented geometrically by a simplex. Accordingly, several measures of association between categorical variables have been proposed and discussed in the literature. Moreover, the standard definitions of covariance and correlation coefficient for continuous random variables have been extended to categorical variables. In this article, we present a geometrical framework where both continuous and categorical data are represented by simplices and lines in a high-dimensional space, respectively. We introduce a function whose direct minimization leads to a single definition of covariance between categorical-categorical, categorical-continuous, and continuous-continuous data. The novelty of this general approach is that a single space and a single distance function can be used for describing both continuous and categorical data. It thus provides a unified geometrical description of the measure of association, in particular between categorical and continuous data. We discuss virtues and limitations of such a geometrical framework and provide examples with possible applications to sociological surveys. (As Provided).
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Erfasst von	ERIC (Education Resources Information Center), Washington, DC
Update	2020/1/01