Michael Smithson, Fuzzy Set Inclusion: Linking Fuzzy Set Methods With Mainstream Techniques, Sociological Methods & Research 2005 33: 431-461.

Abstract: The concept of set inclusion has remained insufficiently developed in the fuzzy set literature to be of much use to social scientists. However, a fully fledged concept of fuzzy set inclusion, along with appropriate statistical methods for evaluating it, could be very useful in the social sciences. This article combines fuzzy set and statistical methods, in the form of a cumulative distribution-based approach to evaluating fuzzy set inclusion without making strong assumptions about measurement levels. It establishes criteria for distinguishing an “inclusion relation” from independence plus skew as well as from other kinds of relationships. A measure of inclusion is developed that is sensitive to the degree to which individual cases violate a strict inclusion rule. A technique for modeling localized inclusion relations in contingency tables and scatter plots is also presented. Finally, the connections between the fuzzy set approach to set inclusion and mainstream statistical techniques are briefly adumbrated.

Key Words: fuzzy set theory • data analysis • set inclusion • cumulative distribution function

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