Set Relations in Social Research: Evaluating Their Consistency and Coverage

TLDR

Set‑theoretic analysis, inherently asymmetric, contrasts with correlation‑based methods, yet social scientists were slow to adopt it due to perceived restrictions to binary variables and low error tolerance; the advent of fuzzy and rough sets has removed these barriers. This paper advances the set‑theoretic approach by presenting simple descriptive measures that can be used to evaluate set‑theoretic relationships, especially between fuzzy sets. The first measure, “consistency,” assesses the degree to which a subset relation has been approximated, while the second measure, “coverage,” evaluates the empirical relevance of a consistent subset. The study further shows that set‑theoretic coverage can be partitioned analogously to the partitioning of explained variation in multiple regression analysis.

Abstract

Because of its inherently asymmetric nature, set-theoretic analysis offers many interesting contrasts with analysis based on correlations. Until recently, however, social scientists have been slow to embrace set-theoretic approaches. The perception was that this type of analysis is restricted to primitive, binary variables and that it has little or no tolerance for error. With the advent of “fuzzy” sets and the recognition that even rough set-theoretic relations are relevant to theory, these old barriers have crumbled. This paper advances the set-theoretic approach by presenting simple descriptive measures that can be used to evaluate set-theoretic relationships, especially relations between fuzzy sets. The first measure, “consistency,” assesses the degree to which a subset relation has been approximated, whereas the second measure, “coverage,” assesses the empirical relevance of a consistent subset. This paper demonstrates further that set-theoretic coverage can be partitioned in a manner somewhat analogous to the partitioning of explained variation in multiple regression analysis.