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“A New Methodology for Calibrating Fuzzy Sets in fsQCA Using Level 2 and Interval Type-2 Fuzzy Sets”
by Jerry M. Mendel and Mohammad M. Korjani
June 2015
This report provides a new methodology for calibrating the fuzzy sets that are used in fsQCA, one that is based on clearly distinguishing between a linguistic variable and the linguistic terms for that variable. The resulting fuzzy sets are reduced-information level 2 fuzzy sets (RI L2 FSs). The major steps for obtaining the RI L2 FSs are: (1) For each linguistic variable, a vocabulary of naturally ordered linguistic terms (words) are chosen; (2) Interval end-point data are collected for each of the linguistic variables, either from a group of subjects or from one expert; (3) The data for each word are mapped into the footprint of uncertainty (FOU) of an interval type-2 fuzzy set (IT2 FS) using the HM approach [10]; (4) An RI L2 FS is created by replacing each word with an uncertainty measure and choosing an appropriate membership grade for it; and, (5) The MF of the RI L2 FS is approximated so that the resulting MF is for x ∈ X . The resulting approximated RI L2 FS MF is for the linguistic variable, and is not the MF of an ordinary FS but instead is the MF of a level 2 FS, one that has an S-shape, the kind of shape that is so widely used by fsQCA scholars, and is so important to fsQCA.
This report also applies its new calibration methodology to Ragin’s Breakdown of Democracy example, using new data provided to us by him, and demonstrates that we are able to obtain his earlier solutions using RI L2 FSs in either type-1 or interval-valued (IV) fsQCA, something that should be reassuring to fsQCA scholars. It also studies the robustness of fsQCA to MF breakpoint location uncertainties as well as to membership grade uncertainties. Finally, because the S-shaped MFs are derived from FOUs for all of the linguistic variable’s terms, this paper shows how to obtain more precise statements of fsQCA causal combinations for their best instances, something that may be of added value to practitioners of fsQCA.