Direct and Indirect Scaling of Membership Functions of Probability Phrases

摘要

A crucial issue in the empirical measurement of membership functions is whether the degree of fuzziness is invariant under different scaling procedures. In this paper a direct and an indirect procedure, magnitude estimation and graded pair-comparison, are compared in the context of establishing member shipfunctions for probability phrases such as probable, rather likely, very unlikely, and so forth. Analyses at the level of individual respondents indicate that (a) membership functions are stable over time, (b) functions for each phrase differ substantially over people, (c) the two procedures yield similarly shaped functions for a given person-phrase combination, (d) the functions from the two procedures differ systematically, in that those obtained directly dominate, or indicate greater fuzziness than do those obtained indirectly, and (e) where the two differ the indirectly obtained function may be the more accurate one. A secondary purpose of the paper is to evaluate the effects of the modifiers very and rather. Very has a general intensifying effect that is described by Zadeh's (1972) concentration model for seven subjects and by a shift model for no one. The effects of rather are unsystematic and not described by an available model. Keywords: Decision making, Psychometrics, Fuzzy sets. (KR)