Fuzzy Moderation and Moderated-Mediation Analysis

Abstract In the causal relationship, a mediator variable is a variable that causes mediation in the dependent and the independent variables. If x is a predictor and y is a response variable, then w is a moderator variable that influences the causal relationship of x and y. A moderator variable is a variable that affects the strength of the relationship between a dependent and independent variable. When there are many complicated causal relations, a mediation analysis or a moderation analysis can be performed considering the existence of mediators or moderators. Moreover, when both mediators and moderators exist, a mediation–moderation analysis can be performed. The existence of these variables occurs in many fields, including social science, medical science, and natural science, etc. However, the values of such variables used are often observed as fuzzy numbers rather than as crisp numbers (real numbers). So in many cases, fuzzy analysis is required because observations are observed with ambiguous values, but in the meantime, only models that use crisp numbers rather than fuzzy numbers have been used. This paper proposes fuzzy moderation analysis and fuzzy moderated-mediation analysis as the first attempts of the moderation and moderated-mediation analysis using fuzzy data. The proposed models can also be used for science and engineering, medical data, but it can also be applied to the humanities fields, where a lot of ambiguous data are observed. For example, data from the humanities fields such as marketing, education or psychology, the data are observed based on a human’s mind. Nevertheless, they have been analyzed using crisp data so far. In this paper, we define several fuzzy moderation models and fuzzy mediation–moderation models considering various situations based on fuzzy least squares estimation (FLSE). In addition, the validity of the proposed model is shown in some examples; it compares the results with existing analysis using crisp data.