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.

在因果关系中，中介变量是导致在因变量和自变量中进行调解的变量。如果x是预测变量，而y是响应变量，则w是影响x和y的因果关系的调节变量。主持人变量是影响因变量和自变量之间关系强度的变量。当存在许多复杂的因果关系时，可以考虑存在中介者或主持人来执行调解分析或仲裁分析。此外，当调解人和主持人同时存在时，可以执行调解-主持人分析。这些变量的存在发生在许多领域，包括社会科学，医学和自然科学等。但是，这些变量的值通常被观察为模糊数，而不是清晰数（实数）。因此，在许多情况下，需要进行模糊分析，因为观察到的观测值含糊不清，但与此同时，仅使用了使用明晰数而不是模糊数的模型。本文提出了模糊适度分析和模糊适度中介分析作为使用模糊数据进行适度和适度中介分析的首次尝试。提出的模型还可以用于科学和工程，医学数据，但是也可以应用于人文科学领域，其中观察到很多模棱两可的数据。例如，来自人文领域（例如市场营销，教育或心理学）的数据是根据人的思想来观察的。不过，到目前为止，已经使用清晰的数据对它们进行了分析。在本文中，我们基于模糊最小二乘估计（FLSE）定义了考虑各种情况的几种模糊调节模型和模糊中介-调节模型。另外，在一些示例中显示了所提出模型的有效性。它将结果与使用清晰数据的现有分析进行比较。