A new adaptive membership function with CUB uncertainty with application to cluster analysis of Likert-type data

Likert-type scales are commonly used in both academia and industry to capture human feelings since they are user-friendly, easy-to-develop and easy-to administer. This kind of scales generate ordinal variables made up of a set of rank ordered items. Since the distance between two consecutive items cannot be either defined or presumed equal, this kind of variable cannot be analysed by either statistical methods defined on a metric space or parametric tests. Therefore, Likert-type variables cannot be used as segmentation variables of a traditional cluster analysis unless pre-transformed. In such context, fuzzy numbers have been suggested as a way to recode Likert-type variables. Fuzzy numbers are defined by a membership function whose form is usually determined by an expert. In practice, researchers usually define one membership function for each Likert-type scale, not considering the peculiar characteristics of neither questions nor respondents. In this way, the individual uncertainty against each question is considered equal and constant. To overcome this limitation and to reduce the expert’s subjectivity, in this study an adaptive membership function based on CUB model is suggested to pre-transform Likert-type variables into fuzzy numbers before the adoption of a clustering algorithm. After a theoretical presentation of the method, an application using real data will be presented to demonstrate how the method works.