A Mixture Model for the Analysis of Categorical Variables Measured on Five-point Semantic Differential Scales

Abstract

Ordered response scales are often used in questionnaires to measure individuals' attitudes or perceptions. Among different response scale formats, we focus on multi-point semantic differential scales, requiring the respondent to position himself/herself on a rating between two bipolar adjectives. The obtained rating data require appropriate statistical models. We resort to the CUM model (Combination of a discrete Uniform and a - linearly transformed - Multinomial random variable), recently proposed in the framework of the CUB (Combination of discrete Uniform and shifted Binomial random variables) class of models. CUM is also suited to all the ordinal response scales with a middle “indifference” option. In the seminal paper on CUM, the methodological approach was developed for an odd number m of response categories, while simulations, case studies and implementation in R were limited to m = 7. The objective of this paper is to extend the original proposal and investigate the model performance in the case of m = 5, which often arises in real situations. The R functions for fitting a CUM model with m = 5 are implemented and made available; simulation studies are developed and compared with results obtained for m = 7 and a case study concerned with the evaluation of museums' visitor experience is proposed.