On the equivalence of two mixture models for rating data

Questionnaires are useful tool for exploring respondents’ perceptions through ratings, assumed to result from a latent decision process (DP). The DP varies when respondents rate on Likert or Semantic Differential scales. A possible paradigm to formalize the DP is based on the presence of a feeling and an uncertainty latent component, originally proposed as the foundations of the CUB (Combination of Uniform and shifted Binomial) class. It can be assumed that with Likert scales, respondents begin reasoning from the bottom, progressing upwards based on their sensations. Conversely, Semantic Differential scale users are assumed to start from the middle and move either upward or downward. The CUM (Combination of Uniform and Multinomial), a new model in the CUB class, derived from this DP, analyzes rating data on a Semantic Differential scale. This paper defines the concept of local and global unidirectional equivalence and studies, from an analytical point of view, the conditions under which CUB and CUM models generate identical theoretical probabilities, in order to enhance the interpretative understanding of the models.