On the interpretation of the uncertainty parameter in CUB models

The rationale for the CUB models is based on the fact that the response of a person to an item is a weighted combination of two factors: a subjective agreement towards the item and some intrinsic fuzziness in the final response. These two components are parametrized as a Shifted Binomial and a discrete Uniform random variable respectively. The final model is a mixture of these two random variables with weights pi and (1-pi ). The last quantity is currently interprets as a measure of the uncertainty that accompanies the choice of a response category made by the subjects who form the population under study. This paper wants to be a warning for this interpretation, when the data of interest derive from a questionnaire designed to measure an overall latent trait. Through a simulation study one will show that in this context there are situations in which a high value of (1-pi ) is connected to the distribution of the overall latent trait among the subjects in the population and to the facet of the overall latent trait that the item wants to represent and not to an high level of fuzziness in the final response.