Treatment of ‘don’t know’ responses in a mixture model for rating data

In recent years the use of questionnaires to investigate and measure human perceptions has hugely intensified and, correspondingly, there has been an increased need for statistical models able to treat ordered categorical data, that usually derive from questions asking for ratings. In this paper we focus on a specific class of models, called CUB (Combination of Uniform and shifted Binomial), based on the assumption that rating data derive from an unconscious decision process composed of two independent components, called feeling and uncertainty. More precisely, we deal with a recently proposed extension in this context, the Nonlinear CUB model, which beyond being able to measure feeling and uncertainty, gives an idea of the state of mind of respondents toward the scale used to express the ratings. The aim of the paper is to show how the parameters of the Nonlinear CUB model can be adjusted in order to take account of the presence of `don't know' responses, following a recent idea developed in the CUB context. In addition, a graphical representation is proposed, able to summarize all the results in a unique graph. A case study is presented, concerned with data from the Eurobarometer survey