Mixture models for ordinal responses to account for uncertainty of choice

Abstract

In CUB models the uncertainty of choice is explicitly modelled as a Combination of discrete Uniform and shifted Binomial random variables. The basic concept to model the response as a mixture of a deliberate choice of a response category and an uncertainty component that is represented by a uniform distribution on the response categories is extended to a much wider class of models. The deliberate choice can in particular be determined by classical ordinal response models as the cumulative and adjacent categories model. Then one obtains the traditional and flexible models as special cases when the uncertainty component is irrelevant. It is shown that the effect of explanatory variables is underestimated if the uncertainty component is neglected in a cumulative type mixture model. Visualization tools for the effects of variables are proposed and the modelling strategies are evaluated by use of real data sets. It is demonstrated that the extended class of models frequently yields better fit than classical ordinal response models without an uncertainty component.

Cumulative and CUB Models for Rating Data: A Comparative Analysis

“…As statisticians, we must consider uncertainty as a real and significant component of human decisions and evaluations. Thus, we support CUB models, or, as an alternative, we advocate the explicit specification of uncertainty in cumulative models as POM: this proposal has been recently pursued by Tutz et al (2016) which showed that the effect of the explanatory variables on the responses tends to be underestimated if uncertainty is neglected.…”

“…It is possible to show that both variance and mean difference of a CUB distribution are not monotone functions of π , although they increase with 1 − π for a large range of π (see Tutz et al , ); on the other side, heterogeneity measures (as Gini, index, for instance) monotonically increase with 1 − π . Thus, this parameter is more correctly related to heterogeneity (Iannario, , pp.…”

Gini heterogeneity index for detecting uncertainty in ordinal data surveys

“…Identity (6) shows that, for a given feeling measure, heterogeneity (as measured by the Gini index) is inversely related to π and increases with uncertainty (that is, 1 − π). This result has been generalized since it has been proved that Gini index is monotonically related to the π parameter in any mixture model that includes a discrete Uniform distribution for the uncertainty [28]. Then, to obtain a larger class of parametric distributions for ordinal data arising in surveys concerning perceptions, opinions, judgements, preferences, etc., a statistical model defined as gecub (=Generalized cub) has been proposed.…”

Uncertain Choices: The Heterogeneous Multinomial Logit Model

“…Explanatory variables can determine the probabilities of the mixture. D’Elia and Piccolo (2005), Iannario and Piccolo (2010), Iannario (2012), Tutz et al (2017), and Iannario et al (2020) all considered models of this type, and Piccolo and Simone (2019) provided an extensive overview.…”