A mixture distribution for modelling bivariate ordinal data

Ordinal responses often arise from surveys which require respondents to rate items on a Likert scale. Since most surveys contain more than one question, the data collected are multivariate in nature, and the associations between different survey items are usually of considerable interest. In this paper, we focus on a mixture distribution, called the combination of uniform and binomial (CUB), under which each response is assumed to originate from either the respondent's uncertainty or the actual feeling towards the survey item. We extend the CUB model to the bivariate case for modelling two correlated ordinal data without using copula-based approaches. The proposed model allows the associations between the unobserved uncertainty and feeling components of the variables to be estimated, a distinctive feature compared to previous attempts. This article describes the underlying logic and deals with both theoretical and practical aspects of the proposed model. In particular, we will show that the model is identifiable under a wide range of conditions. Practical inferential aspects such as parameter estimation, standard error calculations and hypothesis tests will be discussed through simulations and a real case study.