Analyzing bivariate ordinal data with CUB margins

Statistical modelling for ordinal data has received a considerable attention in the literature, and a consolidated theory relying on Generalized Linear Model approach has been developed. In this article, we present an innovative technique for modelling bivariate ordinal data. In particular, we consider the method introduced by Plackett for constructing a one-parameter bivariate distribution from given margins, and we apply it in order to represent correlated ordinal variables which individually follows a CUB model. This is a univariate mixture distribution defined by the convex Combination of a Uniform and a shifted Binomial distribution whose parameters may be related to rater's covariates. The article shows how the bivariate distribution can be defined and how its characterizing parameter, which describes the association between the component random variables, can be related to the subject's covariates. The proposed approach is applied to the study of two key drivers of extra virgin olive oil consumption in Italy. The technique allows a representation of the data whose meaning can be easily interpreted providing useful information for management support.