Joint modelling of two count variables when one of them can be degenerate

Abstract

We formulate a joint statistical model for two variables: one of them is either a count variable or just zero, and the other is a regular count variable. We consider a modelling framework based on switching between a bivariate Poisson regression model and a univariate one, where the switching depends on the observable outcome of the third, dichotomous variable. The ZIP–CP bivariate model (proposed quite recently) and the standard univariate Poisson regression model are used as basic elements of the switching (or mixture) model. Bayesian analysis is advocated; in two special cases of our Bayesian statistical model, consequences for inference are discussed. The empirical part is devoted to joint modelling of the numbers of cash payments and bank card payments in Poland, with the use of data for both cardholders and non-cardholders. Our Bayesian statistical test enables to examine whether it is appropriate to analyse each of two subsamples separately in order to infer on basic parameters. In the case of our data it is so, therefore inference on individual parameters is not affected by the sample selection error. However, inference on the correlation coefficient between two count variables is possible only within the proposed trivariate model.