Bivariate categorical data analysis using normal linear conditional multinomial probability model

Bivariate multinomial data such as the left and right eyes retinopathy status data are analyzed either by using a joint bivariate probability model or by exploiting certain odds ratio‐based association models. However, the joint bivariate probability model yields marginal probabilities, which are complicated functions of marginal and association parameters for both variables, and the odds ratio‐based association model treats the odds ratios involved in the joint probabilities as ‘working’ parameters, which are consequently estimated through certain arbitrary ‘working’ regression models. Also, this later odds ratio‐based model does not provide any easy interpretations of the correlations between two categorical variables. On the basis of pre‐specified marginal probabilities, in this paper, we develop a bivariate normal type linear conditional multinomial probability model to understand the correlations between two categorical variables. The parameters involved in the model are consistently estimated using the optimal likelihood and generalized quasi‐likelihood approaches. The proposed model and the inferences are illustrated through an intensive simulation study as well as an analysis of the well‐known Wisconsin Diabetic Retinopathy status data. Copyright © 2014 John Wiley & Sons, Ltd.