Modeling correlated ordinal response data is usually more complex than the case of continuous and binary responses. Existing literature lacks an appropriate approach to modeling such data. For small sample sizes, however, these models lose their appeal since their inferences are based on large samples. In this work, the Bayesian analysis of an asymmetric bivariate ordinal latent variable model has been developed. The latent response variable has been chosen to follow the generalized bivariate Gumble distribution. Using some specific priors and MCMC algorithms the regression parameters were estimated. As an application, a data set concerning Diabetic Retinopathy in 116 patients have been analyzed. This data set includes the disease status of each eye for patients as an ordinal response and a number of explanatory variables some of which are common to both eyes and the rest are organ-specific.

In this paper a general model is proposed for the joint distribution of nominal, ordinal and continuous variables with and without missing data. Closed forms are presented for likelihood functions of general location models. Also the Joe approximation is used for the parameters of general location models with mixed continuous, ordinal and nominal data with non-ignorable missing responses. To explain the ability of proposed models some simulation studies are performed and some real data are analyzed from a foreign language achievement study.