Bayesian inference for ordinal data

Albert & Chib proposed a Bayesian ordinal probit regression model using the Gibbs sampler to model ordinal data. Their method defines a relationship between latent variables and ordinal outcomes using cutpoint parameters. However, the convergence of this Gibbs sampler is slow when the sample size is large because the cutpoint parameters are not efficiently sampled. Cowles proposed a Gibbs/Metropolis-Hastings (MH) sampler that would update cutpoint parameters more efficiently. In the context of longitudinal ordinal data, this algorithm potentially require the computation of cumulative probability of a multivariate normal distribution to calculate the acceptance probability for the MH sampler. We propose a probit model where the latent variables follow a mixture of normal distributions. This mixture structure can successfully characterize the ordinality of data while holding the cutpoint parameters constant. Gibbs samplings along with reversible jump MCMC are carried out to estimate the size of the mixture. We adopt this idea in modeling ordinal longitudinal data, where the autoregressive error (1) model is proposed to characterize the underlying correlation structure among the repeated measurements. We also propose a Bayesian probabilistic model in estimating the clustering membership using a mixture of Gaussian distributions to tackle the problem of clustering ordinal data. Results are compared with those obtained from K-means method. We further extend the multinomial probit (MNP) model and develop a joint MNP and ordinal probit model to model the cell probabilities for multiple categorical outcomes with ordinal variables nested within each categorical outcome. A hierarchical prior is imposed on the location parameters of the normal kernels in the mixture model associated with the ordinal outcomes. Our model has wide applications in various fields such as clinical trials, marketing research, and social science.