Nonparametric Bayesian models with focused clustering for mixed ordinal and nominal data

De Yoreo, M., J. P. Reiter, and D. S. Hillygus. "Nonparametric Bayesian models with focused clustering for mixed ordinal and nominal data." Bayesian Analysis (2015). DOI: 10.1214/16-BA1020.
Dirichlet process mixtures can be useful models of multivariate categorical data and effective tools for multiple imputation of missing categorical values. In some contexts, however, these models can fit certain variables well at the expense of others in ways beyond the analyst's control. For example, when the data include some variables with non-trivial amounts of missing values, the mixture model may fit the marginal distributions of the nearly and fully complete variables at the expense of the variables with high fractions of missing data. Motivated by this setting, we present a Dirichlet process mixture model for mixed ordinal and nominal data that allows analysts to split variables into two groups: focus variables and remainder variables. The model uses three sets of clusters, one set for ordinal focus variables, one for nominal focus variables, and one for all remainder variables. The model uses a multivariate ordered probit specification for the ordinal variables and independent multinomial kernels for the nominal variables. The three sets of clusters are linked using an infinite tensor factorization prior, as well as via dependence of the means of the latent continuous focus variables on the remainder variables. This effectively specifies a rich, complex model for the focus variables and a simpler model for remainder variables, yet still potentially captures associations among the variables. In the multiple imputation context, focus variables include key variables with high rates of missing values, and remainder variables include variables without much missing data. Using simulations, we illustrate advantages and limitations of using focused clustering compared to mixture models that do not distinguish variables. We apply the model to handle missing values in an analysis of the 2012 American National Election Study.