  Multivariate categorical data are common in many fields. We are motivated by
election polls studies assessing evidence of changes in voters opinions with
their candidates preferences in the 2016 United States Presidential primaries
or caucuses. Similar goals arise routinely in several applications, but current
literature lacks a general methodology which combines flexibility, efficiency,
and tractability in testing for group differences in multivariate categorical
data at different---potentially complex---scales. We address this goal by
leveraging a Bayesian representation which factorizes the joint probability
mass function for the group variable and the multivariate categorical data as
the product of the marginal probabilities for the groups, and the conditional
probability mass function of the multivariate categorical data, given the group
membership. To enhance flexibility, we define the conditional probability mass
function of the multivariate categorical data via a group-dependent mixture of
tensor factorizations, thus facilitating dimensionality reduction and borrowing
of information, while providing tractable procedures for computation, and
accurate tests assessing global and local group differences. We compare our
methods with popular competitors, and discuss improved performance in
simulations and in American election polls studies.
