Bayesian Modeling of Temporal Dependence in Large Sparse Contingency Tables

Abstract

It is of interest in many applications to study trends over time in relationships among categorical variables, such as age group, ethnicity, religious affiliation, political party, and preference for particular policies. At each time point, a sample of individuals provides responses to a set of questions, with different individuals sampled at each time. In such settings, there tend to be an abundance of missing data and the variables being measured may change over time. At each time point, we obtained a large sparse contingency table, with the number of cells often much larger than the number of individuals being surveyed. To borrow information across time in modeling large sparse contingency tables, we propose a Bayesian autoregressive tensor factorization approach. The proposed model relies on a probabilistic Parafac factorization of the joint pmf characterizing the categorical data distribution at each time point, with autocorrelation included across times. We develop efficient computational methods that rely on Markov chain Monte Carlo. The methods are evaluated through simulation examples and applied to social survey data. Supplementary materials for this article are available online.

Keywords:

• Dynamic model
• Multivariate categorical data
• Nonparametric Bayes
• Panel data
• Parafac
• Probabilistic tensor factorization
• Stick-breaking

Acknowledgments

This research was supported by Nakajima Foundation and grants ES017436 and ES017240 from the National Institute of Environmental Health Sciences (NIEHS) of the US National Institutes of Health. The computational results are mainly generated using Ox (Doornik 2006).