Dirichlet Process Mixture Models for Modeling and Generating Synthetic Versions of Nested Categorical Data

Abstract

We present a Bayesian model for estimating the joint distribution of multivariate categorical data when units are nested within groups. Such data arise frequently in social science settings, for example, people living in households. The model assumes that (i) each group is a member of a group-level latent class, and (ii) each unit is a member of a unit-level latent class nested within its group-level latent class. This structure allows the model to capture dependence among units in the same group. It also facilitates simultaneous modeling of variables at both group and unit levels. We develop a version of the model that assigns zero probability to groups and units with physically impossible combinations of variables. We apply the model to estimate multivariate relationships in a subset of the American Community Survey. Using the estimated model, we generate synthetic household data that could be disseminated as redacted public use files. Supplementary materials for this article are available online.

Simulation of Synthetic Complex Data: The R Package simPop

“…For example, random forests algorithms are known to perform well and could be used to simulate variables; see Caiola and Reiter (2010).…”

Simultaneous Edit and Imputation For Household Data with Structural Zeros

“…In theory, f X pX 1 |θ X q can be any multivariate categorical data model that adequately describes the joint distribution of all the variables, has support restricted to C´S, and captures the relevant structure in X 1 . For household data, the truncated NDPMPM model of Hu et al (2018) has those properties. In this section, we briefly review the NDPMPM model.…”

“…For prior distributions, we follow the recommendations of Hu et al (2018). We use independent uniform Dirichlet distributions as priors for λ and φ, and the truncated stick-breaking representation of the Dirichlet process as priors for π and ω (Sethuraman 1994;Dunson and Xing 2009;Si and Reiter 2013;Manrique-Vallier and Reiter 2014).…”

“…We set the parameters for the Dirichlet distributions in (5) to 1 d k (a d k -dimensional vector of ones) and the parameters for the Gamma distributions in (8) to 0.25 to represent vague prior specifications. For further discussions on prior specifications, see Hu et al (2018).…”

“…The hierarchical model includes (i) a multivariate model for the true latent values of the data which has support only on theoretically possible households, (ii) a model for locations of errors given the latent true values, and (iii) a model for the reported values for fields in error. For the multivariate model of the true values, we use the truncated nested data Dirichlet process mixture of products of multinomial distributions (NDPMPM) of Hu et al (2018). This model includes household level and individual level variables, allows for within-household dependence, and puts zero probability on impossible combinations.…”