Multiple imputation of missing values in household data with structural zeros
Section 1. Introduction

In many population censuses and demographic surveys, statistical agencies collect data on individuals grouped within houses. In the U.S. decennial census, for example, the Census Bureau collects the age, race, sex, and relationship to the household head for every individual in the household, as well as whether or not the residents own the house. After collection, agencies share these datasets for secondary analysis, either as tabular summaries, public use microdata samples, or restricted access files.

When creating these data products, agencies typically have to deal with item nonresponse both for individual-level variables and household-level variables. They typically do so using some type of imputation procedure. Ideally, these procedures satisfy three desiderata. First, the imputations preserve the joint distribution of the variables as best as possible. As part of this, the procedure should preserve relationships within households. For example, the missing race of a spouse likely, but certainly not definitely, matches the race of the household head; the imputation procedure should reflect that. Second, the imputations respect structural zeros. For example, a daughter’s age cannot exceed her biological mother’s age. The imputations should not create impossible combinations of individuals in the same household. Third, the imputation procedure allows for appropriate uncertainty to be propagated in subsequent analyses of the data.

Typical approaches to imputation of missing household items use some variant of hot deck imputation (Kalton and Kasprzyk, 1986; Andridge and Little, 2010). However, depending on how the hot deck is implemented, it may not satisfy one or more of the desiderata. Indeed, we are not aware of any hot deck imputation procedure for household data that satisfies all three explicitly. An alternative is to estimate a model that describes the joint distribution of all the variables, and impute missing values from the implied predictive distributions in the model. For household data, one such model is the nested data Dirichlet process mixture of products of multinomial distributions (NDPMPM) model of Hu, Reiter and Wang (2018), which assumes that (i) each household is a member of a household-level latent class, and (ii) each individual is a member of an individual-level latent class nested within its household-level latent class. The model assigns zero probability to combinations corresponding to structural zeros, and also handles both household-level and individual-level variables simultaneously. The NDPMPM is appealing as an imputation engine, as it can preserve multivariate associations while avoiding imputations that result in impossible households. The NDPMPM is related to models proposed by Vermunt (2003, 2008) and Bennink, Croon, Kroon and Vermunt (2016), although these are used for regression rather than multivariate imputation and do not deal with structural zeros.

Hu et al. (2018) use the NDPMPM to generate synthetic datasets (Rubin, 1993; Raghunathan and Rubin, 2001; Reiter and Raghunathan, 2007) for statistical disclosure limitation, but they do not describe how to use it for imputation of missing data. We do so in this article. With structural zeros in the NDPMPM, the conditional distributions of the missing values given the observed values are not available in closed form. We therefore add a rejection sampling step to the Gibbs sampler used by Hu et al. (2018), which generates completed datasets as byproducts of the Markov chain Monte Carlo (MCMC) algorithms used to estimate the model. These completed datasets can be analyzed using multiple imputation inferences (Rubin, 1987). We also present two new strategies for speeding up the computations with NDPMPMs, namely (i) turning data for the household head into household-level variables rather than individual-level variables, and (ii) using an approximation to the likelihood function. These scalable innovations are necessary, as the NDPMPM is computationally quite intensive even without missing data. The speed-up strategies also can be employed when using the NDPMPM to generate synthetic data.

The remainder of this article is organized as follows. In Section 2, we review the NDPMPM model in the presence of structural zeros and the MCMC sampler for fitting the model without missing data. In Section 3, we extend the MCMC sampler for the NDPMPM model to allow for missing data. In Section 4, we present the two strategies for speeding up the MCMC sampler. In Section 5, we present results of simulation studies used to examine the performance of the NDPMPM as a multiple imputation engine, using the two strategies for speeding up the run time. In Section 6, we discuss findings, caveats and future work.

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