Multiple Imputation and Synthetic Data Generation with NPBayesImputeCat

Abstract

In many contexts, missing data and disclosure control are ubiquitous and challenging issues. In particular, at statistical agencies, the respondent-level data they collect from surveys and censuses can suffer from high rates of missingness. Furthermore, agencies are obliged to protect respondents’ privacy when publishing the collected data for public use. The NPBayesImputeCat R package, introduced in this paper, provides routines to i) create multiple imputations for missing data and ii) create synthetic data for statistical disclosure control, for multivariate categorical data, with or without structural zeros. We describe the Dirichlet process mixture of products of the multinomial distributions model used in the package and illustrate various uses of the package using data samples from the American Community Survey (ACS). We also compare results of the missing data imputation to the mice R package and those of the synthetic data generation to the synthpop R package. Introduction and background Multiple imputation for missing data Missing data problems arise in many statistical analyses. To impute missing values, multiple imputation, first proposed by Rubin (1987), has been widely adopted. This approach estimates predictive models based on the observed data, fills in missing values with draws from the predictive models, and produces multiple imputed and completed datasets. Data analysts then apply standard statistical analyses (e.g., regression analysis) on each imputed dataset and use appropriate combining rules to obtain valid point estimates and variance estimates (Rubin, 1987). As a brief review of the multiple imputation combining rules for missing data, let q be the completed data estimator of some estimand of interest Q, and let u be the estimator of the variance of q. For l = 1, . . . , m, let q(l) and u(l) be the values of q and u in the lth completed dataset. The multiple imputation estimate of Q is equal to q̄m = ∑l=1 q (l)/m, and the estimated variance associated with q̄m is equal to Tm = (1 + 1/m)bm + ūm , where bm = ∑l=1(q (l) − q̄m)/(m − 1) and ūm = ∑l=1 u (l)/m. Inferences for Q are based on (q̄m − Q) ∼ tv(0, Tm), where tv is a t-distribution with v = (m − 1)(1 + ūm/[(1 + 1/m)bm]) degrees of freedom. Multiple imputation by chained equations (MICE, Buuren and Groothuis-Oudshoorn (2011)) remains the most popular method for generating multiple completed datasets after multiple imputation. Under MICE, one specifies univariate conditional models separately for each variable, usually using generalized linear models (GLMs) or classification and regression trees (CART Breiman et al. (1984); Burgette and Reiter (2010)), and then iteratively samples plausible predicted values from the sequence of conditional models . For implementing MICE in R, most analysts use the mice package. For an in-depth review of the MICE algorithm, see Buuren and Groothuis-Oudshoorn (2011). For more details and reviews, see Rubin (1996), Harel and Zhou (2007), Reiter and Raghunathan (2007). Synthetic data for statistical disclosure control Statistical agencies regularly collect information from surveys and censuses and make such information publicly available for various purposes, including research and policymaking. In numerous countries around the world, statistical agencies are legally obliged to protect respondents’ privacy when making this information available to the public. Statistical disclosure control (SDC) is the collection of techniques applied to confidential data before public release for privacy protection. Popular SDC techniques for tabular data include cell suppression and adding noise, and popular SDC techniques for respondent-level data (also known as microdata) include swapping, adding noise, and aggregation. Hundepool et al. (2012) provide a comprehensive review of SDC techniques and applications. The multiple imputation methodology has been generalized to SDC. One approach to facilitating microdata release is to provide synthetic data. First proposed by Little (1993) and Rubin (1993), the synthetic data approach estimates predictive models based on the original, confidential data, simulates synthetic values with draws from the predictive models, and produces multiple synthetic datasets. Data analysts then apply standard statistical analyses (e.g., regression analysis) on each synthetic dataset and use appropriate combining rules (different from those in multiple imputation) to obtain valid point estimates and variance estimates (Reiter and Raghunathan, 2007; Drechsler, The R Journal Vol. 13/2, December 2021 ISSN 2073-4859 CONTRIBUTED RESEARCH ARTICLES 91 2011). Moreover, synthetic data comes in two flavors: fully synthetic data (Rubin, 1993), where every variable is deemed sensitive and therefore synthesized, and partially synthetic data (Little, 1993), where only a subset of variables is deemed sensitive and synthesized, while the remaining variables are un-synthesized. Statistical agencies can choose between these two approaches depending on their protection goals, and subsequent analyses also differ. When dealing with fully synthetic data, q̄m estimates Q as in the multiple imputation setting, but the estimated variance associated with q̄m becomes Tf = (1 + 1/m)bm − ūm , where bm and ūm are defined as in previous section on multiple imputation. Inferences for Q are now based on (q̄m − Q) ∼ tv(0, Tf ), where the degrees of freedom is v f = (m − 1)(1 − mūm/((m + 1)bm)). For partially synthetic data, q̄m still estimates Q but the estimated variance associated with q̄m is Tp = bm/m + ūm , where bm and ūm are defined as in the multiple imputation setting. Inferences for Q are based on (q̄m − Q) ∼ tv(0, Tp), where the degrees of freedom is vp = (m − 1)(1 + ūm/[bm/m]). For synthetic data with R, synthpop provides synthetic data generated by drawing from conditional distributions fitted to the confidential data. The conditional distributions are estimated by models chosen by the user, whose choices include parametric or CART models. For more details and reviews of synthetic data for statistical disclosure control, see Drechsler (2011).