1. Introduction
Qi Dong, Michael R. Elliott and Trivellore E. Raghunathan
Statistical methods outside the survey methodology setting have usually been developed without careful consideration for sample design, often implicitly assuming simple random samples, or, occasionally, one-stage cluster samples. Major efforts of modern survey statistics focus on extending methods to analyze complex survey data (Skinner, Holt and Smith 1989), accommodating issues such as stratification, unequal probability of selection, nonresponse bias or calibration. Hinkins, Oh and Scheuren (1997) proposed an inverse sampling design algorithm that connects the survey statistics and the classical statistics from another perspective. Their basic idea is to choose a subsample that has a simple random sample structure unconditionally. The subsample is often much smaller than the original sample, so they propose to repeat the process independently many times and average the results to increase the precision. They also described exact or approximate inverse sampling schemes for stratified simple random sampling, one-stage cluster sampling, and two-stage cluster sampling. However, this new idea is not used widely in practice, perhaps because it is extremely computionally intensive and the precision losses are often substantial. Similarly, generating synthetic populations from a posterior predictive distribution of a population conditional on complex sample data in a fashion that accounts for the complex sample design is not straightforward (Little 1991). However, in recent years demand for synthetic populations has increased, in order to deal with weight trimming or windorization problems (Lazzeroni and Little 1998; Elliott and Little 2000; Elliott 2007; Chen, Elliott and Little 2010), disclosure risk settings (Little 1993; Raghunathan, Reiter and Rubin 2003; Reiter 2004, 2005), or combining data from multiple surveys (Raghunathan, Xie, Schenker, Parsons, Davis, Dodd and Feuer 2007; Dong 2012). Often the synthetic populations are generated under a distributional assumption (normal, binomial, Poisson), with the posterior distribution of the model parameters approximated by the asymptotic normal distribution. The mean and covariance matrix of the normal distribution are estimated after complex sampling design features are taken into account (Raghunathan et al. 2007).
A major weakness of model-based methods is that if the model is seriously misspecified, it may yield invalid inferences (Little 2004). In multivariate settings, we need to consider the relationships among the variables of interest and determine an appropriate model that fits the data, which may be hard if the data contains different types of variables. In this paper we propose a nonparametric method as a counterpart of the model-based method to generate synthetic populations. This work extends the finite population Bayesian bootstrap and related Pólya posterior models of Lo (1988), Ghosh and Meeden (1983), and Cohen (1997) to account for complex sample designs. Since it achieves the same goal of the inverse sampling technique, it can be treated as the Bayesian finite population version of inverse sampling. To make inference using this weighted finite population Bayesian bootstrap, we can either make use of the draws directly, or, for computational efficiency, use results previously derived in the disclosure risk and multiple imputation literature, since these non-parametrically-generated populations can be viewed as multiple imputations of the unobserved elements of the population.
This paper is organized as follows. Section 2 briefly discusses synthetic populations in the context of Bayesian finite population inference. Section 3 reviews and summarizes the Bayesian bootstrap method and its finite population extension, and shows that, for an unequal probability of selection sample, the distribution of synthetic populations generated under a variant of a Pólya urn scheme matches the posterior predictive distribution of a finite population Bayesian bootstrap. Section 4 presents the proposed method under stratified clustering sampling with unequal selection probabilities. Section 5 shows that inference from these non-parametrically-generated synthetic populations can be obtained using results from the disclosure risk and multiple imputation literature, where each synthetic population has zero "within-imputation� variance. Section 6 provides a simulation study to evaluate the performance of the nonparametric method in a repeated sampling context. Section 7 applies the method to generate synthetic populations than can be used to estimate health insurance coverage rates using the 2006 NHIS and MEPS data, and compares the result with a parametric (log-linear) modeling approach. Concluding remarks are provided in Section 8.
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