8. Discussion
Qi Dong, Michael R. Elliott and Trivellore E. Raghunathan
In this paper, we propose and evaluate a nonparametric method to generate synthetic populations. This method adjusts for the complex sampling design features without assuming any models to the observed data so it is robust to model-misspecification. Also, unlike model-based methods that needs to develop separate imputation models for different variables of interest, the nonparametric method only uses the design variables to generate synthetic populations and thus is not variable-specific.
We considered the repeated sampling properties of our non-parametric synthetic estimators in a univariate gamma and bivariate normal setting, estimating means, slopes, and intercepts. Point estimates were unbiased, intervals had approximately nominal coverage, and losses of efficiency relative to the actual data were trivial. We also considered a "real world� setting, generating a predictive distribution for the 2006 NHIS and MEPS and estimating rates and associated variance estimates of health insurance coverage using both the nonparametric method and a fully parametric log-linear modeling approach. When the model fits the data well, the model-based method is more efficient than the nonparametric method. However, when the assumed model does not fit the data well, as was the case in certain small domains, the model-based method may produce invalid inference. In such situations, the nonparametric method is robust to model misspecfication.
In addition to robustness to model misspecification, another advantage is that the nonparametric method only uses the design variables such as stratum, cluster and weight to impute the unobserved part of the population. Unlike model-based methods, it does not need to model the complicated relationships among the variables of interest, which becomes impossible if there are item missing values in the actual data. The synthetic populations generated by the nonparametric method still preserve the item missing values in the actual data. This potentially fills in a gap in the multiple imputation area in that existing imputation methods typically ignore the complex sampling design features in the data and impute the missing values as if they are simple random samples. A related advantage is that, while design variables are used in the nonparametric generation of the synthetic populations, the synthetic populations themselves do not need to contain them, since they can be analyzed as simple random samples. Hence, disclosure risk associated with release of design variables can be eliminated (De Waal and Willenborg 1997; Mitra and Reiter 2006; Reiter and Mitra 2009).
A fourth practical advantage of the nonparametric method is that it is easier to implement in existing statistical software packages because it focuses on the design variables; thus specific strategies for various types of variables and data structures do not need to be developed.
Because use of the weighted FPBB does not require information about the number of clusters in the population or conditional probabilities of selection at each stage of selection in a multistage sample setting, we use an approximate Bayesian bootstrap method to adjust for stratification and clustering. We view this as advantageous in many ways, since public use datasets typically do not break out weights for each stage of the sample. However, it does have the disadvantage that, to ensure positive replicate weights, the Bayesian bootstrap method produces fewer clusters within strata than in the actual data. In the setting where the probabilities of selection are known for all stages of the sample, it seems likely that the weighted FPBB can be implemented at each stage, with the population of unobserved clusters and the population of elements within each cluster imputed in a two-stage fashion, paralleling Meeden (1999) just as the one-stage FPBB parallels Ghosh and Meeden (1983). This remains an area for future research.
Acknowledgements
This research was supported by NCI grant R01CA129101. The authors wish to thank the Editor, Associate Editor, and two anonymous reviewers for their comments. We are especially indebted to the reviewer that helped us to better understand and explain the links between the finite population Bayesian bootstrap and Pólya posterior discussed in Section 3.
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