4. Nonparametric method to generate synthetic populations

Qi Dong, Michael R. Elliott and Trivellore E. Raghunathan

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In this section, we extend the finite population Bayesian bootstrap methods to a stratified, clustered, unequal probability sample design setting to develop a nonparametric method to generate synthetic populations that adjusts for the complex sampling design features. The idea is to treat the unobserved part of the population as missing data and impute it by making draws from the actual data. We do the imputation in such a fashion that the resulting draws from the posterior distribution of the population will capture the complex design features and can be used in a standard fashion to compute posterior distributions of the population quantities of interest.

4.1  Use the Bayesian bootstrap to adjust for stratification and clustering

For a stratified clustering sampling, we first need to resample clusters within the strata. Denote $c$ as the total number of clusters in the actual data, $c={\displaystyle {\sum }_{h=1}^H{c}_h},$ and $C$ as the number of clusters in the population, $C={\displaystyle {\sum }_{h=1}^H{C}_h}.$ One approach is to first apply FPBB Pólya urn scheme to impute the unobserved clusters within each stratum, $c_1^{*},\dots ,c_{C_h-c_h}^{*},$ which together with the observed clusters provide the clusters in stratum $h$  in the population. However, we typically do not know the number of clusters in a stratum from available public use data. Thus we suggest as an alternative to FPBB sample drawing a standard Bayesian bootstrap sample of the clusters within each stratum. Considering the equivalence between the classical bootstrap and Bayesian bootstrap, we follow Rao and Wu (1988), who suggested drawing a simple random sample with replacement (SRSWR) of $m_h$ from the $c_h$ clusters and within each stratum $h$ calculating replicate weights for computation for each bootstrap sample as

$$w^{\text{*}}{}^{\left(l\right)}=\left\{w_{hik}^{\text{*}\left(l\right)},h=1,\dots ,H,i=1,\dots ,c_h,k=1,\dots ,N_{hi}\right\},$$

where

$$w_{hik}^{\text{*}}=w_{hik}\left(\left(1-\sqrt{\frac{m_h}{c_h-1}}\right)+\sqrt{\frac{m_h}{c_h-1}}\frac{c_h}{m_h}{m}_{hi}^{\text{*}}\right)$$

and $m_{hi}^{*}$ denotes the number of times that cluster $i, i=1, \dots , c_h$ is selected. To ensure all the replicate weights are non-negative, $m_h\le \left(c_h-1\right);$ here and below we take $m_h=\left(c_h-1\right).$

Note that, when clustering is not present, we simply draw a standard Bayesian bootstrap sample from the sampled data within each stratum (when stratification is present) or from the entire sample (if stratification not present, so that $H=1$ ) and calculate the replicate weights as $w_{hik}^{*}=w_{hik}{m}_{hi}^{*}.$

This procedure is repeated $L$ times to produce $L$ Bayesian bootstrap (BB) samples denoted by $S_1,\dots ,S_L.$ This step generates $L$ Bayesian bootstrap samples which essentially are $L$ draws from the posterior predictive distribution of the unobserved clusters given the actual data. However, the units for the $L$ Bayesian bootstrap samples still have weights and cannot be analyzed as simple random samples.

4.2  Use weighted FPBB Pólya urn scheme to adjust for weighting

Once we have $L$ BB samples with replicate weights, the second step imputes the unobserved units using the weighted FPBB Pólya urn scheme. In practice, the probability of selecting the $k^{\text{th}}$ unit, $y_k^{*},$ depends on the selection of the first $k-1$ units, $y_1^{*},\dots ,y_{k-1}^{*}.$  In other words, to determine the probability of selecting a new unit, we have to count the number of times that each unit in the sample has been selected among the previous selections. In settings where the population size is extremely large, we need only generate synthetic populations of size $T*n,$ where $T$ is sufficiently large to overwhelm the sample size (e.g., 20-100). To further computational efficiency, we could also draw a moderate sized population $F>1$ times and then pool these $F$  populations to produce one synthetic population, $S_l.$ The size of $S_l$ then is $F*T*n.$

Note that our method only requires knowledge of the final weights in multistage cluster samples, since all stages of unequal probabilities of sampling will be corrected by use of the weighted FPBB Pólya urn scheme. This is a particularly useful feature of the proposed method, as in many public use datasets the components of the probabilities of selection (e.g., cluster-level selection probabilities, non-response weights) are not available.

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