5. Inference from multiple nonparametric synthetic populations

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

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Assume we generate $L$ synthetic populations, $S_l, l=1,\dots ,L$ using the nonparametric method described in Section 4, and that our inferential target is $Q\equiv Q\left(Y\right),$ a function of the population data (e.g., population mean, correlation, population maximum likelihood estimator of a regression parameter, etc.). We can compute $Q_l$ as the estimate of $Q$ obtained from pooling the $F$ synthetic populations that impute the unobserved units of $S_l;$ since these are direct draws from the posterior predictive distribution of the population, we can compute posterior means, quantiles, and credible intervals from the corresponding empirical estimates from the draws, if $L$ is sufficiently large.

However, in many settings, the computational effort required to impute the population may be very large, even if the full population is not required to be synthesized. Hence an alternative approach for inference is to approximate the posterior predictive distribution of a scalar population statistic $Q$ via a $t­$ distribution:

$$Q|S_1,\dots ,S_L\stackrel{·}{~}{t}_{L-1}\left({\overline{Q}}_L,\left(1+L^{-1}\right)V_L\right)$$

where

$${\overline{Q}}_L=\frac{{{\displaystyle \sum }}_{l=1}^L{Q}_l}{L}=\frac{{{\displaystyle \sum }}_{l=1}^L{{\displaystyle \sum }}_{f=1}^F{Q}_{lf}}{LF}\text{ and }{V}_L=\frac{1}{L}{\displaystyle \sum }_{l=1}^L{\left(Q_l-{\overline{Q}}_L\right)}^2.$$

The result follows immediately from Section 4.1 of Raghunathan et al. 2003, and is based on the standard Rubin (1987) multiple imputation combining rules, treating the unobserved units of $S_l$ as missing data and the sampled units as observed data. The average "within� imputation variance is zero, since the entire population is being synthesized; hence the posterior variance of $Q$  is entirely a function of the between-imputation variance, and the degrees of freedom is simply given by the number of FPBB samples. (When the population is extremely large, we need only synthesize a draw sufficiently large for average "within� imputation variance to be trivial relative to the between imputation variance $V_L.$ ) The result assumes that $E\left(Q_{lf}\right)=Q$  - a result guaranteed by our weighted FPBB estimator - as well as a a sufficiently large sample size for Bayesian asymptotics to apply.

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