Appendix B

Inferences Based on Multiple Synthetic Data

Denote the number of synthetic datasets by m, the estimate of the parameter of interest θ is $\stackrel{^\left(j\right)}{\theta }$ and the within-set variability of $\widehat{{\theta }_{\left(j\right)}}$ by w⁽j) (Liu, 2022). Let $\text{W}\text{=}{m}^{-1}{\displaystyle {\sum }_{j=1}^m{w}^{\left(j\right)}}$ (average within-set variability) and $\text{B}\text{=}{{\displaystyle {\sum }_{j=1}^m\left(\stackrel{^(j)}{\theta }-\hat{\theta }\right)}}^2/\left(m-1\right)$ (between-set variability).

Fully synthetic data: The final estimate of θ over m synthetic sets is $\hat{\theta }=m^{-1}{\displaystyle {\sum }_{j=1}^m{\hat{\theta }}^{\left(j\right)}}$ and its estimated variability is given by T = (1 + m⁻¹) B − W. Hypothesis testing and confidence interval construction are based on the asymptotic assumption of $T\frac{1}{2}\left(\hat{\theta }-\theta \right)~N\left(0,1\right)$.

Partial synthetic data with or without differential privacy (DP): The final estimate of θ over m synthetic sets is $\hat{\theta }=m^{-1}{\displaystyle {\sum }_{j=1}^m{\hat{\theta }}^{\left(j\right)}}$ and the variance estimator is T = W + m⁻¹B. Hypothesis testing and confidence interval construction are based on the asymptotic assumption of $T\frac{1}{2}\left(\hat{\theta }-\theta \right)~t_v\left(0,1\right)$, where the degrees of freedom are $v=\left(m-1\right){\left(1+\frac{w}{m^{-1}B}\right)}^{-1}$. Though the inferential approaches based on multiple synthetic datasets are the same with or without DP, what is captured in the between-set variance component B is different between the two. For DP data synthesis, B has the extra variability in the synthetic data due to the employment of randomized mechanisms for achieving DP guarantees, compared to the case without DP.

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