A Roadmap for Disclosure Avoidance in the Survey of Income and Program Participation (2024)

Chapter: Appendix B: Inferences Based on Multiple Synthetic Data

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Suggested Citation: "Appendix B: Inferences Based on Multiple Synthetic Data." National Academies of Sciences, Engineering, and Medicine. 2024. A Roadmap for Disclosure Avoidance in the Survey of Income and Program Participation. Washington, DC: The National Academies Press. doi: 10.17226/27169.
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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 θ ^ ( j ) and the within-set variability of θ ( j ) ^ by w(j) (Liu, 2022). Let W   =   m − 1 ∑ j = 1 m   w ( j ) (average within-set variability) and B   =   ∑ j = 1 m ( θ ^ ( j )   −   θ ^ ) 2 / ( m   −   1 ) (between-set variability).

Fully synthetic data: The final estimate of θ over m synthetic sets is θ ^   =   m − 1 ∑ j = 1 m   θ ^ ( j ) 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 1 2 ( θ ^   −   θ )   ~ N ( 0 , 1 ) .

Partial synthetic data with or without differential privacy (DP): The final estimate of θ over m synthetic sets is θ ^   =   m − 1 ∑ j = 1 m   θ ^ ( j ) and the variance estimator is T = W + m−1B. Hypothesis testing and confidence interval construction are based on the asymptotic assumption of T 1 2 ( θ ^   −   θ )   ~ t v ( 0 , 1 ) , where the degrees of freedom are v = ( m − 1 ) ( 1 + w m − 1 B ) − 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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Suggested Citation: "Appendix B: Inferences Based on Multiple Synthetic Data." National Academies of Sciences, Engineering, and Medicine. 2024. A Roadmap for Disclosure Avoidance in the Survey of Income and Program Participation. Washington, DC: The National Academies Press. doi: 10.17226/27169.
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Page 211 Cite Bookmark
Suggested Citation: "Appendix B: Inferences Based on Multiple Synthetic Data." National Academies of Sciences, Engineering, and Medicine. 2024. A Roadmap for Disclosure Avoidance in the Survey of Income and Program Participation. Washington, DC: The National Academies Press. doi: 10.17226/27169.
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Page 212 Cite Bookmark
Suggested Citation: "Appendix B: Inferences Based on Multiple Synthetic Data." National Academies of Sciences, Engineering, and Medicine. 2024. A Roadmap for Disclosure Avoidance in the Survey of Income and Program Participation. Washington, DC: The National Academies Press. doi: 10.17226/27169.
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Page 212
Next Chapter: Appendix C: Technical Details for Differential Privacy Table Builder
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