Most of you know that the parameters of the imputation model are usually drawn from a Bayesian posterior distribution. That's necessary to ensure that Rubin's within-between (WB) formula will consistently estimate the standard error of a multiple imputation point estimate. Unfortunately, getting posterior draws can substantially increase runtime and computational complexity.
What if you don't take posterior draws? What if you just estimate the parameters of the imputation model using maximum likelihood (or something equally efficient). Everything will run faster, with less coding, but Rubin's WB formula will underestimate the standard errors. Is there another way to estimate the standard errors? In a new article for *Statistical Science* <https://urldefense.com/v3/__https://www.e-publications.org/ims/submission/STS/user/submissionFile/30892?confirm=5d63f917__;!!Dq0X2DkFhyF93HkjWTBQKhk!Dd1B2VYgozF-ixE63Ez1UAHi8fHP9vW0p9FvFbd-UhSp_Cb6hYSURd7OpiWZi2eamIBTRhSwIrGvcg$ >, Jonathan Bartlett and I derive three standard error estimators that work when the imputation parameters are estimated by maximum likelihood. One standard error estimator is a WB formula that combines the within and between variances in a different way than Rubin. One is a decomposition of the score function inspired by Robins and Wang. And one -- probably the most useful -- is a new application of the bootstrap to calculate variance components due to sampling and imputation. Jonathan implemented the methods in R, and I just finished implementing them in Stata. Theoretically, it's the deepest statistical article I've ever written, and I hope you learn as much from reading it as we did from writing it. It does not appear to be paywalled. Have a look! von Hippel, P. T., & Bartlett, J. (2020). Maximum likelihood multiple imputation: Faster imputations and consistent standard errors without posterior draws <https://urldefense.com/v3/__https://www.e-publications.org/ims/submission/STS/user/submissionFile/30892?confirm=5d63f917__;!!Dq0X2DkFhyF93HkjWTBQKhk!Dd1B2VYgozF-ixE63Ez1UAHi8fHP9vW0p9FvFbd-UhSp_Cb6hYSURd7OpiWZi2eamIBTRhSwIrGvcg$ >. *Statistical Science*. -- Best wishes, Paul von Hippel Associate Professor, LBJ School of Public Affairs University of Texas, Austin PaulvonHippel.com <https://urldefense.com/v3/__http://paulvonhippel.com__;!!Dq0X2DkFhyF93HkjWTBQKhk!Dd1B2VYgozF-ixE63Ez1UAHi8fHP9vW0p9FvFbd-UhSp_Cb6hYSURd7OpiWZi2eamIBTRhQcARMIrA$ > @PaulvonHippel <https://urldefense.com/v3/__https://twitter.com/paulvonhippel__;!!Dq0X2DkFhyF93HkjWTBQKhk!Dd1B2VYgozF-ixE63Ez1UAHi8fHP9vW0p9FvFbd-UhSp_Cb6hYSURd7OpiWZi2eamIBTRhSMIUHkzw$ >
