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Analysts often use Rubin's rule (RR) to obtain a pooled estimate of a popular quantity from multiple (imputed) datasets. While popular statistical software (such as the R survey package or Stata's mi) will apply RR to any set of inputs, this may lead to invalid inferences when the underlying assumptions are violated. For example, RR assumes "congenial" sources of input. Usually this means that the missing data are modeled drawn from some multivariate or joint distribution that also features the analysis model.

In many cases, this assumption is unjustified. For example, multiple imputation with chained equations (MICE) is a noncongenial imputation model with possibly-flexible functions of predictors of missingness. Many machine learning methods also try to predict missingness with added noise but their sampling distributions are unknown and are non-congenial. Most notably, in cases where complex (e.g., stratified, clustered) sampling is employed, clustering errors with the Horwitz-Thompson estimator basically violates all of the underlying assumptions - post-hoc adjustments to covariance matrices do not lend to congeniality. Of course, congeniality is often not enough!

How then might one combine multiple imputations of such data for valid inferences? Specifically, how should one pool estimates from multiply-imputed data with complex sampling designs to ensure consistency (at a minimum) and efficiency/unbiasedness (at best)?

I found Barlett and Hughes (2020) propose some options but am not sure if there is a more analytical result to rely on or if the bootstrap they recommend is valid for complex samples.

Relevant Readings

  • Bartlett JW, Hughes RA. Bootstrap inference for multiple imputation under uncongeniality and misspecification. Statistical Methods in Medical Research. 2020;29(12):3533-3546.

  • Meng, Xiao-Li. “Multiple-Imputation Inferences with Uncongenial Sources of Input.” Statistical Science, vol. 9, no. 4, 1994, pp. 538–58.

  • Jared S. Murray. "Multiple Imputation: A Review of Practical and Theoretical Findings." Statist. Sci. 33 (2) 142 - 159, May 2018.

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