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X.-L. Meng 547subsequent analysts can then apply their favorite complete-data analysis proceduresto reach valid inferences. This same separation creates the issue ofuncongeniality. The consequences of uncongeniality can be severe, from boththeoretical and practical points of view. Perhaps the most striking exampleis that the very appealing variance combining rule for MI inference derivedunder congeniality (and another application of the aforementioned EVE law),namely,var Total = var Between−imputation + var Within−imputation (45.4)can lead to seriously invalid results in the presence of uncongeniality, as reportedinitially by Fay (1992) and Kott (1995).Specifically, the so-called Rubin’s variance combining rule is based on(45.4), wherevar Between−imputation and var Within−imputationare estimated by (1 + m −1 )B m and Ūm, respectively (Rubin, 1987). Here the(1 + m −1 ) factor accounts for the Monte Carlo error due to finite m, B m isthe sampling variance of ˆθ (l) ≡ ˆθ A (Y com) (l) and Ūm is the sample average ofU(Y com),l=1, (l) . . . , m, where ˆθ A (Y com )istheanalyst’scomplete-dataestimatorfor θ, U(Y com ) is its associated variance (estimator), and Y (l)misare i.i.d.draws from an imputation model P I (Y mis |Y obs ). Here, for notational convenience,we assume the complete data Y com can be decomposed into the missingdata Y mis and observed data Y obs . The left-hand side of (45.4) then is meantto be an estimator, denoted by T m , of the variance of the MI estimator of θ,i.e., ¯θ m ,theaverageof{ˆθ (l) ,l=1,...,m}.To understand the behavior of ¯θ m and T m ,letusconsiderarelativelysimple case where the missing data are missing at random (Rubin, 1976), andthe imputer does not have any additional data. Yet the imputer has adopteda Bayesian model uncongenial to the analyst’s complete-data likelihood function,P A (Y com |θ), even though both contain the true data-generating modelas a special case. For example, the analyst may have correctly assumed thattwo subpopulations share the same mean, an assumption that is not in theimputation model; see Meng (1994) and Xie and Meng (2013). Furthermore,we assume the analyst’s complete-data procedure is the fully efficient MLEˆθ A (Y com ), and U A (Y com ), say, is the usual inverse of Fisher information.Clearly we need to take into account both the sampling variability andimputation uncertainty, and for consistency we need to take both imputationsize m →∞and data size n →∞.Thatis,weneedtoconsiderreplicationsgenerated by the hybrid model (note P I (Y mis |Y obs ) is free of θ):P H (Y mis ,Y obs |θ) =P I (Y mis |Y obs )P A (Y obs |θ), (45.5)where P A (Y obs |θ) is derived from the analyst’s complete-data modelP A (Y com |θ).