Meng’s concept of congeniality in multiple imputation (MI) is I think a tricky one (for me anyway!). Loosely speaking congeniality is about whether the imputation and analysis models make different assumptions about the data. Meng gave a definition in his 1994 paper, but I prefer the one given in a more recent paper by Xie and Meng, which is what I and Rachael Hughes used in our paper this year on different methods of combining bootstrapping with MI. In words (see the papers for the same in equations) it is that there exists a Bayesian model for the data such that:
- given complete/full data, the posterior mean of the parameter of interest matches the point estimate given by fitting our analysis model of interest to that data, and the posterior variance matches the variance estimator calculated by our analysis model fit.
- the conditional distribution of the missing data given the observed in this Bayesian model matches that used by our imputation model.
If they are congenial and the models are correctly specified, Rubin’s variance estimator is (asymptotically) unbiased for the true repeated sampling variance of the MI point estimator(s).
One of the potentially useful features of MI are that we can include variables in the imputation stage which we then don’t use in the analysis model. Including such auxiliary variables in the imputation model can increase the likelihood that the MAR assumption holds when the auxiliary variable is associated with the probability of missingness, and can increase efficiency according to how strongly it is correlated with the variable(s) being imputed. A nice paper (among many) on the potential of including auxiliary variables in MI is Hardt et al 2012. In this post, I’ll consider whether including auxiliary variables in the imputation model leads to uncongeniality. The post was prompted following a discussion earlier in the year with my colleague Paul von Hippel.