Multiple Imputation With Survey Weights: A Multilevel Approach

Multiple imputation is now well established as a practical and flexible method for analyzing partially observed data, particularly under the missing at random assumption. However, when the substantive model is a weighted analysis, there is concern about the empirical performance of Rubin's rules and also about how to appropriately incorporate possible interaction between the weights and the distribution of the study variables. One approach that has been suggested is to include the weights in the imputation model, potentially also allowing for interactions with the other variables. We show that the theoretical criterion justifying this approach can be approximately satisfied if we stratify the weights to define level-two units in our data set and include random intercepts in the imputation model. Further, if we let the covariance matrix of the variables have a random distribution across the level-two units, we also allow imputation to reflect any interaction between weight strata and the distribution of the variables. We evaluate our proposal in a number of simulation scenarios, showing it has promising performance both in terms of coverage levels of the model parameters and bias of the associated Rubin's variance estimates. We illustrate its application to a weighted analysis of factors predicting reception-year readiness in children in the UK Millennium Cohort Study.