Multiple imputation of interaction terms and nonlinear associations in multilevel models when model assumptions are violated

Abstract

Unfortunately it is quite usual for missing data to occur in several variables. In multilevel models, for example, response variables as well as level-1, level-2, or even higher-level predictors may be missing. Multiple imputation (MI) is one state-of-the-art strategy to address missing data in multilevel data sets. Two popular approaches to impute interaction terms and non-linear associations (obtained from incomplete predictors) are passive imputation (PI) or to include the respective terms as just another variable (JAV) into the imputation model. Seaman, Bartlett, and White (2012) have shown that both approaches are suboptimal: JAV seems to work only under missing completely at random mechanisms (Rubin, 1976), while PI leads to a misspecified imputation model and biased results in many scenarios. Bartlett, Seaman, White, and Carpenter (2015) have proposed a solution to impute interaction terms and nonlinear associations under the correct model based on the popular chained equations MI framework (MICE). However, an extension of their proposed approach to multilevel data sets is not yet available. Goldstein, Carpenter, and Browne (2014) on the other hand have proposed a Bayesian joint modelling MI method that also works for multilevel data. The purpose of the paper is to discuss and evaluate the quality and practical applicability of the currently availabe solutions (including available software packages) to impute interaction terms and nonlinear associations in multilevel models. By means of Monte Carlo Simulation, I show that the newer approaches work reasonably well when all model assumptions are more or less met. Unfortunately, results can get quite heavily biased, when model assumptions are violated. Future research should therefore also focus on methods that are either robust against violations of model assumptions like multivariate normality or homoscedasticity or that allow more modelling flexibility.