Multiple imputation approaches for handling incomplete three‐level data with time‐varying cluster‐memberships

Three‐level data arising from repeated measures on individuals clustered within higher‐level units are common in medical research. A complexity arises when individuals change clusters over time, resulting in a cross‐classified data structure. Missing values in these studies are commonly handled via multiple imputation (MI). If the three‐level, cross‐classified structure is modeled in the analysis, it also needs to be accommodated in the imputation model to ensure valid results. While incomplete three‐level data can be handled using various approaches within MI, the performance of these in the cross‐classified data setting remains unclear. We conducted simulations under a range of scenarios to compare these approaches in the context of an acute‐effects cross‐classified random effects substantive model, which models the time‐varying cluster membership via simple additive random effects. The simulation study was based on a case study in a longitudinal cohort of students clustered within schools. We evaluated methods that ignore the time‐varying cluster memberships by taking the first or most common cluster for each individual; pragmatic extensions of single‐ and two‐level MI approaches within the joint modeling (JM) and the fully conditional specification (FCS) frameworks, using dummy indicators (DI) and/or imputing repeated measures in wide format to account for the cross‐classified structure; and a three‐level FCS MI approach developed specifically for cross‐classified data. Results indicated that the FCS implementations performed well in terms of bias and precision while JM approaches performed poorly. Under both frameworks approaches using the DI extension should be used with caution in the presence of sparse data.