How should variable selection be performed with multiply imputed data?

Multiple imputation is a popular technique for analysing incomplete data. Given the imputed data and a particular model, Rubin's rules (RR) for estimating parameters and standard errors are well established. However, there are currently no guidelines for variable selection in multiply imputed data sets. The usual practice is to perform variable selection amongst the complete cases, a simple but inefficient and potentially biased procedure. Alternatively, variable selection can be performed by repeated use of RR, which is more computationally demanding. An approximation can be obtained by a simple 'stacked' method that combines the multiply imputed data sets into one and uses a weighting scheme to account for the fraction of missing data in each covariate. We compare these and other approaches using simulations based around a trial in community psychiatry. Most methods improve on the naïve complete-case analysis for variable selection, but importantly the type 1 error is only preserved if selection is based on RR, which is our recommended approach.