Model selection and model averaging after multiple imputation

Model selection and model averaging are two important techniques to obtain practical and useful models in applied research. However, it is now well-known that many complex issues arise, especially in the context of model selection, when the stochastic nature of the selection process is ignored and estimates, standard errors, and confidence intervals are calculated as if the selected model was known a priori. While model averaging aims to incorporate the uncertainty associated with the model selection process by combining estimates over a set of models, there is still some debate over appropriate interpretation and confidence interval construction. These problems become even more complex in the presence of missing data and it is currently not entirely clear how to proceed. To deal with such situations, a framework for model selection and model averaging in the context of missing data is proposed. The focus lies on multiple imputation as a strategy to deal with the missingness: a consequent combination with model averaging aims to incorporate both the uncertainty associated with the model selection and with the imputation process. Furthermore, the performance of bootstrapping as a flexible extension to our framework is evaluated. Monte Carlo simulations are used to reveal the nature of the proposed estimators in the context of the linear regression model. The practical implications of our approach are illustrated by means of a recent survival study on sputum culture conversion in pulmonary tuberculosis.