Multiple imputation for the analysis of incomplete compound variables

In many settings interest lies in modelling a compound variable defined as a function of two or more component variables. When one or more of the components are missing, the compound variable is not observed and a strategy for handling incomplete data is required. Analyses based on individuals with complete data are inefficient and yield potentially inconsistent estimators. We develop a multiple imputation strategy in this setting with an auxiliary model for imputing the compound variable directly, and one based on a multivariate imputation model for the component variables. Asymptotic properties of the imputation-based estimators are presented for the case in which the imputation model is correctly specified, and a shrinkage estimator is proposed to reduce the bias arising from misspecification of the imputation model. Finite sample properties of the various estimators are examined through simulations. An application to data from the Canadian Youth Smoking Survey involving a study of body mass index illustrates the approach.