Application of Auxiliary Variables in Two-Step Semi-Parametric Multiple Imputation Procedure in Estimation of Population Mean

Multiple imputation procedure is used in handling of item non-response. The imputation procedure is affected by model misspecification and leads to loss in efficiency and biased results. The inclusion of auxiliary variables in the sampling design helps to avoid sensitivity of inference to model misspecification and improves the precision of estimate of population mean. The main aim of this study is to incorporate auxiliary variables in the multiple imputations to improve the accuracy of the values imputed and the efficiency of point estimators. The two-step semi-parametric multiple imputation procedure is considered and modified to incorporate the auxiliary variables. The two-step semi-parametric multiple imputation procedure accounts for unequal probabilities of selection and reduces misspecification in the imputation model. In the first step a non-parametric model is used to generate a posterior predictive model that includes both item level missingness and auxiliary information. The size variables in a sample are replicated using a constrained Bayesian Bootstrap. A constrained Weighted Finite Population Bayesian Bootstrap is then used to create a population of size variables which is considered to be the value of an auxiliary variable that is closely associated with the survey variable. The imputed size variables are then used in a linear regression model to predict the survey variables for the synthetic population. A parametric model is used to impute the missing data on the survey variables in the second step. A simulation study was conducted using single stage probability-proportionate-to-size without replacement sampling design to compare the asymptotic properties of the estimator of the population mean to those obtained using the existing two-step semi-parametric multiple imputation procedure. The proposed procedure reduced bias and resulted in gain in efficiency. The 95 % confidence interval coverage rates of the proposed estimator are close to nominal level when the sample size is small.