Variance Estimation for Imputed Survey Data with Non-negligible Sampling Fractions

We consider variance estimation for Horvitz-Thompson type estimated totals based on survey data with imputed nonrespondents and with non-negligible sampling fractions. A method based on a variance decomposition is proposed. Our method can be applied to complicated situations where a composite of some deterministic and/or random imputation methods is used, including using imputed data to impute. We mainly adopt the linearization or Taylor expansion type techniques, but replication methods, such as the jackknife, balanced repeated replication, and random groups, can also be used in applying our method to derive variance estimators. Using our method, variance estimators can be derived under either the customary design-based approach or the model-based approach, and are asymptotically unbiased and consistent. The Transportation Annual Survey conducted at the U.S. Census Bureau, in which nonrespondents are imputed using a composite of cold deck and ratio type imputation methods, is used as an example as well as the motivation for our study.