The all-cases imputation variance estimator: A new approach to variance estimation for imputed data

For the past several decades, imputation methods have been used to compensate for item nonresponse. Traditionally, the imputed values have been treated as if they had actually been observed or reported, and variance estimates have been computed using standard complete data methods. This approach leads to underestimation of the variance of the estimator. Even for items with relatively low nonresponse, this downward bias in the variance estimates may be significant. Here, we propose a model-assisted method for accounting for imputation error in variance estimates for imputed data. Our proposed method, which we call "all-cases imputation (or ACI)," imputes a value of the characteristic of interest for all cases, including those with actual (observed or reported) values. The difference between the imputed value and the actual value (imputation error) for respondents is used to estimate the imputation error variance and covariance for nonrespondents. We present the all-cases imputation variance estimator for simple random sampling and for stratified random sampling and describe the approach for the extension to stratified cluster sampling. We demonstrate, both analytically and empirically, the properties of our proposed variance estimator. A Monte Carlo simulation study compares the proposed variance estimator to the naive variance estimator, the Rao-Shao jackknife, and the Shao bootstrap, under simple random sampling. A second Monte Carlo simulation study examines the properties of the ACI variance estimator for stratified random sampling. In order to make valid inferences, it is important to ascertain the distribution of the ACI variance estimator. We examine analytically the exact distribution, and then consider an approximation. A simulation study verifies that the approximation provides a good fit to the empirical distribution.