Bruch, C. Applying the rescaling bootstrap under imputation for a multistage sampling design. Computational Statistics 37, 1461–1494 (2022). https://doi.org/10.1007/s00180-021-01164-6
This paper proposes a method for variance estimation that accounts for the use of imputed values in population estimation when using a multistage sampling design. Failure to account for the imputation process and sampling design in variance estimation and standard error estimation, respectively, can be associated with large biases. The proposed method is based on the rescaling bootstrap for multistage sample designs developed by Preston (Surv Methodol 35(2):227-234, 2009). In its original version, this method assumes that the dataset contains only complete cases and no missing values. We propose two modifications to this method in the presence of nonresponse and the use of imputation. These modifications are compared to other modifications of the rescaling bootstrap in a Monte Carlo simulation study. The results of the simulation study show that, unlike other modifications of the rescaling bootstrap, the two proposed methods provide good estimates of the variance of the population estimator under imputation in multistage sample designs in many situations.