Balanced Repeated Replication for Stratified Multistage Survey Data under Imputation

Balanced repeated replication (BRR) is a popular method for variance estimation in surveys. The standard BRR method works by rst creating a set of \balanced" pseudo-replicated data sets from the original data set. For a survey estimator ^ , the BRR variance estimator is the average of squared deviations ^ (r) ? ^ , where ^ (r) is the same as ^ but based on the data in the rth pseudo-replicated data set only. When there are a large number of imputed missing values (nonrespondents), however, treating the imputed values as observed data and applying the standard BRR variance estimation formula does not produce valid variance estimators. Intuitively, the variation due to imputation can be captured by the BRR method if every pseudo-replicated data set is imputed in exactly the same way as the original data set is imputed (assuming that the data set contains ags for nonrespondents). When a random imputation method (such as random hot deck imputation, random ratio imputation, or random regression imputation) is used, however, imputing every pseudo-replicated data set requires the generation of many random variates and is computationally expensive. We propose an adjusted BRR variance estimator which is exactly the same as the BRR variance estimator obtained by imputing every pseudo-replicated data set when a deterministic imputation method (e.g., ratio or regression imputation) is used. For random imputation methods, the proposed adjustment does not require the generation of additional random variates, but it still captures the variation due to imputation. Under a general stratiied multistage sampling design, consistency of the adjusted BRR variance estimators for functions of estimated totals (smooth statistics) or for sample quantiles (nonsmooth statistics) is established. A simulation study shows that the adjusted BRR method works well and is much better than the unadjusted BRR method. An example with real data is also presented. The authors wish to thank two referees and an associate editor for their very helpful comments and suggestions. The authors also would like to thank Graham Kalton and Mary Nixon from Westat for providing a real-data example.