The Effect of Multiple Weighting Steps on Variance Estimation

1. Introduction Multiple steps in weighting are common in survey estimation. Each step usually introduces a source of variability in an estimator that may be important to reflect when estimating variances. A typical sequence of weighting steps in a probability sample is this: 1. Compute base weights. 2. Adjust weights to account for units with unknown eligibility. 3. Adjust weights for nonresponse. 4. Use auxiliary data. The variance of an estimator is affected by the population structure of the variables being estimated, the complexity of the design used to collect data, and the form of the estimator itself, including the weighting steps above. Intuition may lead us to believe that a variance estimator that somehow incorporates all of these complications is better than one that does not. However, literature that directly addresses this question is limited. The collection on survey nonresponse by Groves, Dillman, Eltinge, and Little (2002), for example, does not include any articles on the effect on variance estimates of multiple steps in weighting. The two major competitors in finite population variance estimation are replication and linearization. For replication variance estimators there is evidence in particular cases that it is necessary to repeat each step of estimation separately for each replicate subsample in order to produce a consistent or approximately unbiased variance estimate. Empirical results, however, are not uniform. Lemeshow (1979) is an early paper illustrating by simulation that this is necessary for the BRR method. Valliant (1993) showed theoretically and empirically that poststratification factors must be recomputed for every replicate in order for the BRR or jackknife estimators to be consistent in two-stage sampling. Yung and Rao (1996) obtained similar results for the jackknife in stratified, multistage sampling. Yung and Rao (2000) studied the poststratified estimator when weighting class nonresponse adjustments were made. They proved that the jackknife is consistent if the nonresponse adjustment factors and the poststratification factors are recomputed for each replicate subsample. There are a number of articles that cover some, but not all, of the four steps when applying Taylor series variance estimators. Lundström and Särndal (1999) study the use of a linearization estimator for the general regression (GREG) estimator when there is nonresponse. Rao (1996) derived a modified linearization variance estimator that accounted for mean imputation. Shortcut implementations of linearization estimators, that ignore some steps in weighting, are fairly common in practice for at least two reasons. First, linearizing complex estimators is …