1 Introduction
Jae Kwang Kim and Changbao Wu
Variance estimation is an important practical problem in sample surveys. In addition to analytic use of variances such as testing statistical hypotheses and constructing confidence intervals, variance estimation can also be used to provide descriptive measures on the accuracy of survey estimates and the efficiency of the given sampling design. There are two types of commonly used techniques for variance estimation under the design-based framework. The first is called the linearization method, which uses the standard variance formula applied either directly to the estimator if the parameter is a population total or to the linearized one-step Taylor series expansion of the estimator if the parameter is a nonlinear function of one or several population totals. The second is called the replication method, which constructs variance estimators in a simple systematic way using multiple sets of replication weights along with the original survey data set.
Replication variance estimation techniques have become very popular for design-based inferences using complex survey data. Some early practices using replication weights go back to 1970s at the U.S. Bureau of the Census, Bureau of Labor Statistics and Westat (Dippo, Fay and Morganstein 1984). It is now a routine practice for survey organizations to provide replication weights together with survey data. The most attractive feature of this approach is that it works the same way regardless of the complexity of the parameter. For parameters that are smooth functions of population means or totals, the "linearization� step has been automatically built into the estimation process and computation of partial derivatives involved in the Taylor series expansion is not required. It is extremely user-friendly for multi-purpose data analyses once the survey data set is released together with replication weights. Furthermore, the use of replication methods reduces concerns on confidentiality issues since detailed design information such as stratum or cluster identifier is not released (Lu and Sitter 2008).
Replication weights are typically constructed by the bootstrap, the jackknife or the balanced repeated replication (BRR) methods. Rust and Rao (1996), Shao (1996, 2003) and Wolter (2007) provided excellent overviews on the topic. There are three major issues in the construction of replication weights: validity, efficiency and sparsity. Validity refers to the asymptotic unbiasedness of replication variance estimators under the given sampling design. The asymptotic unbiasedness of an estimator is generally a weaker concept than the estimator being consistent. If the coefficient of variation of the variance estimator goes to zero, then the asymptotically unbiased variance estimator is also consistent. Efficiency is measured by the relative performance of the replication variance estimator to the standard linearization variance estimator which is viewed as fully efficient. Sparsity refers to the number of sets of replication weights required to achieve fully efficient variance estimation.
Validity of replication variance estimators was discussed by Krewski and Rao (1981), Shao and Tu (1995) and Fuller (2009a), among others. Efficiency and stability of replication variance estimators were discussed by Rust and Kalton (1987) and Jang and Eltinge (2009). For sparsity, Kott (2001) considered using delete-a-group jackknife to achieve sparsity under certain designs, and Lu, Brick and Sitter (2006) also discussed combining strata for sparse replication variance estimation.
Most replication methods discussed in the literature are only valid for certain sampling designs. For example, the jackknife method is commonly used for stratified random sampling (Krewski and Rao 1981). The bootstrap method has several popular procedures, including the without-replacement bootstrap method (Gross 1980; McCarthy and Snowden 1985), the re-scaling bootstrap method (Rao and Wu 1988; Preston 2009) and the mirror-match bootstrap method (Sitter 1992). These procedures, however, are only applicable for certain types of sampling designs.
The sparsity of a replication method depends on how the replication weights are constructed. The number of sets of the jackknife replication weights is related to the number of units in the sample and can be very large if the sample size is large. Bootstrap methods typically require at least 1,000 sets of replication weights in order to achieve the desired level of efficiency. As a compromise, most survey organizations provide 500 sets of bootstrap weights alongside the main survey variables. The resulting data sets are still too big for data users to have visual checks and can be very cumbersome to manipulate in practice.
This paper presents methods for constructing efficient and sparse replication weights for variance estimation under the design-based framework. By maintaining full efficiency of the resulting variance estimator for key variables with a smaller number of sets of replication weights, our methods address one of the major tasks at the data file preparation stage and can easily be applied by survey runners to reduce the burden of data users in dealing with excessively large data files. A major limitation of our proposed method is that it does not directly handle situations where design weights are adjusted for nonresponse or calibrated to known auxiliary population information.
In Section 2, we present a general procedure for constructing replication weights based on the method of Fay (1984) and Fay and Dippo (1989), which provides fully efficient replication weights for arbitrary sampling designs. In Section 3, we discuss two strategies, random sampling and calibration weighting, for constructing sparse replication weights. By using a novel application of the calibration technique, our proposed methods allow the use of a small number of sets of replication weights while the resulting replication variance estimators remain efficient. In Section 4, some asymptotic theory for the validity of the replication variance estimator is presented. In Section 5, extensions to some balanced sampling designs are discussed. In Section 6, we report results from a simulation study, using real data from Statistics Canada's Family Expenditure Survey, to evaluate the effectiveness of the proposed strategies for replication variance estimation. Some concluding remarks are given in Section 7.
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