Balanced repeated replication (BRR) is a method to estimate the sampling variability of a statistic that takes into account the properties of the sample design. It provides unbiased estimates of the sampling error arising from complex sample selection procedures. The estimates capture the effects of stratification, clustering, and unequal probabilities of selection.
BRR works by repeatedly re-estimating the target statistics using half the sample at a time. When each sampling stratum contains two PSUs, replicates are formed by including one of the two PSUs from a stratum. If the sample includes H strata, such replicates can be formed. The variability of the target statistics across these replicates offers an estimate of the sampling variance of the estimates.
Fay's method is a variant of Balanced Repeated Replication in which less extreme adjustments are made to the weights for each replicate. For example, the "included" PSUs may have their weights increased by 50%, and the excluded PSUs would have their weights decreased by a corresponding amount.
The BRR procedure consists of three steps:
- Forming balanced half-samples from the full sample ;
- Constructing the replicate weights to be used in calculating the estimate of the parameter of interest for the subsamples; and
- Computing the estimates of variance for the parameter of interest.
1. Forming Random Groups
The random groups comprise all possible balanced half-samples of PSUs. The sample design must include two PSUs per stratum (although many actual samples using this method form psuedo strata including psuedo-PSUs that meet this requirement). Each half sample is formed by deleting one PSU from each stratum. All possible combinations of such half-samples comprise the set of replicates.
2. Constructing Replicate Weights
Typically, BRR is implemented using replicate weights. Weights are set to zero for elements in excluded PSUs, and corresponding adjustments made to the weights of elements in the remaining PSUs. One set of such weights is constructed for each replicate.
Fay's method uses a milder adjustment to each weight. In some instances this approach improves the estimates of statistics such as medians and percentiles.
3. Computing Estimates of Variance
Assume that A such groups have been constructed. Then, for each group (a = 1,... ,A), the target statistics ô(a) is calculated based on data from half-sample a.The point estimate for the statistic may be estimated as the average of these half-sample estimates, or based on the single overall sample. The variance estimate is given by V = (1/A)S(ôa-ô)2 where the summation is done over a = 1 to A.
When using Fay's method the formula becomes V =[1/(A(1-k)2 )]S(ôa-ô)2
McCarthy, P. J. (1969) Psuedo-replication: half samples. Review of the International Statistical Institute. 37:239-264.
Särndal, C. E., Swensson, B., & Wretman, J. (1992). Model Assisted Survey Sampling. New York: Springer-Verlag.