Balanced Repeated Replication (BRR)

Balanced repeated replication (BRR) is a technique for computing standard errors of survey estimates. It is a special form of the replicate weights technique. The basic form of BRR is for a stratified sample with two primary sampling units (PSUs) sampled with replacement in each stratum, although variations have been constructed for some other sample designs. BRR is attractive because it requires slightly less computational effect than the jackknife method for constructing replicate weights and it is valid for a wider range of statistics. In particular, BRR standard errors are valid for the median and other quantiles, whereas the jackknife method can give invalid results.

A sample with two PSUs in each stratum can be split into halves consisting of one PSU from each stratum. The PSU that is excluded from a half-sample is given weight zero, and the PSU that is included is given weight equal to 2 times its sampling weight. Under sampling with replacement or sampling from an infinite population, these two halves are independent stratified samples. Computing a statistic on each half and taking the square of the difference gives an unbiased estimate of the variance of the statistic. Averaging this estimate over many possible ways of choosing one PSU from each stratum gives a more precise estimate of the variance.

If the sample has L strata there are 2L ways to take one PSU from each stratum, but this would be computationally prohibitive even for moderately large L. The same estimate of the variance of a population mean or population total can be obtained from a much smaller set of “splittings” as long as the following conditions are satisfied:

• 1.
  Each PSU is in the first half in exactly 50% of the splittings.
• 2.
  Any pair of PSUs from different strata is in the same half in exactly 50% of the splittings.

A set of replicates constructed in this way is said to be in full orthogonal balance. It is clearly necessary for these conditions that the number of splittings, R, is a multiple of 4.

An important open question in coding theory, the Hadamard conjecture, implies that a suitable set of splittings is possible whenever R is a multiple of 4 that is larger than L. Although the Hadamard conjecture is unproven, sets of replicates with full orthogonal balance are known for all values of R that are likely to be of interest in survey statistics. The construction is especially simple when R is a power of 2, which results in at most twice as many replicates as necessary.

All sets of replicates with full orthogonal balance give the same standard errors as the full set of 2Lreplicates for the estimated population mean or population total, and thus it does not matter which set is chosen. For a statistic other than the mean or total, on the other hand, different sets of replicates in full orthogonal balance will typically not give exactly the same standard error. The difference is usually small, and analyses often do not report how the set of replicates was constructed.

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