The SURVEYMEANS Procedure

Fay’s BRR Method

Fay’s method is a modification of the BRR method, and it requires a stratified sample design with two primary sampling units (PSUs) per stratum. The total number of replicates R is the smallest multiple of 4 that is greater than the total number of strata H. However, if you prefer a larger number of replicates, you can specify the REPS= method-option.

For each replicate, Fay’s method uses a Fay coefficient $0\le \epsilon <1$ to impose a perturbation of the original weights in the full sample that is gentler than using only half-samples, as in the traditional BRR method. The Fay coefficient $0\le \epsilon <1$ can be set by specifying the FAY = $\epsilon$ method-option. By default, $\epsilon =0.5$ if the FAY method-option is specified without providing a value for $\epsilon$ (Judkins, 1990; Rao and Shao, 1999). When $\epsilon =0$ , Fay’s method becomes the traditional BRR method. For more details, see Dippo, Fay, and Morganstein (1984); Fay (1984, 1989); Judkins (1990).

Let H be the number of strata. Replicates are constructed by using the first H columns of the $R\times R$ Hadamard matrix, where R is the number of replicates, $R>H$ . The rth ( $r=1, 2, ..., R$ ) replicate is created from the full sample according to the rth row of the Hadamard matrix as follows:

If the $(r,h)$ element of the Hadamard matrix is 1, then the full sample weight of the first PSU in stratum h is multiplied by $\epsilon$ and the full sample weight of the second PSU is multiplied by $2-\epsilon$ to obtain the rth replicate weights.
If the $(r,h)$ element of the Hadamard matrix is –1, then the full sample weight of the first PSU in stratum h is multiplied by $2-\epsilon$ and the full sample weight of the second PSU is multiplied by $\epsilon$ to obtain the rth replicate weights.

You can use the VARMETHOD=BRR(OUTWEIGHTS=) method-option to save the replicate weights into a SAS data set.

By default, an appropriate Hadamard matrix is generated automatically to create the replicates. You can request that the Hadamard matrix be displayed by specifying the VARMETHOD=BRR(PRINTH) method-option. If you provide a Hadamard matrix by specifying the VARMETHOD=BRR(HADAMARD=) method-option, then the replicates are generated according to the provided Hadamard matrix.

Suppose that $\theta$ is a population parameter of interest. Let $\hat{\theta }$ be the estimate from the full sample for $\theta$ . Let $\hat{\theta _ r}$ be the estimate from the rth replicate subsample by using replicate weights. PROC SURVEYMEANS estimates the variance of $\hat{\theta }$ by

$\widehat{V}(\hat{\theta }) = \frac{1}{R{(1-\epsilon )}^2} \sum _{r=1}^ R \left( \hat{\theta _ r} - \hat{\theta } \right)^2$

with H degrees of freedom, where H is the number of strata.