### 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 is the smallest multiple of 4 that is greater than the total number of strata . 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 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 can be set by specifying the FAY = method-option. By default, if the FAY method-option is specified without providing a value for (Judkins 1990; Rao and Shao 1999). When , Fay’s method becomes the traditional BRR method. For more details, see Dippo, Fay, and Morganstein (1984), Fay (1984), Fay (1989), and Judkins (1990).

Let be the number of strata. Replicates are constructed by using the first columns of the Hadamard matrix, where is the number of replicates, . The th () replicate is created from the full sample according to the th row of the Hadamard matrix as follows:

• If the th element of the Hadamard matrix is 1, then the full sample weight of the first PSU in stratum is multiplied by and the full sample weight of the second PSU is multiplied by to obtain the th replicate weights.

• If the th element of the Hadamard matrix is –1, then the full sample weight of the first PSU in stratum is multiplied by and the full sample weight of the second PSU is multiplied by to obtain the th 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 is a population parameter of interest. Let be the estimate from the full sample for . Let be the estimate from the th replicate subsample by using replicate weights. PROC SURVEYMEANS  estimates the variance of by

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