# 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 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, 1989); Judkins (1990).

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

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

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

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