Replication Methods for Variance Estimation :: SAS/STAT(R) 13.2 User's Guide

Balanced Repeated Replication (BRR) Method

The balanced repeated replication (BRR) method requires that the full sample be drawn by using a stratified sample design with two primary sampling units (PSUs) per stratum. Let H be the total number of strata. The total number of replicates R is the smallest multiple of 4 that is greater than H. However, if you prefer a larger number of replicates, you can specify the REPS=number option. If a number $\times$ number Hadamard matrix cannot be constructed, the number of replicates is increased until a Hadamard matrix becomes available.

Each replicate is obtained by deleting one PSU per stratum according to the corresponding Hadamard matrix and adjusting the original weights for the remaining PSUs. The new weights are called replicate weights.

Replicates are constructed by using the first H columns of the $R\times R$ Hadamard matrix . The rth ( $r=1, 2, ..., R$ ) replicate is drawn 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 first PSU of stratum h is included in the rth replicate and the second PSU of stratum h is excluded.
If the $(r,h)$ element of the Hadamard matrix is –1, then the second PSU of stratum h is included in the rth replicate and the first PSU of stratum h is excluded.

Note that the "first" and "second" PSUs are determined by data order in the input data set. Thus, if you reorder the data set and perform the same analysis by using BRR method, you might get slightly different results, because the contents in each replicate sample might change.

The replicate weights of the remaining PSUs in each half-sample are then doubled to their original weights. For more details about the BRR method, see Wolter (2007) and Lohr (2010).

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.

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

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

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

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

If a parameter cannot be computed from one or more replicates, then the variance estimate is computed by using those replicates from which the parameter can be estimated. For example, suppose the parameter is a ratio. If a replicate r contains observations such that the denominator of the ratio is zero, then the ratio cannot be computed from replicate r. In this case, the BRR variance estimate is computed as

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

where the summation is over the replicates where the parameter $\theta$ can be computed, and $R’$ is the number of those replicates.

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.

Jackknife Method

The jackknife method of variance estimation deletes one PSU at a time from the full sample to create replicates. The total number of replicates R is the same as the total number of PSUs. In each replicate, the sample weights of the remaining PSUs are modified by the jackknife coefficient $\alpha _ r$ . The modified weights are called replicate weights.

The jackknife coefficient and replicate weights are described as follows.

Without Stratification

If there is no stratification in the sample design (no STRATA statement), the jackknife coefficients $\alpha _ r$ are the same for all replicates:

$\alpha _ r=\frac{R-1}{R} \, \, \, \, \, \mbox{where } r=1, 2, ..., R$

Denote the original weight in the full sample for the jth member of the ith PSU as $w_{ij}$ . If the ith PSU is included in the rth replicate ( $r=1, 2, ..., R$ ), then the corresponding replicate weight for the jth member of the ith PSU is defined as

$w^{(r)}_{ij}={w_{ij}}/{\alpha _ r}$

With Stratification

If the sample design involves stratification, each stratum must have at least two PSUs to use the jackknife method.

Let stratum ${{\tilde h}_ r}$ be the stratum from which a PSU is deleted for the rth replicate. Stratum ${{\tilde h}_ r}$ is called the donor stratum. Let $n_{{{\tilde h}_ r}}$ be the total number of PSUs in the donor stratum ${\tilde h}_ r$ . The jackknife coefficients are defined as

$\alpha _ r=\frac{n_{{{\tilde h}_ r}}-1}{n_{{\tilde h}_ r}} \, \, \, \, \, \mbox{where } r=1, 2, ..., R$

Denote the original weight in the full sample for the jth member of the ith PSU as $w_{ij}$ . If the ith PSU is included in the rth replicate ( $r=1, 2, ..., R$ ), then the corresponding replicate weight for the jth member of the ith PSU is defined as

$w^{(r)}_{ij}=\left\{ \begin{array}{ll} w_{ij} & \mbox{if }\mi{i}\mbox{th PSU is not in the donor stratum }\tilde h_ r \\ w_{ij}/\alpha _ r & \mbox{if }\mi{i}\mbox{th PSU is in the donor stratum }\tilde h_ r \end{array} \right.$

You can use the VARMETHOD=JACKKNIFE(OUTJKCOEFS=) method-option to save the jackknife coefficients into a SAS data set and use the VARMETHOD=JACKKNIFE(OUTWEIGHTS=) method-option to save the replicate weights into a SAS data set.

If you provide your own replicate weights with a REPWEIGHTS statement, then you can also provide corresponding jackknife coefficients with the JKCOEFS= option. If you provide replicate weights but do not provide jackknife coefficients, PROC SURVEYMEANS uses $\alpha _ r = (R-1)/ R$ as the jackknife coefficient for all replicates.

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

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

with $R-H$ degrees of freedom, where R is the number of replicates and H is the number of strata, or R – 1 when there is no stratification.

Hadamard Matrix

A Hadamard matrix $\mb{H}$ is a square matrix whose elements are either 1 or –1 such that

$\mb{H}\mb{H}’=k\mb{I}$

where k is the dimension of $\mb{H}$ and $\mb{I}$ is the identity matrix of order k. The order k is necessarily 1, 2, or a positive integer that is a multiple of 4.

For example, the following matrix is a Hadamard matrix of dimension k = 8:

$\begin{array}{rrrrrrrr} 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1\\ 1 & -1 & 1 & -1 & 1 & -1 & 1 & -1\\ 1 & 1 & -1 & -1 & 1 & 1 & -1 & -1\\ 1 & -1 & -1 & 1 & 1 & -1 & -1 & 1\\ 1 & 1 & 1 & 1 & -1 & -1 & -1 & -1\\ 1 & -1 & 1 & -1 & -1 & 1 & -1 & 1\\ 1 & 1 & -1 & -1 & -1 & -1 & 1 & 1\\ 1 & -1 & -1 & 1 & -1 & 1 & 1 & -1 \end{array}$

The SURVEYMEANS Procedure