Taylor Series (Linearization)

The Taylor series (linearization) method is the most commonly used method to estimate the covariance matrix of the regression coefficients for complex survey data. It is the default variance estimation method used by PROC SURVEYREG.

Use the notation described in the section Notation to denote the residuals from the linear regression as

with as its elements. Let the matrix be defined as

where

The Taylor series estimate of the covariance matrix of is

The factor in the computation of the matrix reduces the small sample bias associated with using the estimated function to calculate deviations (Hidiroglou, Fuller, and Hickman 1980). For simple random sampling, this factor contributes to the degrees of freedom correction applied to the residual mean square for ordinary least squares in which parameters are estimated. By default, the procedure use this adjustment in the variance estimation. If you do not want to use this multiplier in variance estimation, you can specify the VADJUST=NONE option in the MODEL statement to suppress this factor.

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 be the total number of strata. The total number of replicates is the smallest multiple of 4 that is greater than . However, if you prefer a larger number of replicates, you can specify the REPS=number option. If a 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 columns of the Hadamard matrix. The th () replicate is drawn from the full sample according to the th row of the Hadamard matrix as follows:

• If the th element of the Hadamard matrix is , then the first PSU of stratum is included in the th replicate and the second PSU of stratum is excluded.

• If the th element of the Hadamard matrix is , then the second PSU of stratum is included in the th replicate and the first PSU of stratum is excluded.

The replicate weights of the remaining PSUs in each half sample are then doubled to their original weights. For more detail about the BRR method, see Wolter (1985) and Lohr (1999).

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.

Let be the estimated regression coefficients from the full sample for , and let be the estimated regression coefficient from the th replicate by using replicate weights. PROC SURVEYREG estimates the covariance matrix of by

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

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

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= 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 optionally set by the FAY = method-option. By default, if only the FAY method-option is used without specifying 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 , then the full sample weight of the first PSU in stratum is multiplied by and that of the second PSU is multiplied by to obtain the th replicate weights.

• If the th element of the Hadamard matrix is , then the full sample weight of the first PSU in stratum is multiplied by and that 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.

Let be the estimated regression coefficients from the full sample for . Let be the estimated regression coefficient obtained from the th replicate by using replicate weights. PROC SURVEYREG estimates the covariance matrix of by

with degrees of freedom, where 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 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 . 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 are the same for all replicates:

Denote the original weight in the full sample for the th member of the th PSU as . If the th PSU is included in the th replicate (), then the corresponding replicate weight for the th member of the th PSU is defined as

With Stratification

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

Let stratum be the stratum from which a PSU is deleted for the th replicate. Stratum is called the donor stratum. Let be the total number of PSUs in the donor stratum . The jackknife coefficients are defined as

Denote the original weight in the full sample for the th member of the th PSU as . If the th PSU is included in the th replicate (), then the corresponding replicate weight for the th member of the th PSU is defined as

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.

Let be the estimated regression coefficients from the full sample for . Let be the estimated regression coefficient obtained from the th replicate by using replicate weights. PROC SURVEYREG estimates the covariance matrix of by

with degrees of freedom, where is the number of replicates and is the number of strata, or when there is no stratification.