Taylor Series Variance Estimation

The Taylor series linearization method can be used to estimate standard errors of proportions and other statistics for crosstabulation tables. For sample survey data, the proportion estimator is a ratio estimator formed from estimators of totals. For example, to estimate the proportion in a crosstabulation table cell, the procedure uses the ratio of the estimator of the cell total frequency to the estimator of the overall population total, where these totals are linear statistics computed from the survey data. The Taylor series expansion method obtains a first-order linear approximation for the ratio estimator and then uses the variance estimate for this approximation to estimate the variance of the estimate itself (Woodruff 1971; Fuller 1975). For more information about Taylor series variance estimation for sample survey data, see Lohr (2009), Särndal, Swensson, and Wretman (1992), Lee, Forthoffer, and Lorimor (1989), and Wolter (1985).

When there are clusters (PSUs) in the sample design, the Taylor series method estimates variance from the variance among PSUs. When the design is stratified, the procedure combines stratum variance estimates to compute the overall variance estimate. For a multistage sample design, the variance estimation depends only on the first stage of the sample design. So the required input includes only first-stage cluster (PSU) and first-stage stratum identification. You do not need to input design information about any additional stages of sampling. This variance estimation method assumes that the first-stage sampling fraction is small, or the first-stage sample is drawn with replacement, as it often is in practice.

See the sections Proportions, Row and Column Proportions, Risks and Risk Difference, and Odds Ratio and Relative Risks for details and formulas for Taylor series variance estimates.

Replication-Based Variance Estimation

Replication-based methods for variance estimation draw multiple replicates (subsamples) from the full sample by following a specific resampling scheme. Commonly used resampling schemes include balanced repeated replication (BRR) and the jackknife. PROC SURVEYFREQ estimates the parameter of interest (a proportion, total, odds ratio, or other statistic) from each replicate, and then uses the variability among replicate estimates to estimate the overall variance of the parameter estimate. See Wolter (1985) and Lohr (2009) for more information.

The BRR variance estimation method requires a stratified sample design with two PSUs in each stratum. Each replicate is obtained by deleting one PSU per stratum according to the corresponding Hadamard matrix and adjusting the original weights of the remaining PSUs. The adjusted weights are called replicate weights. PROC SURVEYFREQ also provides Fay’s method, which is a modification of the BRR method. See the section Balanced Repeated Replication (BRR) for details.

The jackknife method deletes one PSU at a time from the full sample to create replicates, and modifies the original weights to obtain replicate weights. The total number of replicates equals the number of PSUs. If the sample design is stratified, each stratum must contain at least two PSUs, and the jackknife is applied separately within each stratum. See the section The Jackknife Method for details.

Instead of having PROC SURVEYFREQ generate replicate weights for the analysis, you can input your own replicate weights with a REPWEIGHTS statement. This can be useful if you need to do multiple analyses with the same set of replicate weights, or if you have access to replicate weights instead of design information. See the section Replicate Weights Output Data Set for more information.