- Definitions and Notation
- Mean
- Variance and Standard Error of the Mean
- t Test for the Mean
- Degrees of Freedom
- Confidence Limits for the Mean
- Coefficient of Variation
- Proportions
- Total
- Variance and Standard Deviation of the Total
- Confidence Limits for the Total
- Ratio
- Domain Statistics
- Quantiles
- Geometric Mean
- Poststratification

The SURVEYMEANS procedure uses the Taylor series (linearization) method or replication (resampling) methods to estimate sampling errors of estimators based on complex sample designs. For more information, see Fuller (2009); Wolter (2007); Lohr (2010); Kalton (1983); Hidiroglou, Fuller, and Hickman (1980); Fuller et al. (1989); Lee, Forthoffer, and Lorimor (1989); Cochran (1977); Kish (1965); Hansen, Hurwitz, and Madow (1953); Rust (1985); Dippo, Fay, and Morganstein (1984); Rao and Shao (1999); Rao, Wu, and Yue (1992); Rao and Shao (1996). You can use the VARMETHOD= option to specify a variance estimation method to use. By default, the Taylor series method is used.

The Taylor series method obtains a linear approximation for the estimator and then uses the variance estimate for this approximation to estimate the variance of the estimate itself (Woodruff, 1971; Fuller, 1975). When there are clusters, or PSUs, in the sample design, the procedure estimates variance from the variation among PSUs. When the design is stratified, the procedure pools stratum variance estimates to compute the overall variance estimate. For t tests of the estimates, the degrees of freedom equal the number of clusters minus the number of strata in the sample design.

For a multistage sample design, the Taylor series estimation depends only on the first stage of the sample design. Therefore, 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 that the first-stage sample is drawn with replacement, as it often is in practice.

Quite often in complex surveys, respondents have unequal weights, which reflect unequal selection probabilities and adjustments for nonresponse. In such surveys, the appropriate sampling weights must be used to obtain valid estimates for the study population.

However, replication methods have recently gained popularity for estimating variances in complex survey data analysis. One reason for this popularity is the relative simplicity of replication-based estimates, especially for nonlinear estimators; another is that modern computational capacity has made replication methods feasible for practical survey analysis.

Replication methods draw multiple replicates (also called subsamples) from a full sample according to a specific resampling
scheme. The most commonly used resampling schemes are the *balanced repeated replication* (BRR) method and the *jackknife* method. For each replicate, the original weights are modified for the PSUs in the replicates to create replicate weights.
The population parameters of interest are estimated by using the replicate weights for each replicate. Then the variances
of parameters of interest are estimated by the variability among the estimates derived from these replicates. You can use
a REPWEIGHTS statement to provide your own replicate weights for variance estimation. For more information about using replication methods
to analyze sample survey data, see the section Replication Methods for Variance Estimation.