  ### Appendix F (Estimating Complex Statistics When Sample Size Is Not Fixed)

Whenever the sample size is not fixed, use the Taylor Series estimation or one of the replicated methods such as balanced repeated replication (BRR) or jackknife repeated replication (JRR) to estimate ratio means or other complex statistics.

• Taylor Series estimation:
• This method computes the sampling variance of an approximation to a complex function like a ratio or regression coefficient. (See  for the exact formulas.)
• Used by most statistical software packages.
• Requires analytic manipulations and computation of derivatives (however, these already have been done by the developers of the software packages for common type of estimates).
• Not useful if estimate cannot be expressed as a function of sample totals.
• Taylor Series estimates in most software packages do not account for the variability of nonresponse adjustments.
• Balanced repeated replication (or half-sample replication):
• This method assumes a paired selection design (i.e., 2 PSUs per stratum) and selects $$H^*$$ half sample replicates ($$H^*$$ is the smallest multiple of 4 greater than or equal to the number of strata) by deleting one primary sampling unit (PSU) from each stratum according to the pattern in a Hadamard matrix. Each remaining element in the half sample receives a replicate weight of two. Fay’s method of BRR is an alternative that retains both PSUs in a pair but modifies their survey weights .
• More useful for complex estimates such as medians than Taylor Series.
• Easily applied to user-specified statistics like differences or ratios of domain means.
• Accounts for variability due to multiple steps in adjustment more easily than does Taylor Series.
• Best used only with a paired selection stratification design.
• Appending replicate weights to each record increases file size.
• Combining of strata and PSUs is sometimes done to reduce number of replicates. This must be done carefully to avoid biased variance estimates.
• Jackknife repeated replication:
• This method creates a replicate by dropping a PSU from one stratum and weights up the other PSUs in the stratum to maintain the sampling distribution across the strata.