Convert a survey design object to a successive differences replicate design

Converts a survey design object to a replicate design object with replicate weights formed using the successive differences replication (SDR) method. The SDR method is suitable for designs that use systematic sampling or finely-stratified sampling designs.

Usage

as_sdr_design(
  design,
  replicates,
  sort_variable = NULL,
  use_normal_hadamard = FALSE,
  compress = TRUE,
  mse = TRUE
)

Arguments

design: A survey design object created using the 'survey' (or 'srvyr') package, with class 'survey.design' or 'svyimputationList'.
replicates: The target number of replicates to create. This will determine the order of the Hadamard matrix to use when creating replicate factors. If use_normal_hadamard = TRUE, then the actual number of replicates will be greater than or equal to replicates and determined by identifying the smallest available Hadamard matrix available from the 'survey' package. If use_normal_hadamard = FALSE, then the actual number of replicates will be the smallest power of 4 that is greater or equal to the specified value of replicates.
sort_variable: A character string specifying the name of a sorting variable. This variable should give the sort order used in sampling. If the design includes strata, then the replicate factors will be assigned after first sorting by the first-stage strata identifier and then sorting by the value of sort_variable within each stratum.
use_normal_hadamard: Whether to use a normal Hadamard matrix: that is, a matrix whose first row and first column only have entries equal to 1. This means that one of the replicates will be an "inactive" replicate. See the "Details" section for more information.
compress: Use a compressed representation of the replicate weights matrix. This reduces the computer memory required to represent the replicate weights and has no impact on estimates.
mse: If TRUE, compute variances from sums of squares around the point estimate from the full-sample weights, If FALSE, compute variances from sums of squares around the mean estimate from the replicate weights.

Value

A replicate design object, with class svyrep.design, which can be used with the usual functions, such as svymean() or svyglm().

Use weights(..., type = 'analysis') to extract the matrix of replicate weights.
Use as_data_frame_with_weights() to convert the design object to a data frame with columns for the full-sample and replicate weights.

Statistical Overview

The successive difference replication method was proposed by Fay and Train (1995) as a replication method appropriate for samples selected using systematic sampling. It is designed to yield variance estimates for totals that are equivalent to successive difference variance estimators described in Fay and Train (1995). There are different methods for forming the replicate factors depending on whether the replicate variance estimator is meant to be equivalent to the SD2 variance estimator (i.e., the circular successive difference estimator) or the SD1 variance estimator (the non-circular successive difference estimator) described in Ash (2014). This function uses the approach based on the SD2 variance estimator. For multistage designs, this replication method only takes into account information about the first stage of sampling.

The scale factor to be used for variance estimation with the replicate weights is \(4/R\), where \(R\) is the number of replicates. This scale factor will be used even when there are finite population corrections; see the subsection below.

As an alternative to the successive difference replication estimator, one can use a generalized replication method where the target variance estimator is the "SD1" or "SD2" estimator. See the functions as_gen_boot_design or as_fays_gen_rep_design for more details on generalized replication and see the help section variance-estimators for more details on the "SD1" and "SD2" variance estimators.

Details on Stratification and Finite Population Corrections

If the design includes strata, then the replicate factors will be assigned after first sorting by the first-stage strata identifier and then sorting by the value of sort_variable within each stratum.

If there are finite population correction factors, then these finite population correction factors will be applied to the replicate factors. This means that variance estimates with the finite population correction do not require any adjustment to the overall scale factor used in variance estimation. This is the approach used by the U.S. Census Bureau for the 5-year American Community Survey (ACS) replicate weights (U.S. Census Bureau, 2022, p. 12-8). This approach is used regardless of whether the design has one overall finite population correction factor or has different finite population correction factors for different strata.

Details on Row Assignments for Creating Replicate Factors

The number of replicates must match the order of an available Hadamard matrix. A Hadamard matrix can either be normal or non-normal: a normal Hadamard matrix is one where the entries in the first row and in the first column are all equal to one. If the user specifies use_normal_hadamard = TRUE, then there are more choices of Hadamard matrix sizes available, and so greater flexibility in choosing the number of replicates to create. When a normal Hadamard matrix is used, this will result in the creation of an inactive replicate (sometimes referred to as a "dead" replicate), which is a replicate where all the replicate factors equal one. Inactive replicates are perfectly valid for variance estimation, though some users may find them confusing.

An important part of the process of creating replicate weights is the assignment of rows of the Hadamard matrix to primary sampling units. The method of Ash (2014) referred to as "RA1" is used for row assignments, which means that the replication-based variance estimates for totals will be equivalent to the SD2 variance estimator described by Ash (2014). The number of cycles used with the "RA1" method is the smallest integer greater than \(n/R\), where \(n\) is the number of primary sample units and \(R\) is the number of replicates.

References

Ash, S. (2014). "Using successive difference replication for estimating variances." Survey Methodology, Statistics Canada, 40(1), 47–59.

Fay, R.E. and Train, G.F. (1995). "Aspects of Survey and Model-Based Postcensal Estimation of Income and Poverty Characteristics for States and Counties." Joint Statistical Meetings, Proceedings of the Section on Government Statistics, 154-159.

U.S. Census Bureau. (2022). "American Community Survey and Puerto Rico Community Survey Design and Methodology, Version 3.0."

Examples

library(survey)

# Load example stratified systematic sample
data('library_stsys_sample', package = 'svrep')

## First, ensure data are sorted in same order as was used in sampling
library_stsys_sample <- library_stsys_sample[
  order(library_stsys_sample$SAMPLING_SORT_ORDER),
]

## Create a survey design object
design_obj <- svydesign(
  data   = library_stsys_sample,
  strata = ~ SAMPLING_STRATUM,
  ids    = ~ 1,
  fpc    = ~ STRATUM_POP_SIZE
)

## Convert to SDR replicate design
sdr_design <- as_sdr_design(
  design              = design_obj,
  replicates          = 180,
  sort_variable       = "SAMPLING_SORT_ORDER",
  use_normal_hadamard = TRUE
)
#> Using Hadamard matrix of order 180. If `use_normal_hadamard=TRUE`, the smallest possible order is 180.
#> Finite population corrections are incorporated into the replicate factors.

## Compare to generalized bootstrap
## based on the SD2 estimator that SDR approximates
gen_boot_design <- as_gen_boot_design(
  design             = design_obj,
  variance_estimator = "SD2",
  replicates         = 180,
  exact_vcov         = TRUE
)
#> For `variance_estimator='SD2', assumes rows of data are sorted in the same order used in sampling.

## Estimate sampling variances
svytotal(x = ~ TOTSTAFF, na.rm = TRUE, design = sdr_design)
#>           total    SE
#> TOTSTAFF 180977 39077
svytotal(x = ~ TOTSTAFF, na.rm = TRUE, design = gen_boot_design)
#>           total    SE
#> TOTSTAFF 180977 38893