Package {bootsurv}

Title:	Bootstrap Methods for Complete Survey Data
Version:	0.1.0
Depends:	R (≥ 3.5.0)
Description:	Bootstrap resampling methods have been widely studied in the context of survey data. This package implements various bootstrap resampling techniques tailored for survey data, with a focus on stratified simple random sampling and stratified two-stage cluster sampling. It provides tools for precise and consistent bootstrap variance estimation for population totals, means, and quartiles. Additionally, it enables easy generation of bootstrap samples for in-depth analysis.
License:	GPL-3
Imports:	MASS, stats
Encoding:	UTF-8
LazyData:	true
Config/roxygen2/version:	8.1.0
Suggests:	testthat (≥ 3.0.0)
Config/testthat/edition:	3
NeedsCompilation:	no
Packaged:	2026-08-27 15:43:16 UTC; zeinabmashreghi
Author:	Zeinab Mashreghi [aut, cre]
Maintainer:	Zeinab Mashreghi <z.mashreghi@uwinnipeg.ca>
Repository:	CRAN
Date/Publication:	2026-08-28 07:01:39 UTC

Bootstrap Replicate Weights from bootsurv Output

Description

The function boot.replicates turns the output of supported bootstrap functions in the package into a matrix of bootstrap replicate weights, with one row per unit of the original sample and one column per bootstrap replicate. This is the form in which replicate weights are stored on public-use survey files and supplied to survey packages.

Usage

boot.replicates(object)

Arguments

Details

With this matrix, bootstrap variances can be estimated for estimators beyond the totals, means and quartiles returned by the package. The estimator is computed once with each column of weights, and the resulting values are used to estimate its bootstrap variance. The rows of the matrix follow the order of the original sample, so the replicate weights can be applied directly to any variable defined on the original sample. This allows bootstrap variances to be estimated for estimators such as totals, ratios, calibrated estimators, and domain estimates.

For the bootstrap weight methods, the replicate weight is the bootstrap weight itself. For the methods that resample units, a unit's replicate weight is its design weight multiplied by the number of times it is selected in the bootstrap sample, and zero if it is not selected. In both cases, the bootstrap statistics returned by the package are reproduced exactly, so the estimator can be computed in the same way whether the units are resampled or only reweighted.

The method of Rao and Wu (1988) is a special case. It rescales the study-variable values toward the stratum mean. Under one-stage simple random sampling without replacement, this rescaling can still be represented by replicate weights, which reproduce estimators that are functions of totals. Under a one-stage stratified design, it cannot be represented by a per-unit replicate-weight matrix that can be applied to arbitrary variables, so the function stops. The bootstrap can instead be reproduced directly from boot.sample and rescale.factor by rescaling the values within each stratum, as shown in the examples.

When the method of Rao and Wu (1988) is used under the two-stage design of boot.twostage(), the function also stops. In this case, the rescaling is variable-dependent and cannot be converted into a fixed matrix of replicate weights. To reproduce the bootstrap for a new variable, the rescaling must be recomputed for that variable using its bootstrap cluster totals and the corresponding stratum means. This requires information from both the original sample and each bootstrap sample, including the cluster sizes and stratum population sizes. The output does not contain all the information needed to perform this recomputation for an arbitrary new variable. The study variable is an exception: its rescaled values are already stored in boot.sample, so an estimator based on the study variable can be computed from the output.

Value

A numeric matrix of bootstrap replicate weights with one row per unit of the original sample, and one column per bootstrap replicate. For the pseudo-population methods the columns are ordered by pseudo-population, that is, all the bootstrap samples drawn from the first pseudo-population, followed by those drawn from the second, and so on.

References

Rao, J. N. K. and C. F. J. Wu (1988). Resampling inference with complex survey data. Journal of the American Statistical Association 83 (401), 231–241.

See Also

boot.twostage(), direct.boot.stsrs(), boot.weights.stsrs() and pseudopop.boot.stsrs(), whose output is turned into bootstrap replicate weights by this function.

Examples

data(data_samp_srs)
population_size<- 6000
y<- data_samp_srs$study.variable

# Showing that the bootstrap variance estimators from the
# boot.weights.stsrs and boot.replicates functions are equal:
boot.CP<- boot.weights.stsrs(data_samp_srs, population_size, R = 20,
           bootstrap.method = "Chipperfield.Preston")

rep.weights<- boot.replicates(boot.CP)
dim(rep.weights)

# The replicate weights reproduce the bootstrap variance of the package:
total.star<- apply(rep.weights, 2, function(w) sum(w * y))

all.equal(var(total.star), boot.CP$boot.var)

# Variance of a ratio of two study variables:
u<- data_samp_srs$study.variable^2
ratio.star<- apply(rep.weights, 2, function(w) sum(w * y)/sum(w * u))
var(ratio.star)

# Variance of a total within a sub-group (domain):
domain<- data_samp_srs$study.variable > 50
domain.star<- apply(rep.weights, 2, function(w) sum(w[domain] * y[domain]))
var(domain.star)

# Several study variables are estimated from the same set of replicate weights:
Y<- cbind(y = y, u = u)
total.several<- crossprod(rep.weights, Y)
apply(total.several, 2, var)

# A direct method, where the units are resampled. The estimator is written in
# the same way, and the replicate weights reproduce the bootstrap statistics:
boot.Efron<- direct.boot.stsrs(data_samp_srs, population_size, R = 20,
              bootstrap.method = "Efron")

rep.weights.Efron<- boot.replicates(boot.Efron)
total.Efron<- apply(rep.weights.Efron, 2, function(w) sum(w * y))
all.equal(total.Efron, as.vector(boot.Efron$boot.statistic))

# A pseudo-population method (Booth, Butler and Hall, 1994). The columns of
# the replicate-weight matrix are ordered by pseudo-population, with R.samp
# columns per pseudo-population:
boot.BBH<- pseudopop.boot.stsrs(data_samp_srs, population_size,
                R.pop = 3, R.samp = 5)

rep.weights.BBH<- boot.replicates(boot.BBH)
total.BBH<- apply(rep.weights.BBH, 2, function(w) sum(w * y))
all.equal(total.BBH, as.vector(t(boot.BBH$boot.statistic)))

# For a pseudo-population method the bootstrap variance is the average of the
# within-pseudo-population variances, not the variance pooled over all the
# replicates. Grouping the columns by pseudo-population reproduces boot.var,
# and the same recipe gives the variance of any other estimator, such as a
# ratio:
pop<- rep(1:3, each = 5)                  # R.pop groups of R.samp columns
mean(tapply(total.BBH, pop, var))         # equal to boot.BBH$boot.var

ratio.star<- apply(rep.weights.BBH, 2, function(w) sum(w * y) / sum(w * u))
mean(tapply(ratio.star, pop, var))        # bootstrap variance of a ratio

# The rescaling method of Rao and Wu (1988). Under simple random sampling the
# replicate weights are available directly:
boot.RW<- direct.boot.stsrs(data_samp_srs, population_size, R = 20,
           bootstrap.method = "Rao.Wu")
rep.weights.RW<- boot.replicates(boot.RW)
all.equal(apply(rep.weights.RW, 2, function(w) sum(w * y)),
          as.vector(boot.RW$boot.statistic))

# Under a stratified design the rescaling is reproduced directly from the
# bootstrap sample: within each stratum a study variable is rescaled by the
# factor in 'rescale.factor', and any other variable is rescaled the
# same way.
data(data_samp_stsrs)
boot.RW.st<- direct.boot.stsrs(data_samp_stsrs, rep(3000, 5), R = 20,
              stratum = "stratum", bootstrap.method = "Rao.Wu")

y.st<- data_samp_stsrs$study.variable
C<- boot.RW.st$rescale.factor              # one rescaling factor per stratum
strata<- sort(unique(data_samp_stsrs$stratum))
ybar<- tapply(y.st, data_samp_stsrs$stratum, mean)   # stratum means

total.RW.st<- sapply(boot.RW.st$boot.sample, function(one.rep){
  h<- match(one.rep$stratum, strata)
  y.star<- ybar[h] + C[h] * (y.st[one.rep$row.id] - ybar[h])
  sum(one.rep$bootstrap.weight * y.star)
})
all.equal(total.RW.st, as.vector(boot.RW.st$boot.statistic))

Bootstrap Methods for Two-Stage Sampling Designs

Description

The function boot.twostage applies one of the following bootstrap methods on complete (full response) survey data selected under stratified two-stage cluster sampling with simple random sampling without replacement at both stages (SRSWOR/SRSWOR): Rao and Wu (1988), Rao, Wu and Yue (1992), the modified version of Sitter (1992, CJS) (see Chen, Haziza and Mashreghi, 2022), Funaoka, Saigo, Sitter and Toida (2006), Chauvet (2007) and Preston (2009). This function also applies the method of Rao, Wu and Yue (1992) on complete survey data selected under stratified two-stage cluster sampling with inclusion-probability-proportional-to-size sampling without replacement at the first stage and SRSWOR at the second stage (IPPSWOR/SRSWOR), or the method of Chauvet (2007) under conditional Poisson sampling at the first stage and SRSWOR at the second stage (CPS/SRSWOR).

Usage

boot.twostage(
  data,
  no.cluster,
  cluster.size,
  R,
  study.variable = "study.variable",
  stratum = NULL,
  cluster = "cluster",
  Pi1 = NULL,
  parameter = "total",
  bootstrap.method = "Rao.Wu.Yue",
  survey.design = "SRSWOR",
  population.size = NULL,
  boot.sample.size = NULL
)

Arguments

Details

One-stage cluster sampling, where all units in a selected cluster are enumerated, is handled as the special case in which the within-cluster sample equals the cluster size.

Value

boot.statistic A vector of bootstrap statistics of size R

boot.var The bootstrap variance estimator of the estimator of the parameter of interest

boot.mean The average of the bootstrap estimator of the parameter of interest

boot.sample A list of results for each bootstrap replicate. This includes the bootstrap values (bootstrap.sample for the methods that resample units, or study.variable for the bootstrap weight methods, where the original values are kept and only the weights change), the original-sample row id of each selected unit (row.id), a cluster identifier (cluster), a stratum identifier (stratum), and the bootstrap weight (bootstrap.weight). Some methods return additional columns, such as the number of times a cluster is selected in the bootstrap sample (No.selected.psu).

rescale.factor The rescaling factor within each stratum. It is returned by the method of Rao and Wu (1988) only.

sample.size The total number of sampled elements across clusters within each stratum

References

Chauvet, G. (2007). Méthodes de bootstrap en population finie. PhD thesis, École Nationale de Statistique et Analyse de l’Information, Bruz, France.

Chen, S., Haziza, D. and Mashreghi, Z. (2022). A Comparison of Existing Bootstrap Algorithms for Multi-Stage Sampling Designs. Stats 5 (2), 521–537.

Funaoka, F., Saigo, H., Sitter, R. R. and Toida, T. (2006). Bernoulli bootstrap for stratified multistage sampling. Survey Methodology 32 (2), 151–156.

Rao, J. N. K. and C. F. J. Wu (1988). Resampling inference with complex survey data. Journal of the American Statistical Association 83 (401), 231–241.

Rao, J. N. K., Wu, C. F. J. and Yue, K. (1992). Some recent work on resampling methods for complex surveys. Survey Methodology 18 (2), 209–217.

Särndal, C.-E., Swensson, B. and Wretman, J. (1992). Model-Assisted Survey Sampling. New York: Springer.

Sitter, R. R. (1992). Comparing three bootstrap methods for survey data. The Canadian Journal of Statistics 20 (2), 135–154.

Preston, J. (2009). Rescaled bootstrap for stratified multistage sampling. Survey Methodology 35 (2), 227–234.

Examples

R<- 20

data(data_samp_clust)
data(data_pop_clust)
no_cluster<- 200
cluster_size<- table(data_pop_clust$cluster)[
                 unique(data_samp_clust$cluster)]

# The first stage sampling fraction is about 20% and the overall second
# stage sampling is about 15%.
# data_samp_clust is a sample taken from data_pop_clust available in
# the package.

boot.RWY<- boot.twostage(data_samp_clust, no_cluster, cluster_size, R)
boot.RWY$boot.var

boot.Pr<- boot.twostage(data_samp_clust, no_cluster, cluster_size, R,
           bootstrap.method="Preston")
boot.Pr$boot.var

boot.RWY.med<- boot.twostage(data_samp_clust, no_cluster, cluster_size,
              R, parameter="median")
boot.RWY.med$boot.var
boot.RWY.med$boot.sample[[5]]

boot.Ch<- boot.twostage(data_samp_clust, no_cluster, cluster_size,
           R=c(3, 10), bootstrap.method="Chauvet")
boot.Ch$boot.mean

data(data_samp_stclust)
data(data_pop_stclust)
# The first stage sampling fraction is about 20% and the overall second
# stage sampling is about 15%.
# data_samp_stclust is a sample taken from data_pop_stclust available
# in the package.

no_cluster_stclust<- c(100, 125, 65)
cluster_size_pop_st<- aggregate(data_pop_stclust$cluster,
 by=list(data_pop_stclust$stratum), table)[[2]]
L<- length(unique(data_samp_stclust$stratum))
cluster_size_st<- NULL
for(h in 1:L) cluster_size_st<- c(cluster_size_st,
 cluster_size_pop_st[[h]][unique(data_samp_stclust$cluster[
   data_samp_stclust$stratum==h])])

boot.RWY.st<- boot.twostage(data_samp_stclust, no_cluster_stclust,
              cluster_size_st, R, stratum="stratum")
boot.RWY.st$boot.statistic

# The output already stores the resampling (row.id and bootstrap.weight), so
# the bootstrap of another variable can be obtained by reusing it, without
# re-running the bootstrap and without boot.replicates. The new variable is
# defined on the original sample, and each replicate reuses the stored weights
# through row.id:
new.variable<- data_samp_clust$study.variable^2
total.new<- sapply(boot.RWY$boot.sample,
             function(one.rep) sum(new.variable[one.rep$row.id] *
                                   one.rep$bootstrap.weight))
# Bootstrap variance of the new variable
var(total.new)

Bootstrap Weight Methods for Survey Data

Description

The function boot.weights.stsrs applies one of the following bootstrap weight methods on complete (full response) survey data selected under either simple random sampling without replacement (SRSWOR) or stratified simple random sampling without replacement (STSRSWOR): Rao, Wu and Yue (1992), Bertail and Combris (1997), Chipperfield and Preston (2007) and Beaumont and Patak (2012).

Usage

boot.weights.stsrs(
  data,
  population.size,
  R,
  study.variable = "study.variable",
  stratum = NULL,
  parameter = "total",
  bootstrap.method = "Rao.Wu.Yue",
  boot.sample.size = NULL,
  distribution.adjust = NULL,
  epsilon = NULL
)

Arguments

Value

boot.statistic A vector of bootstrap statistics

boot.var The bootstrap variance estimator of the estimator of the parameter of interest

boot.mean The average of the bootstrap estimator of the parameter of interest

boot.sample A list of size R. Each element is a data frame containing the study-variable values (study.variable), the original-sample row id of each unit (row.id), a stratum identifier (stratum), the bootstrap weight (bootstrap.weight) and the bootstrap weight adjustment (weight.adjustment).

sample.size The number of sampled units in each stratum.

References

Beaumont, J.-F. and Patak, Z. (2012). On the generalized bootstrap for sample surveys with special attention to Poisson sampling. International Statistical Review 80 (1), 127–148.

Bertail, P. and Combris, P. (1997). Bootstrap généralisé d’un sondage. Annales d’économie et de statistique 46, 49–83.

Chipperfield, J. and Preston, J. (2007). Efficient bootstrap for business surveys. Survey Methodology 33 (2), 167–172.

Rao, J. N. K., Wu, C. F. J. and Yue, K. (1992). Some recent work on resampling methods for complex surveys. Survey Methodology 18 (2), 209–217.

Särndal, C.-E., Swensson, B. and Wretman, J. (1992). Model-Assisted Survey Sampling. New York: Springer.

Examples

R<- 20

data(data_samp_srs)
population_size<- 6000
# The sampling fraction is about 30%.
# data_samp_srs is a sample taken from data_pop available in the package.

boot.RWY<- boot.weights.stsrs(data_samp_srs, population_size, R)
boot.RWY$boot.var

boot.CP<- boot.weights.stsrs(data_samp_srs, population_size, R,
           bootstrap.method="Chipperfield.Preston")
boot.CP$boot.var

boot.BP.med<- boot.weights.stsrs(data_samp_srs, population_size, R,
               parameter="median", bootstrap.method="Beaumont.Patak",
               distribution.adjust="Exponential")
boot.BP.med$boot.var
boot.BP.med$boot.sample[[5]]


data(data_samp_stsrs)
population_size_st<- c(4500, 6300, 3500, 2000, 1500)
# The overall sampling fraction is about 30%.
# data_samp_stsrs is a sample taken from data_pop_st available in the
# package.

boot.RWY.st<- boot.weights.stsrs(data_samp_stsrs, population_size_st, R,
                stratum="stratum")
boot.RWY.st$boot.var
boot.RWY.st$boot.statistic

# The output already stores the resampling (row.id and bootstrap.weight), so
# the bootstrap of another variable can be obtained by reusing it, without
# re-running the bootstrap and without boot.replicates(). The new variable is
# defined on the original sample, and each replicate reuses the stored weights
# through row.id:
new.variable<- data_samp_srs$study.variable^2
total.new<- sapply(boot.RWY$boot.sample,
             function(one.rep) sum(new.variable[one.rep$row.id] *
                                   one.rep$bootstrap.weight))
# Bootstrap variance of the new variable
var(total.new)

Simulated Populations and Samples in the bootsurv Package

Description

This package contains multiple datasets described below.

Datasets

data_pop

This is a population of size 6,000. This dataset contains a column of the generated study variable, labeled as study.variable.

data_pop_st

This dataset represents a population of size 17,800, divided into 5 strata. It includes a column for the generated study variable, labeled as study.variable, and a column identifying the strata, labeled as stratum. The subpopulation sizes within each stratum are as follows: 4,500, 6,300, 3,500, 2,000, and 1,500, respectively.

data_pop_clust

This dataset represents a population consisting of 10,048 units distributed across 200 clusters. The number of units within each cluster was generated using a Poisson distribution with a mean of 50. It includes columns for the generated study variable, labeled as study.variable, and cluster identification, denoted as cluster.

data_pop_stclust

This dataset represents a population with 14,511 units distributed across three strata, consisting of 100, 125, and 65 clusters, respectively. The number of units within each cluster was generated using a Poisson distribution with a mean of 50. It includes columns of the generated study variable, labeled as study.variable, stratum identification, labeled as stratum, and cluster identification within each stratum, labeled as cluster.

data_samp_srs

This dataset comprises a sample of size 1,850, obtained through simple random sampling without replacement from the data_pop dataset.

data_samp_stsrs

This dataset represents a sample of size 5,350 obtained through stratified simple random sampling without replacement from the stratified population data_pop_st. The sample consists of subsample sizes of 1,350, 1,900, 1,050, 600, and 450.

data_samp_clust

This sample was drawn using a two-stage cluster sampling method, with simple random sampling without replacement applied at each stage. The sample is drawn from the data_pop_clust dataset. In the first stage, approximately 20% of clusters were selected. Subsequently, within each selected cluster, approximately 15% of units were sampled.

data_samp_stclust

A stratified two-stage cluster sampling method is applied to draw this sample from the data_pop_stclust dataset. In each stratum, simple random sampling without replacement is applied at each stage. The first stage sampling fraction is approximately 20%, and the overall second stage sampling is approximately 15%.

Direct Bootstrap Methods for Survey Data

Description

The function direct.boot.stsrs applies one of the following bootstrap methods on complete (full response) survey data selected under either simple random sampling without replacement (SRSWOR) or stratified simple random sampling without replacement (STSRSWOR): Efron (1979), McCarthy and Snowden (1985), Rao and Wu (1988) and Sitter (1992, JASA).

Usage

direct.boot.stsrs(
  data,
  population.size,
  R,
  study.variable = "study.variable",
  stratum = NULL,
  parameter = "total",
  bootstrap.method = "Rao.Wu",
  boot.sample.size = NULL
)

Arguments

Value

boot.statistic A vector of bootstrap statistics

boot.var The bootstrap variance estimator of the estimator of the parameter of interest

boot.mean The average of the bootstrap estimator of the parameter of interest

boot.sample A list of size R. Each element is a data frame with four columns: the bootstrap values (bootstrap.sample; rescaled for the Rao and Wu (1988) method), the original-sample row id of each selected unit (row.id), a stratum identifier (stratum), and the bootstrap weight (bootstrap.weight).

rescale.factor The per-stratum rescaling factor applied to the study variable. It equals 1 for every method except Rao and Wu (1988). For Rao and Wu, it can be used together with row.id and the original sample to reproduce the bootstrap rescaling for another variable or estimator.

sample.size The number of sampled units in each stratum.

References

Efron, B. (1979). Bootstrap methods: another look at the jackknife. The Annals of Statistics 7 (1), 1–26.

McCarthy, P. J. and C. B. Snowden (1985). The bootstrap and finite population sampling. Vital and Health Statistics, Series 2, No. 95. DHHS Publication No. (PHS) 85–1369. Public Health Service. Washington. U.S. Government Printing Office.

Rao, J. N. K. and C. F. J. Wu (1988). Resampling inference with complex survey data. Journal of the American Statistical Association 83 (401), 231–241.

Särndal, C.-E., Swensson, B. and Wretman, J. (1992). Model-Assisted Survey Sampling. New York: Springer.

Sitter, R. R. (1992). A resampling procedure for complex survey data. Journal of the American Statistical Association 87 (419), 755–765.

Examples

R<- 20

data(data_samp_srs)
population_size<- 6000
# The sampling fraction is about 30%.
# data_samp_srs is a sample taken from data_pop available in the package.

boot.RW<- direct.boot.stsrs(data_samp_srs, population_size, R)
boot.RW$boot.var

boot.Efron<- direct.boot.stsrs(data_samp_srs, population_size, R,
              parameter="total", bootstrap.method="Efron")
boot.Efron$boot.var

boot.RW.med<- direct.boot.stsrs(data_samp_srs, population_size, R,
               parameter="median")
boot.RW.med$boot.var

data(data_samp_stsrs)
population_size_st<- c(4500, 6300, 3500, 2000, 1500)
# The overall sampling fraction is about 30%.
# data_samp_stsrs is a sample taken from data_pop_st available in the
# package.

boot.RW.st<- direct.boot.stsrs(data_samp_stsrs, population_size_st, R,
              stratum="stratum", parameter="total",
              bootstrap.method="Rao.Wu")
boot.RW.st$boot.statistic

# The output already stores the resampling (row.id and bootstrap.weight), so
# the bootstrap of another variable can be obtained by reusing it, without
# re-running the bootstrap and without boot.replicates(). The new variable is
# defined on the original sample, and each replicate reuses the stored weights
# through row.id:
new.variable<- data_samp_srs$study.variable^2
total.new<- sapply(boot.Efron$boot.sample,
             function(one.rep) sum(new.variable[one.rep$row.id] *
                                   one.rep$bootstrap.weight))
# Bootstrap variance of the new variable:
var(total.new)

Pseudo-Population Bootstrap Methods for Survey Data

Description

The function pseudopop.boot.stsrs applies one of the following pseudo-population bootstrap methods on complete (full response) survey data selected under either simple random sampling without replacement (SRSWOR) or stratified simple random sampling without replacement (STSRSWOR): Bickel and Freedman (1984), Chao and Lo (1985), Sitter (1992, CJS), Booth, Butler and Hall (1994) and Chao and Lo (1994).

Usage

pseudopop.boot.stsrs(
  data,
  population.size,
  R.pop,
  R.samp,
  study.variable = "study.variable",
  stratum = NULL,
  parameter = "total",
  bootstrap.method = "Booth.Butler.Hall"
)

Arguments

Value

boot.statistic A vector of bootstrap statistics

boot.parameter A vector of bootstrap parameters computed from each generated bootstrap pseudo-population

boot.var The bootstrap variance estimator of the estimator of the parameter of interest

boot.mean The average of the bootstrap estimator of the parameter of interest

boot.sample A list of size R.pop. Each element contains a list of results for the bootstrap samples from the corresponding generated bootstrap pseudo-population. Each bootstrap sample includes four columns: the bootstrap values (bootstrap.sample), the original-sample row id of each selected unit (row.id), a stratum identifier (stratum), and the bootstrap weight (bootstrap.weight).

sample.size The number of sampled units in each stratum.

References

Bickel, P. J. and Freedman, D. A. (1984). Asymptotic normality and the bootstrap in stratified sampling. The Annals of Statistics 12, 470–82.

Booth, J. G., Butler, R. W. and Hall, P. (1994). Bootstrap methods for finite populations. Journal of the American Statistical Association 89 (428), 1282–1289.

Chao, M. T. and Lo, S.-H. (1985). A bootstrap method for finite population. Sankhya: The Indian Journal of Statistics, Series A 47, 399–405.

Chao, M. T. and Lo, S.-H. (1994). Maximum likelihood summary and the bootstrap method in structured finite populations. Statistica Sinica 4 (2), 389–406.

Särndal, C.-E., Swensson, B. and Wretman, J. (1992). Model-Assisted Survey Sampling. New York: Springer.

Sitter, R. R. (1992). Comparing three bootstrap methods for survey data. The Canadian Journal of Statistics 20 (2), 135–154.

Examples

R.pop<- 5
R.samp<- 10

data(data_samp_srs)
population_size<- 6000
# The sampling fraction is about 30%.
# data_samp_srs is a sample taken from data_pop available in the package.

boot.Booth<- pseudopop.boot.stsrs(data_samp_srs, population_size,
                  R.pop, R.samp)
boot.Booth$boot.var

boot.BF<- pseudopop.boot.stsrs(data_samp_srs, population_size, R.pop, R.samp,
           bootstrap.method="Bickel.Freedman")
boot.BF$boot.var

boot.Sitter.med<- pseudopop.boot.stsrs(data_samp_srs, population_size, R.pop,
                   R.samp, parameter="median", bootstrap.method="Sitter.BWO")
boot.Sitter.med$boot.var
boot.Sitter.med$boot.sample[[2]][[5]]

data(data_samp_stsrs)
population_size_st<- c(4500, 6300, 3500, 2000, 1500)
# The overall sampling fraction is about 30%.
# data_samp_stsrs is a sample taken from data_pop_st available in the
# package.

boot.Booth.st<- pseudopop.boot.stsrs(data_samp_stsrs, population_size_st,
                  R.pop, R.samp, stratum="stratum")
boot.Booth.st$boot.statistic

# The output already stores the resampling (row.id and bootstrap.weight), so
# the bootstrap of another variable can be obtained by reusing it, without
# re-running the bootstrap and without boot.replicates(). For a pseudo-population
# method the variance is the average of the within-pseudo-population variances:
new.variable<- data_samp_srs$study.variable^2
total.new<- lapply(boot.Booth$boot.sample,
             function(one.pop) sapply(one.pop,
               function(one.rep) sum(new.variable[one.rep$row.id] *
                                     one.rep$bootstrap.weight)))
# Bootstrap variance of the new variable:
mean(sapply(total.new, var))