Functions for Optimum Sample Allocation in Stratified Sampling

Description

Optimum Sample Allocation in Stratified Sampling

Author(s)

Wojciech Wójciak wojciech.wojciak@gmail.com

References

Stenger, H., Gabler, S. (2005). Combining random sampling and census strategies - Justification of inclusion probabilities equal to 1. Metrika, 61(2), pp. 137–156. doi:10.1007/s001840400328

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

Wesołowski, J., Wieczorkowski, R., Wójciak, W. (2021). Optimality of the Recursive Neyman Allocation. Journal of Survey Statistics and Methodology, 10(5), pp. 1263–1275. doi:10.1093/jssam/smab018, doi:10.48550/arXiv.2105.14486

Wesołowski, J., Wieczorkowski, R., Wójciak, W. (2023). R Package stratallo - source code (Version 2.2.0). https://github.com/wwojciech/stratallo

Wesołowski, J., Wieczorkowski, R., Wójciak, W. (2023). Numerical Performance of the RNABOX Algorithm (Version 1.0.1). https://github.com/rwieczor/recursive_Neyman_rnabox

Wójciak, W. (2023). Another Solution of Some Optimum Allocation Problem. Statistics in Transition new series, 24(5) (in press). https://arxiv.org/abs/2204.04035

Wójciak, W. (2019). Optimal Allocation in Stratified Sampling Schemes. MSc Thesis, Warsaw University of Technology, Warsaw, Poland. http://home.elka.pw.edu.pl/~wwojciak/msc_optimal_allocation.pdf

Integer-valued Optimal Univariate Allocation Under Constraints for Stratified Sampling

Description

Better algorithm from paper Friedrich et al. (2015) for integer-valued optimal allocation in stratified sampling.

Usage

CapacityScaling(n, Ah, mh = rep(1, length(Ah)), Mh = rep(Inf, length(Ah)))

CapacityScaling2(
  v0,
  Nh,
  Sh,
  mh = rep(1, length(Nh)),
  Mh = rep(Inf, length(Nh))
)

Arguments

Value

A vector of optimal allocation sizes.

Functions

• CapacityScaling2():

References

Friedrich, U., Münnich, R., de Vries, S. and Wagner, M. (2015) Fast integer-valued algorithms for optimal allocations under constraints in stratified sampling, Computational Statistics and Data Analysis, 92, pp. 1–12. https://www.sciencedirect.com/science/article/pii/S0167947315001413

Integer-valued Optimal Univariate Allocation Under Constraints for Stratified Sampling

Description

Simple algorithm from paper Friedrich et al. (2015) for integer-valued optimal allocation in stratified sampling.

Usage

SimpleGreedy(
  n,
  Ah,
  mh = rep(1, length(Ah)),
  Mh = rep(Inf, length(Ah)),
  nh = mh
)

SimpleGreedy2(v0, Nh, Sh, mh = rep(1, length(Nh)), Mh = Nh, nh = mh)

Arguments

Value

A vector of optimal allocation sizes.

Functions

• SimpleGreedy2():

References

Friedrich, U., Münnich, R., de Vries, S. and Wagner, M. (2015) Fast integer-valued algorithms for optimal allocations under constraints in stratified sampling, Computational Statistics and Data Analysis, 92, pp. 1–12. https://www.sciencedirect.com/science/article/pii/S0167947315001413

Summarizing the Allocation

Description

A helper function that returns a simple data.frame with summary of the allocation as returned by the opt() or optcost(). See the illustrate example below.

Usage

asummary(x, A, m = NULL, M = NULL)

Arguments

Value

A data.frame with as many rows as number of strata H + 1, and up to 7 variables. A single row corresponds to a given stratum h \in \{1,\ldots,H\}, whilst the last row contains sums of all of the numerical values from the above rows (wherever feasible). Summary table has the following columns (* indicates that the column may not be present):

A

population constant A_h

m*

lower bound imposed on sample size in stratum

M*

upper bound imposed on sample size in stratum

allocation

sample size for a given stratum

take_min*

indication whether the allocation is of take-min type, i.e. x_h = m_h

take_max*

indication whether the allocation is of take-max type, i.e. x_h = M_h

take_Neyman

indication whether the allocation is of take-Neyman type, i.e. m_h < x_h < M_h

See Also

opt(), optcost().

Examples

A <- c(3000, 4000, 5000, 2000)
m <- c(100, 90, 70, 80)
M <- c(200, 150, 300, 210)

xopt_1 <- opt(n = 400, A, m)
asummary(xopt_1, A, m)

xopt_2 <- opt(n = 540, A, m, M)
asummary(xopt_2, A, m, M)

Optimal Univariate Allocation Under Constraints for Stratified Sampling

Description

Algorithm for optimal allocation in stratified sampling with lower and upper constraints based on fixed point iteration.

Usage

fpia(
  n,
  Ah,
  mh = NULL,
  Mh = NULL,
  lambda0 = NULL,
  maxiter = 100,
  tol = .Machine$double.eps * 1000
)

fpia2(v0, Nh, Sh, mh = NULL, Mh = NULL, lambda0 = NULL, maxiter = 100)

Arguments

Value

A vector of optimal allocation sizes, and number of iterations.

Functions

• fpia2():

References

Münnich, R. T., Sachs, E.W. and Wagner, M. (2012) Numerical solution of optimal allocation problems in stratified sampling under box constraints, AStA Advances in Statistical Analysis, 96(3), pp. 435-450. doi:10.1007/s10182-011-0176-z

Optimum Sample Allocation in Stratified Sampling

Description

A classical problem in survey methodology in stratified sampling is optimum sample allocation. This problem is formulated as determination of strata sample sizes that minimize the variance of the stratified \pi estimator of the population total (or mean) of a given study variable, under certain constraints on sample sizes in strata.

The opt() user function solves the following optimum sample allocation problem, formulated below in the language of mathematical optimization.

Minimize

f(x_1,\ldots,x_H) = \sum_{h=1}^H \frac{A^2_h}{x_h}

subject to

\sum_{h=1}^H x_h = n

m_h \leq x_h \leq M_h, \quad h = 1,\ldots,H,

where n > 0,\, A_h > 0,\, m_h > 0,\, M_h > 0, such that m_h < M_h,\, h = 1,\ldots,H, and \sum_{h=1}^H m_h \leq n \leq \sum_{h=1}^H M_h, are given numbers. The minimization is on \mathbb R_+^H.

The inequality constraints are optional and user can choose whether and how they are to be added to the optimization problem. This is achieved by the proper use of m and M arguments of this function, according to the following rules:

• no inequality constraints imposed: both m and M must be both set to NULL (default).

• one-sided lower bounds m_h,\, h = 1,\ldots,H, imposed: lower bounds are specified with m, while M is set to NULL.

• one-sided upper bounds M_h,\, h = 1,\ldots,H, imposed: upper bounds are specified with M, while m is set to NULL.

• box-constraints imposed: lower and upper bounds must be specified with m and M, respectively.

Usage

opt(n, A, m = NULL, M = NULL, M_algorithm = "rna")

Arguments

Details

The opt() function makes use of several allocation algorithms, depending on which of the inequality constraints should be taken into account in the optimization problem. Each algorithm is implemented in a separate R function that in general should not be used directly by the end user. The following is the list with the algorithms that are used along with the name of the function that implements a given algorithm. See the description of a specific function to find out more about the corresponding algorithm.

• one-sided lower-bounds m_h,\, h = 1,\ldots,H:

  • LRNA - rna()

• one-sided upper-bounds M_h,\, h = 1,\ldots,H:

  • RNA - rna()

  • SGA - sga()

  • SGAPLUS - sgaplus()

  • COMA - coma()

• box constraints m_h, M_h,\, h = 1,\ldots,H:

  • RNABOX - rnabox()

Value

Numeric vector with optimal sample allocations in strata.

Note

If no inequality constraints are added, the allocation is given by the Neyman allocation as:

x_h = A_h \frac{n}{\sum_{i=1}^H A_i}, \quad h = 1,\ldots,H.

For stratified \pi estimator of the population total with stratified simple random sampling without replacement design in use, the parameters of the objective function f are:

A_h = N_h S_h, \quad h = 1,\ldots,H,

where N_h is the size of stratum h and S_h denotes standard deviation of a given study variable in stratum h.

References

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

See Also

optcost(), rna(), sga(), sgaplus(), coma(), rnabox().

Examples

A <- c(3000, 4000, 5000, 2000)
m <- c(100, 90, 70, 50)
M <- c(300, 400, 200, 90)

# One-sided lower bounds.
opt(n = 340, A = A, m = m)
opt(n = 400, A = A, m = m)
opt(n = 700, A = A, m = m)

# One-sided upper bounds.
opt(n = 190, A = A, M = M)
opt(n = 700, A = A, M = M)

# Box-constraints.
opt(n = 340, A = A, m = m, M = M)
opt(n = 500, A = A, m = m, M = M)
xopt <- opt(n = 800, A = A, m = m, M = M)
xopt
var_st(x = xopt, A = A, A0 = 45000) # Value of the variance for allocation xopt.

# Execution-time comparisons of different algorithms with microbenchmark R package.
## Not run: 
N <- pop969[, "N"]
S <- pop969[, "S"]
A <- N * S
nfrac <- c(0.005, seq(0.05, 0.95, 0.05))
n <- setNames(as.integer(nfrac * sum(N)), nfrac)
lapply(
  n,
  function(ni) {
    microbenchmark::microbenchmark(
      RNA = opt(ni, A, M = N, M_algorithm = "rna"),
      SGA = opt(ni, A, M = N, M_algorithm = "sga"),
      SGAPLUS = opt(ni, A, M = N, M_algorithm = "sgaplus"),
      COMA = opt(ni, A, M = N, M_algorithm = "coma"),
      times = 200,
      unit = "us"
    )
  }
)

## End(Not run)

Algorithms for Optimum Sample Allocation Under One-Sided Bounds

Description

Functions that implement selected optimal allocation algorithms that compute a solution to the optimal allocation problem defined in the language of mathematical optimization as follows.

Minimize

f(x_1,\ldots,x_H) = \sum_{h=1}^H \frac{A^2_h}{x_h}

subject to

\sum_{h=1}^H c_h x_h = c

and either

x_h \leq M_h, \quad h = 1,\ldots,H

or

x_h \geq m_h, \quad h = 1,\ldots,H,

where c > 0,\, c_h > 0,\, A_h > 0,\, m_h > 0,\, M_h > 0,\, h = 1,\ldots,H, are given numbers. The minimization is on \mathbb R_+^H.

The inequality constraints are optional and user can choose whether and how they are to be added to the optimization problem. If one-sided lower bounds m_h,\, h = 1,\ldots,H, must be imposed, it is then required that c \geq \sum_{h=1}^H c_h m_h. If one-sided upper bounds M_h,\, h = 1,\ldots,H, must be imposed, it is then required that 0 < c \leq \sum_{h=1}^H c_h M_h. Lower bounds can be specified instead of the upper bounds only in case of the LRNA algorithm. All other algorithms allow only for specification of the upper bounds. For the sake of clarity, we emphasize that in the optimization problem consider here, the lower and upper bounds cannot be imposed jointly.

Costs c_h,\, h = 1,\ldots,H, of surveying one element in stratum, can be specified by the user only in case of the RNA and LRNA algorithms. For remaining algorithms, these costs are fixed at 1, i.e. c_h = 1,\, h = 1,\ldots,H.

The following is the list of all the algorithms available to use along with the name of the function that implements a given algorithm. See the description of a specific function to find out more about the corresponding algorithm.

• RNA - rna()

• LRNA- rna()

• SGA- sga()

• SGAPLUS - sgaplus()

• COMA - coma()

Functions in this family should not be called directly by the user. Use opt() or optcost() instead.

Usage

rna(
  total_cost,
  A,
  bounds = NULL,
  unit_costs = 1,
  check_violations = .Primitive(">="),
  details = FALSE
)

sga(total_cost, A, M)

sgaplus(total_cost, A, M)

coma(total_cost, A, M)

Arguments

Value

Numeric vector with optimal sample allocations in strata. In case of the rna() only, it can also be a list with optimal sample allocations and strata assignments (either to take-Neyman or take-bound).

Functions

• rna(): Recursive Neyman Algorithm (RNA) and its twin version, Lower Recursive Neyman Algorithm (LRNA) dedicated to the allocation problem with one-sided lower-bounds constraints. The RNA is described in Wesołowski et al. (2021), while LRNA is introduced in Wójciak (2023).

• sga(): Stenger-Gabler type algorithm SGA, described in Wesołowski et al. (2021) and in Stenger and Gabler (2005). This algorithm solves the problem with one-sided upper-bounds constraints. It also assumes unit costs are constant and equal to 1, i.e. c_h = 1,\, h = 1,\ldots,H.

• sgaplus(): modified Stenger-Gabler type algorithm, described in Wójciak (2019) as Sequential Allocation (version 1) algorithm. This algorithm solves the problem with one-sided upper-bounds constraints. It also assumes unit costs are constant and equal to 1, i.e. c_h = 1,\, h = 1,\ldots,H.

• coma(): Change of Monotonicity Algorithm (COMA), described in Wesołowski et al. (2021). This algorithm solves the problem with one-sided upper-bounds constraints. It also assumes unit costs are constant and equal to 1, i.e. c_h = 1,\, h = 1,\ldots,H.

Note

If no inequality constraints are added, the allocation is given by the Neyman allocation as:

x_h = \frac{A_h}{\sqrt{c_h}} \frac{n}{\sum_{i=1}^H A_i \sqrt{c_i}}, \quad h = 1,\ldots,H.

For stratified \pi estimator of the population total with stratified simple random sampling without replacement design in use, the parameters of the objective function f are:

A_h = N_h S_h, \quad h = 1,\ldots,H,

where N_h is the size of stratum h and S_h denotes standard deviation of a given study variable in stratum h.

References

Wójciak, W. (2023). Another Solution of Some Optimum Allocation Problem. Statistics in Transition new series, 24(5) (in press). https://arxiv.org/abs/2204.04035

Wesołowski, J., Wieczorkowski, R., Wójciak, W. (2021). Optimality of the Recursive Neyman Allocation. Journal of Survey Statistics and Methodology, 10(5), pp. 1263–1275. doi:10.1093/jssam/smab018, doi:10.48550/arXiv.2105.14486

Wójciak, W. (2019). Optimal Allocation in Stratified Sampling Schemes. MSc Thesis, Warsaw University of Technology, Warsaw, Poland. http://home.elka.pw.edu.pl/~wwojciak/msc_optimal_allocation.pdf

Stenger, H., Gabler, S. (2005). Combining random sampling and census strategies - Justification of inclusion probabilities equal to 1. Metrika, 61(2), pp. 137–156. doi:10.1007/s001840400328

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

See Also

opt(), optcost(), rnabox().

Examples

A <- c(3000, 4000, 5000, 2000)
m <- c(50, 40, 10, 30) # lower bounds
M <- c(100, 90, 70, 80) # upper bounds

rna(total_cost = 190, A = A, bounds = M)
rna(total_cost = 190, A = A, bounds = m, check_violations = .Primitive("<="))
sga(total_cost = 190, A = A, M = M)
sgaplus(total_cost = 190, A = A, M = M)
coma(total_cost = 190, A = A, M = M)

Minimum Cost Allocation in Stratified Sampling

Description

Function that determines fixed strata sample sizes that minimize total cost of the survey, under assumed level of the variance of the stratified estimator and under optional one-sided upper bounds imposed on strata sample sizes. Namely, the following optimization problem, formulated below in the language of mathematical optimization, is solved by optcost() function.

Minimize

c(x_1,\ldots,x_H) = \sum_{h=1}^H c_h x_h

subject to

\sum_{h=1}^H \frac{A^2_h}{x_h} - A_0 = V

x_h \leq M_h, \quad h = 1,\ldots,H,

where A_0,\, A_h > 0,\, c_h > 0,\, M_h > 0,\, h = 1,\ldots,H, and V > \sum_{h=1}^H \frac{A^2_h}{M_h} - A_0 are given numbers. The minimization is on \mathbb R_+^H. The upper-bounds constraints x_h \leq M_h,\, h = 1,\ldots,H, are optional and can be skipped. In such a case, it is only required that V > 0.

Usage

optcost(V, A, A0, M = NULL, unit_costs = 1)

Arguments

Details

The algorithm that is used by optcost() is the LRNA and it is described in Wójciak (2023). The allocation computed is valid for all stratified sampling schemes for which the variance of the stratified estimator is of the form:

\sum_{h=1}^H \frac{A^2_h}{x_h} - A_0,

where H denotes total number of strata, x_1,\ldots,x_H are strata sample sizes and A_0,\, A_h > 0,\, h = 1,\ldots,H, do not depend on x_h,\, h = 1,\ldots,H.

Value

Numeric vector with optimal sample allocations in strata.

Note

For stratified \pi estimator of the population total and for stratified simple random sampling without replacement design, the population parameters are as follows:

A_h = N_h S_h, \quad h = 1,\ldots,H,

A_0 = \sum_{h=1}^H N_h S_h^2,

where N_h is the size of stratum h and S_h denotes standard deviation of a given study variable in stratum h.

References

Wójciak, W. (2023). Another Solution of Some Optimum Allocation Problem. Statistics in Transition new series, 24(5) (in press). https://arxiv.org/abs/2204.04035

See Also

rna(), opt().

Examples

A <- c(3000, 4000, 5000, 2000)
M <- c(100, 90, 70, 80)
xopt <- optcost(1017579, A = A, A0 = 579, M = M)
xopt

Example Population with 10 Strata and Lower and Upper Bounds

Description

A dataset containing the artificial population with 10 strata. Additionally, the lower and upper bounds for samples in strata are specified.

Usage

pop10_mM

Format

A matrix with 10 rows and 5 variables:

N

stratum size

S

standard deviation of study variable in stratum

m

lower bound for sample size in stratum

M

upper bound for sample size in stratum

unit_cost

cost of surveying one element in stratum

Example Population with 507 Strata

Description

A dataset containing the artificial population with 507 strata.

Usage

pop507

Format

A matrix with 507 rows and 3 variables:

N

stratum size

S

standard deviation of study variable in stratum

unit_cost

cost of surveying one element in stratum

Example Population with 969 Strata

Description

A dataset containing the artificial population with 969 strata.

Usage

pop969

Format

A matrix with 969 rows and 3 variables:

N

stratum size

S

standard deviation of study variable in stratum

unit_cost

cost of surveying one element in stratum

Random Rounding of Numbers

Description

A number x is rounded to integer y according to the following rule:

y = \left\lfloor{x}\right\rfloor + I(u < (x - \left\lfloor{x}\right\rfloor)),

where function I:\{TRUE, FALSE\} \to \{0, 1\}, is defined as:

I(x) = \begin{cases} 0, & x \text{ is } FALSE \\ 1, & x \text{ is } TRUE, \end{cases}

and u is number that is generated from Uniform(0, 1) distribution.

Usage

ran_round(x)

Arguments

Value

An integer vector.

Examples

x <- c(4.5, 4.1, 4.9)
set.seed(5)
ran_round(x) # 5 4 4
set.seed(6)
ran_round(x) # 4 4 5

RNA in version that uses prior information about violations

Description

This is the version of the RNA that makes use of additional information about strata for which the allocation can possibly be violated. For all other strata allocation will not be violated.

Usage

rna_prior(
  total_cost,
  A,
  bounds = NULL,
  check = NULL,
  check_violations = .Primitive(">="),
  details = FALSE
)

Arguments

Note

this coded was not extensively tested.

RNA - Recursive Implementation

Description

Usage

rna_rec(
  total_cost,
  A,
  bounds = NULL,
  unit_costs = rep(1, length(A)),
  check_violations = .Primitive(">=")
)

Arguments

Note

this coded was not extensively tested.

Examples

A <- c(3000, 4000, 5000, 2000)
M <- c(100, 90, 70, 80) # upper bounds.
rna_rec(total_cost = 190, A = A, bounds = M)
rna_rec(total_cost = 312, A = A, bounds = M)
rna_rec(total_cost = 339, A = A, bounds = M)
rna_rec(total_cost = 340, A = A, bounds = M)

Recursive Neyman Algorithm for Optimal Sample Allocation Under Box Constraints

Description

An internal function that implements the RNABOX algorithm that solves the following optimal allocation problem, formulated below in the language of mathematical optimization.

Minimize

f(x_1,\ldots,x_H) = \sum_{h=1}^H \frac{A^2_h}{x_h}

subject to

\sum_{h=1}^H x_h = n

m_h \leq x_h \leq M_h, \quad h = 1,\ldots,H,

where n > 0,\, A_h > 0,\, m_h > 0,\, M_h > 0, such that m_h < M_h,\, h = 1,\ldots,H, and \sum_{h=1}^H m_h \leq n \leq \sum_{h=1}^H M_h, are given numbers. The minimization is on \mathbb R_+^H. Inequality constraints are optional and can be skipped.

rnabox() function should not be called directly by the user. Use opt() instead.

Usage

rnabox(
  n,
  A,
  bounds1 = NULL,
  bounds2 = NULL,
  check_violations1 = .Primitive(">="),
  check_violations2 = .Primitive("<=")
)

Arguments

Value

Numeric vector with optimal sample allocations in strata.

References

To be added soon.

See Also

opt(), optcost(), sga(), sgaplus(), coma().

Examples

N <- c(454, 10, 116, 2500, 2240, 260, 39, 3000, 2500, 400)
S <- c(0.9, 5000, 32, 0.1, 3, 5, 300, 13, 20, 7)
A <- N * S
m <- c(322, 3, 57, 207, 715, 121, 9, 1246, 1095, 294) # lower bounds
M <- N # upper bounds

# Regular allocation.
n <- 6000
opt_regular <- rnabox(n, A, M, m)

# Vertex allocation.
n <- 4076
opt_vertex <- rnabox(n, A, M, m)

Optimal Rounding under Integer Constraints

Description

Usage

round_oric(x)

Arguments

Value

An integer vector.

References

Cont, R., Heidari, M. (2014). Optimal rounding under integer constraints. doi:10.48550/arXiv.1501.00014

Examples

x <- c(4.5, 4.1, 4.9)
round_oric(x) # 4 4 5

Variance of the Stratified Estimator

Description

Compute the value of the variance function V of the stratified estimator, which is of the following generic form:

\sum_{h=1}^H \frac{A^2_h}{x_h} - A_0,

where H denotes total number of strata, x_1,\ldots,x_H are strata sample sizes and A_0,\, A_h > 0,\, h = 1,\ldots,H, are population constants.

Usage

var_st(x, A, A0)

var_st_tsi(x, N, S)

Arguments

Value

Value of the variance V for a given allocation vector x_1,\ldots,x_H.

Functions

• var_st_tsi(): computes value of variance V for the case of stratified \pi estimator of the population total and stratified simple random sampling without replacement design. This particular case yields:

  A_h = N_h S_h, \quad h = 1,\ldots,H,

  A_0 = \sum_{h=1}^H N_h S_h^2,

  where N_h is the size of stratum h, and S_h is stratum standard deviation of a study variable, h = 1,\ldots,H.

References

Särndal, C.-E., Swensson, B. and Wretman, J. (1992). Model Assisted Survey Sampling, Chapter 3.7 Stratified Sampling, Springer, New York.

Examples

N <- c(3000, 4000, 5000, 2000)
S <- rep(1, 4)
M <- c(100, 90, 70, 80)
xopt <- opt(n = 190, A = N * S, M = M)
var_st_tsi(x = xopt, N, S) # 1017579