User functions

The package provides three main user functions for solving optimum allocation problems:

The package also includes several helper functions:

Datasets

The package includes several artificial populations that can be used for examples and benchmarking. The available datasets can be listed with

data(package = "stratallo")

1. Optimum allocation with fixed sample size

This is the classical optimum allocation problem of determining stratum sample sizes that minimize the variance of the stratified \(\pi\) estimator of a population total for a given total sample size. Optionally, lower and/or upper bounds on the stratum sample sizes may be imposed.

Let the following be given:

• \(H\) - total number of strata,
• population parameters \(A_h > 0,\, h \in \{1,\, \ldots,\, H\}\),
• (optional) lower bounds on sample sizes in strata \(m_h \in (0,\, M_h),\, h \in \{1,\, \ldots,\, H\}\),
• (optional) upper bounds on sample sizes in strata \(M_h \in (m_h,\, N_h],\, h \in \{1,\, \ldots,\, H\}\), where \(N_h\) is the stratum size,
• total sample size \(n \in \bigl[ \sum_{h=1}^H m_h,\, \sum_{h=1}^H M_h \bigr]\).

Determine an allocation vector \(\boldsymbol{x}= (x_1,\, \ldots,\, x_H)\) that solves the following optimization problem:

\[\begin{alignat}{3} & \underset{\boldsymbol{x}~\in~{\mathbb R}_+^H}{\text{minimize}} \quad && \sum_{h=1}^H \tfrac{A_h^2}{x_h} & & \\ & \text{subject to} \quad && \sum_{h=1}^H x_h = n, & & \\ & && m_h \leq x_h \leq M_h, & \qquad h &\in \{1,\, \ldots,\, H\}, \end{alignat}\]

where the inequality constraints are optional.

This problem is solved with

opt()

The imposing of inequality constraints is achieved with the proper use of the m and M arguments of the function. For more details, see the help page for opt() function.

The opt() function uses different allocation algorithms depending on which inequality constraints are applied. Each algorithm is implemented in a separate R function, which is generally not intended to be called directly by the end user. The algorithms used for different types of constraints are:

• Lower bounds only (\(m_1,\, \ldots,\, m_H\)):
  • LRNA - rna()
• Upper bounds only (\(M_1,\, \ldots,\, M_H\)):
  • RNA - rna()
  • SGA - sga()
  • SGAPLUS - sgaplus()
  • COMA - coma()
• Box constraints (\(m_h, M_h,\, h = 1,\ldots,H\)):
  • RNABOX - rnabox().

See the documentation of each specific function for more details about the corresponding algorithm.

N <- c(3000, 4000, 5000, 2000) # Strata sizes.
S <- c(48, 179, 176, 16) # Standard deviations of a study variable in strata.
A <- N * S
n <- 190 # Total sample size.

x <- opt(n = n, A = A)
x
#> [1] 15.440181 76.772009 94.356659  3.431151

# Variance of the st. estimator that corresponds to the optimum allocation.
var_stsi(x, N, S)
#> [1] 16235763579

# Round non-integer allocation.

x_int <- round(x)
x_int
#> [1] 15 77 94  3
sum(x_int)
#> [1] 189

x_int_oric <- round_oric(x)
x_int_oric
#> [1] 16 77 94  3
sum(x_int_oric)
#> [1] 190
x_int_oric <= N
#> [1] TRUE TRUE TRUE TRUE
var_stsi(x_int_oric, N, S)
#> [1] 16243033336

M <- c(100, 90, 70, 80) # Upper bounds.
all(M <= N)
#> [1] TRUE
n <= sum(M)
#> [1] TRUE

x <- opt(n = n, A = A, M = M)
x
#> [1] 24.545455 90.000000 70.000000  5.454545

# Variance of the st. estimator that corresponds to the optimum allocation.
var_stsi(x, N, S)
#> [1] 17501100254

m <- c(100, 90, 500, 50) # Lower bounds.
M <- c(300, 400, 800, 90) # Upper bounds.
n <- 1284
n >= sum(m) && n <= sum(M)
#> [1] TRUE

x <- opt(n = n, A = A, m = m, M = M)
x
#> [1] 117.2812 400.0000 716.7188  50.0000

var_stsi(x, N, S)
#> [1] 2268937372

2. Minimum-cost allocation under a variance constraint

A minor modification of the classical optimum sample allocation problem leads to the minimum-cost allocation. This problem involves determining a vector of stratum sample sizes that minimizes the total survey cost while maintaining a fixed level of the variance of the stratified \(\pi\) estimator of a population total. As in the case of the classical optimum allocation, upper bounds on stratum sample sizes can optionally be imposed.

Let the following be given:

• \(H\) - total number of strata,
• population parameters \(A_0,\, A_h > 0,\, h \in \{1,\, \ldots,\, H\}\),
• costs \(c_h > 0,\, h \in \{1,\, \ldots,\, H\}\) of surveying one element in each stratum,
• (optional) upper bounds on sample sizes in strata \(M_h \in (0,\, N_h],\, h \in \{1,\, \ldots,\, H\}\), where \(N_h\) is the stratum size.
• level of variance \(V\). If upper bounds are imposed, it must satisfy: \(V > \sum_{h=1}^H A_h^2/M_h - A_0\). Otherwise, \(V > 0\).

Determine an allocation vector \(\boldsymbol{x}= (x_1,\, \ldots,\, x_H)\) that solves:

\[\begin{alignat}{3} & \underset{\boldsymbol{x}\in {\mathbb R}_+^H}{\text{minimize}} \quad && \sum_{h=1}^H c_h x_h & & \\ & \text{subject to} \quad && \sum_{h=1}^H \frac{A_h^2}{x_h} - A_0 = V, & & \\ & && x_h \le M_h, & \qquad h &\in \{1,\, \ldots,\, H\}, \end{alignat}\]

where the inequality constraints are optional.

This problem is solved with

optcost()

The algorithm used for this problem is the LRNA, described in Wójciak (2023).

A <- c(3000, 4000, 5000, 2000)
A0 <- 70000
unit_costs <- c(0.5, 0.6, 0.6, 0.3) # c_h, h = 1,...,4.
M <- c(100, 90, 70, 80)
V <- 1e6 # Variance constraint.
V >= sum(A^2 / M) - A0
#> [1] TRUE

optcost(V = V, A = A, A0 = A0, M = M, unit_costs = unit_costs)
#> [1] 40.39682 49.16944 61.46181 34.76805

3. Multi-domain optimum allocation with controlled precision

In many survey applications, estimates are required not only for the population as a whole but also for specific subpopulations (domains). The objective of this allocation is the simultaneous minimization of the variance of the global total estimator and the variances of domain total estimators under a fixed total sample size, while ensuring that allocations do not exceed the population sizes within strata.

The settings of this problem allow for using relative variances (instead of absolute variances) and provide explicit control of domain-wise precisions. This is achieved through the parameter \(\boldsymbol{\rho} = (\rho_d,\, d \in \mathcal D)\), where each element represents the product of the domain total \(t_d\) and the square root of the priority weight \(\kappa_d\), i.e., \(\rho_d = t_d \, \sqrt{\kappa_d},\, d \in \mathcal D\), where \(\mathcal D\) denotes the set of domain indices (see Wójciak (2026) for more details).

Let the following be given:

• \(\mathcal H\subset {\mathbb N}^2\) - the set of domain-stratum indices,
• \(\mathcal D:= \{d \in {\mathbb N}\colon\; \exists h \in {\mathbb N},\; (d,h) \in \mathcal H\}\) - the set of domain indices,
• \(\mathcal H_d := \{h \in {\mathbb N}\colon\; (d,h) \in \mathcal H\}\) - the set of strata indices in domain \(d \in \mathcal D\),
• strata sizes \(N_{d,h},\, (d,h) \in \mathcal H\),
• standard deviations of study variable in strata \(S_{d,h},\, (d,h) \in \mathcal H\),
• domain totals \(t_d > 0,\, d \in \mathcal D\),
• priority weights for domains \(\kappa_d > 0,\, d \in \mathcal D\), such that \(\sum_{d \in \mathcal D} \kappa_d = 1\),
• total sample size \(n \in (0,\, \sum_{(d,h) \in \mathcal H} N_{d,h}]\).

Determine an allocation vector \(\boldsymbol{x}= (x_{d,h},\, (d,h) \in \mathcal H)\) and scalar \(T\) that solve the following optimization problem:

\[\begin{alignat}{3} & \underset{(T,\, \boldsymbol{x})~\in~{\mathbb R}\times {\mathbb R}_+^{\lvert \mathcal H \rvert}}{\text{minimize}\quad} \quad && T & & \\ & \text{subject to} \quad && \sum_{(d,h) \in \mathcal H} x_{d,h} - n = 0, & & \label{cpda:cnstr_n} \\ & && \sum_{h \in \mathcal H_d} \Bigl( \frac{1}{x_{d,h}} - \frac{1}{N_{d,h}} \Bigr) \frac{N_{d,h}^2 S_{d,h}^2}{\rho_d^2} - T = 0, & \qquad d &\in \mathcal D, \\ & && x_{d,h} - N_{d,h} \le 0, & \qquad (d,h) &\in \mathcal H, \label{cpda:cnstr_N} \end{alignat}\]

where \((T,\, \boldsymbol{x}) = \bigl( T,\, (x_{d,h},\, (d,h) \in \mathcal H) \bigr)\) is the optimization variable, and \(\rho_d := t_d\, \sqrt{\kappa_d},\, d \in \mathcal D\).

This problem is solved with

dopt()

The algorithm used is the RDCA, described in Wójciak (2026) and implemented in rdca().

# Three domains with 2, 2, and 3 strata, respectively.
H_counts <- c(2, 2, 3)
N <- c(140, 110, 135, 190, 200, 40, 70)
S <- c(180, 20, 5, 4, 35, 9, 40)
total <- c(2, 3, 5)
kappa <- c(0.5, 0.2, 0.3)
n <- 828

dopt(n, H_counts, N, S, total, kappa)
#> [1] 140.00000 108.06261 135.00000 154.02807 200.00000  20.90933  70.00000