opt: Optimum Sample Allocation in Stratified Sampling

In stratallo: Optimum Sample Allocation in Stratified Sampling

Optimum Sample Allocation in Stratified Sampling

Description

Computes the optimum allocation for the following optimum allocation problem, formulated in mathematical optimization terms:

Minimize

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

over \mathbb R_+^H, subject to

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

m_h \leq x_h \leq M_h, \qquad 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. Inequality constraints are optional and may be omitted.

Inequality constraints are optional, and the user can choose whether and how they are applied to the optimization problem. This is controlled using the m and M arguments as follows:

• No inequality constraints: both m and M must be NULL (default).

• Lower bounds only (m_1,\, \ldots,\, m_H): specify m, and set M = NULL.

• Upper bounds only (M_1,\, \ldots,\, M_H): specify M, and set m = NULL.

• Box constraints (m_h, M_h,\, h = 1,\ldots,H): specify both m and M.

Usage

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

Arguments

n	(integerish(1)) total sample size. Must satisfy n > 0. Additionally: If bounds_inner is not NULL, then n >= sum(bounds_inner) when bounds_inner are treated as lower bounds, or n <= sum(bounds_inner) when treated as upper bounds. If bounds_outer is not NULL, then n >= sum(bounds_outer) when bounds_outer are treated as lower bounds, or n <= sum(bounds_outer) when treated as upper bounds.
A	(numeric) population constants A_1,\ldots,A_H. All values must be strictly positive.
m	(numeric or NULL) optional lower bounds m_1,\ldots,m_H for the stratum sample sizes. If no lower bounds are desired, set m = NULL. If M is not NULL, it is required that m_h < M_h for all strata.
M	(numeric or NULL) optional upper bounds M_1,\ldots,M_H for the stratum sample sizes. If no upper bounds are desired, set M = NULL. If m is not NULL, it is required that m_h < M_h for all strata.
M_algorithm	(string) Name of the algorithm to use for computing the sample allocation when only upper-bound constraints are imposed. Must be one of "rna" (default), "sga", "sgaplus", or "coma". This parameter is used only when H > 1 and n < sum(M).

Details

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 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.

Value

A numeric vector of the optimal sample allocations for each stratum.

Note

If no inequality constraints are applied, the allocation follows the Neyman allocation:

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

For a stratified \pi estimator of the population total using stratified simple random sampling without replacement design, the objective function parameters A_h are:

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

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

References

\insertRef

Sarndalstratallo

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)
x <- opt(n = 800, A = A, m = m, M = M)
x

# Variance corresponding to the allocation x.
var_st(x = x, A = A, A0 = 45000)

# Execution-time comparison of different algorithms using the microbenchmark package.
## Not run: 
N <- pop969s_ucost[, "N"]
S <- pop969s_ucost[, "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)

stratallo documentation built on March 12, 2026, 5:06 p.m.