# Optimum Sample Allocation in Stratified Sampling with stratallo Package

Functions in this package provide solution to classical problem in survey methodology - an optimum sample allocation in stratified sampling schemes. In this context, the optimal allocation is in the classical Tschuprov-Neyman’s sense and it satisfies additional lower or upper bounds restrictions imposed on sample sizes in strata. There are few different algorithms available to use, and one them is based on popular sample allocation method that applies Neyman allocation to recursively reduced set of strata.

A minor modification of the classical optimum sample allocation problem leads to the minimum cost allocation. This problem lies in the determination of a vector of strata sample sizes that minimizes total cost of the survey, under assumed fixed level of the stratified $$\pi$$-estimator’s variance. As in the case of the classical optimum allocation, the problem of minimum cost allocation can be complemented by imposing upper bounds on sample sizes in strata.

stratallo provides two user functions:

• opt()
• optcost()

that solve sample allocation problems briefly characterized above. In this context, it is assumed that the variance of the stratified estimator is of the following generic form: $V_{st}(\mathbf n) = \sum_{h=1}^{H} \frac{A_h^2}{n_h} - A_0,$ where $$H$$ denotes total number of strata, $$\mathbf n= (n_h)_{h \in \{1,\ldots,H\}}$$ is the allocation vector with strata sample sizes, and population parameters $$A_0$$, and $$A_h > 0,\, h = 1,\ldots,H$$, do not depend on the $$x_h,\, h = 1,\ldots,H$$.

Among stratified estimators and stratified sampling designs that jointly give rise to a variance of the above form, is the so called stratified $$\pi$$ estimator of the population total with stratified simple random sampling without replacement design, which is one of the most basic and commonly used stratified sampling designs. This case yields $$A_0 = \sum_{h = 1}^H N_h S_h^2$$, $$A_h = N_h S_h,\, h = 1,\ldots,H$$, where $$S_h$$ denotes stratum standard deviation of study variable and $$N_h$$ is the stratum size (see e.g. @sarndal, Result 3.7.2, p.103).

Apart from opt() and optcost(), stratallo provides the following helpers functions:

• var_st(),
• var_st_tsi(),
• asummary(),
• ran_round(),
• round_oric().

Functions var_st() and var_st_tsi() compute a value of the variance $$V_{st}$$. The var_st_tsi() is a simple wrapper of var_st() that is dedicated for the case when $$A_0 = \sum_{h = 1}^H N_h S_h^2$$ and $$A_h = N_h S_h,\, h = 1,\ldots,H$$. asummary() creates a data.frame object with summary of the allocation. Functions ran_round() and round_oric() are the rounding functions that can be used to round non-integers allocations (see section Rounding, below). The package comes with three predefined, artificial populations with 10, 507 and 969 strata. These are stored under pop10_mM, pop507andpop969 objects, respectively.

See package’s vignette for more details.

## Installation

You can install the released version of stratallo package from CRAN with:

install.packages("stratallo")

## Examples

These are basic examples that show how to use opt() and optcost() functions to solve different versions of optimum sample allocation problem for an example population with 4 strata.

library(stratallo)

Define example population

N <- c(3000, 4000, 5000, 2000) # Strata sizes.
S <- c(48, 79, 76, 16) # Standard deviations of a study variable in strata.
a <- N * S
n <- 190 # Total sample size.

Tschuprov-Neyman allocation (no inequality constraints).

opt <- opt(n = n, a = a)
opt
#> [1] 31.376147 68.853211 82.798165  6.972477
sum(opt) == n
#> [1] TRUE
# Variance of the stratified estimator that corresponds to optimum allocation.
var_st_tsi(opt, N, S)
#> [1] 3940753053

One-sided upper-bounds constraints.

M <- c(100, 90, 70, 80) # Upper bounds imposed on the sample sizes in strata.
all(M <= N)
#> [1] TRUE
n <= sum(M)
#> [1] TRUE

# Solution to Problem 1.
opt <- opt(n = n, a = a, M = M)
opt
#> [1] 35.121951 77.073171 70.000000  7.804878
sum(opt) == n
#> [1] TRUE
all(opt <= M) # Does not violate upper-bounds constraints.
#> [1] TRUE
# Variance of the stratified estimator that corresponds to optimum allocation.
var_st_tsi(opt, N, S)
#> [1] 4018789143

One-sided lower-bounds constraints.

m <- c(50, 120, 1, 2) # Lower bounds imposed on the sample sizes in strata.
n >= sum(m)
#> [1] TRUE

# Solution to Problem 2.
opt <- opt(n = n, a = a, m = m)
opt
#> [1]  50 120  18   2
sum(opt) == n
#> [1] TRUE
all(opt >= m) # Does not violate lower-bounds constraints.
#> [1] TRUE
# Variance of the stratified estimator that corresponds to optimum allocation.
var_st_tsi(opt, N, S)
#> [1] 9719807556

Box constraints.

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

# Optimum allocation under box-constraints.
opt <- opt(n = n, a = a, m = m, M = M)
opt
#> [1] 228.9496 400.0000 604.1727  50.8777
sum(opt) == n
#> [1] TRUE
all(opt >= m & opt <= M) # Does not violate any lower or upper bounds constraints.
#> [1] TRUE
# Variance of the stratified estimator that corresponds to optimum allocation.
var_st_tsi(opt, N, S)
#> [1] 538073357

Minimization of the total cost with optcost() function

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

opt <- optcost(V = V, a = a, a0 = a0, M = M, unit_costs = unit_costs)
opt
#> [1] 40.39682 49.16944 61.46181 34.76805
sum(a^2 / opt) - a0 == V
#> [1] TRUE
all(opt <= M)
#> [1] TRUE

Rounding.

m <- c(100, 90, 500, 50)
M <- c(300, 400, 800, 90)
n <- 1284

# Optimum, non-integer allocation under box-constraints.
opt <- opt(n = n, a = a, m = m, M = M)
opt
#> [1] 297.4286 396.5714 500.0000  90.0000

opt_int <- round_oric(opt)
opt_int
#> [1] 297 397 500  90`