`stratallo`

PackageFunctions 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`

, pop507`and`

pop969` objects,
respectively.

See package’s vignette for more details.

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

`install.packages("stratallo")`

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

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

Tschuprov-Neyman allocation (no inequality constraints).

```
<- opt(n = n, a = a)
opt
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.

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

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

```
<- 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.
M <- 1284
n >= sum(m) && n <= sum(M)
n #> [1] TRUE
# Optimum allocation under box-constraints.
<- opt(n = n, a = a, m = m, M = M)
opt
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

```
<- c(3000, 4000, 5000, 2000)
a <- 70000
a0 <- c(0.5, 0.6, 0.6, 0.3) # c_h, h = 1,...4.
unit_costs <- c(100, 90, 70, 80)
M <- 1e6 # Variance constraint.
V >= sum(a^2 / M) - a0
V #> [1] TRUE
<- optcost(V = V, a = a, a0 = a0, M = M, unit_costs = unit_costs)
opt
opt#> [1] 40.39682 49.16944 61.46181 34.76805
sum(a^2 / opt) - a0 == V
#> [1] TRUE
all(opt <= M)
#> [1] TRUE
```

Rounding.

```
<- c(100, 90, 500, 50)
m <- c(300, 400, 800, 90)
M <- 1284
n
# Optimum, non-integer allocation under box-constraints.
<- opt(n = n, a = a, m = m, M = M)
opt
opt#> [1] 297.4286 396.5714 500.0000 90.0000
<- round_oric(opt)
opt_int
opt_int#> [1] 297 397 500 90
```