stratallo

https://github.com/wwojciech/stratallo

Repository

README.Rmd

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output: github_document
---



```{r, include = FALSE}
knitr::opts_chunk$set(
  collapse = TRUE,
  comment = "#>",
  fig.path = "man/figures/README-",
  out.width = "100%"
)
```

\def\R{{\mathbb R}}
\def\x{\mathbf x}
\def\n{\mathbf n}
\newcommand{\texteq}{\mathrm}

# Optimum Sample Allocation in Stratified Sampling with `stratallo`




Functions in this package provide solution to classical problem in survey
methodology - an optimum sample allocation in stratified sampling. In this
context, the optimum allocation is in the classical Tschuprow-Neyman's sense and
it satisfies additional lower or upper bounds restrictions imposed on sample
sizes in strata. In particular, it is assumed that the variance of the
stratified $\pi$ estimator is of the following generic form:

$$
  V_{st} = \sum_{h=1}^{H} \frac{A_h^2}{n_h} - A_0,
$$

where $H$ denotes total number of strata, $(n_1,\ldots,n_H)$ is the allocation
vector with strata sample sizes, and population parameters
$A_0$, $A_h > 0$, $h = 1,\ldots,H$, do not depend on the $x_h$, $h = 1,\ldots,H$.

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.

There are few different algorithms available to use, and one them is based on
a popular sample allocation method that applies Neyman allocation to recursively
reduced set of strata.
  
Package *stratallo* provides two **user functions**:

* `opt()`
* `optcost()`

that solve sample allocation problems briefly characterized above as well as 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 of *stratified* $\pi$ *estimator* of the population total with *stratified
simple random sampling without replacement* design in use.
Helper `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.

## Installation

You can install the released version of *stratallo* package from
[CRAN](https://CRAN.R-project.org) with:

``` r
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.

```{r load_package}
library(stratallo)
```

Define example population.

```{r pop}
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.
```

Tschuprow-Neyman allocation (no inequality constraints).

```{r opt_Neyman}
xopt <- opt(n = n, A = A)
xopt
sum(xopt) == n
# Variance of the st. estimator that corresponds to the optimum allocation.
var_st_tsi(xopt, N, S)
```

One-sided upper bounds.

```{r opt_M}
M <- c(100, 90, 70, 80) # Upper bounds imposed on the sample sizes in strata.
all(M <= N)
n <= sum(M)

xopt <- opt(n = n, A = A, M = M)
xopt
sum(xopt) == n
all(xopt <= M) # Does not violate upper-bounds constraints.
# Variance of the st. estimator that corresponds to the optimum allocation.
var_st_tsi(xopt, N, S)
```

One-sided lower bounds.

```{r opt_m}
m <- c(50, 120, 1, 2) # Lower bounds imposed on the sample sizes in strata.
n >= sum(m)

xopt <- opt(n = n, A = A, m = m)
xopt
sum(xopt) == n
all(xopt >= m) # Does not violate lower-bounds constraints.
# Variance of the st. estimator that corresponds to the optimum allocation.
var_st_tsi(xopt, N, S)
```

Box constraints.

```{r opt_box}
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)

xopt <- opt(n = n, A = A, m = m, M = M)
xopt
sum(xopt) == n
all(xopt >= m & xopt <= M) # Does not violate any lower or upper bounds constraints.
# Variance of the st. estimator that corresponds to the optimum allocation.
var_st_tsi(xopt, N, S)
```

Minimization of the total cost with `optcost()` function

```{r optcost}
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

xopt <- optcost(V = V, A = A, A0 = A0, M = M, unit_costs = unit_costs)
xopt
sum(A^2 / xopt) - A0 == V
all(xopt <= M)
```

Rounding.

```{r rounding}
m <- c(100, 90, 500, 50)
M <- c(300, 400, 800, 90)
n <- 1284

# Optimum, non-integer allocation under box constraints.
xopt <- opt(n = n, A = A, m = m, M = M)
xopt

xopt_int <- round_oric(xopt)
xopt_int
```

Owner

Citation (CITATION.cff)

cff-version: 1.0.0
message: "If you use this software, please cite it as below."
authors:
  - family-names: "Wójciak"
    given-names: "Wojciech"
    orcid: "https://orcid.org/0000-0002-5042-160X"
  - family-names: "Wesołowski"
    given-names: "Jacek"
    orcid: "https://orcid.org/0000-0001-7615-694X"
  - family-names: "Wieczorkowski"
    given-names: "Robert"
title: "R Package stratallo - source code"
version: 2.2.1
date-released: 2023-06-24
url: "https://github.com/wwojciech/stratallo"

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