Optimum Sample Allocation in Stratified Sampling Schemes 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 either 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 optimium sample allocation problem leads to the minimum sample size allocation. This problems lies in the determination of a vector of strata sample sizes that minimizes total sample size, under assumed fixed level of the pi-estimator’s variance. As in the case of the classical optimal allocation, the problem of minimum sample size allocation can be complemented by imposing upper bounds constraints on sample sizes in strata.

Stratallo provides two user functions, dopt and nopt that solve sample allocation problems briefly characterized above. In this context, it is assumed that the sampling designs in strata are chosen so that the variance of the pi-estimator of the population total is of the following generic form:

where H denotes total number of strata, x₁,…,x_H are the strata sample sizes, and parameters b and a_w > 0 do not depend on x_w, w = 1,…,H.

Apart from dopt and nopt, stratallo provides var_tst and var_tst_si functions that compute a value of variance D²_st. The var_tst_si is a simple wrapper of var_tst that is dedicated for the case of simple random sampling without replacement design in each stratum. Furthermore, the package comes with two predefined, artificial populations with 507 and 969 strata. These are stored in pop507 and pop969 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 dopt and nopt functions to solve optimal sample allocation problems for an example population with 4 strata.

library(stratallo)

Function dopt

# Example population.
N <- c(3000, 4000, 5000, 2000) # Strata sizes.
S <- c(48, 79, 76, 17) # Standard deviations of a study variable in strata.
a <- N * S

M <- c(100, 90, 70, 80) # Upper bounds constraints imposed on sample sizes in strata.
all(M <= N)
n <- 190 # Total sample size.
n < sum(M)

# Optimal allocation under one-sided upper bounds constraints.
opt <- dopt(n = n, a = a, M = M)
opt
sum(opt) # Equals to n.
all(opt <= M) # Does not violate upper bounds constraints.
# Variance of the pi-estimator that corresponds to a given optimal allocation.
var_tst_si(opt, N, S)

m <- c(50, 120, 1, 1) # Lower bounds constraints imposed on sample sizes in strata.
n > sum(m)

# Optimal allocation under one-sided lower bounds constraints.
opt <- dopt(n = n, a = a, m = m)
opt
sum(opt) # Equals to n.
all(opt >= m) # Does not violate lower bounds constraints.
# Variance of the pi-estimator that corresponds to a given optimal allocation.
var_tst_si(opt, N, S)

# Tschuprov-Neyman allocation (no inequality constraints).
opt <- dopt(n = n, a = a)
opt
sum(opt) # Equals to n.
# Variance of the pi-estimator that corresponds to a given optimal allocation.
var_tst_si(opt, N, S)

Function nopt

a <- c(3000, 4000, 5000, 2000)
b <- 70000
M <- c(100, 90, 70, 80)
D <- 1e6 # Variance constraint.

opt <- nopt(D, a, b, M)
sum(opt)