23.5 Balanced Sampling

Balanced sampling ensures that the means of auxiliary variables in the sample match those in the population. This method improves estimation efficiency and reduces variability without introducing bias.

Balanced sampling differs from purposive selection because it still involves randomization, ensuring statistical validity.

The balancing equation is given by: $$\sum_{k \in S} \frac{\mathbf{x}_k}{\pi_k} = \sum_{k \in U} \mathbf{x}_k$$ where:

• $\mathbf{x}_k$ is a vector of auxiliary variables (e.g., income, age, household size).

• $\pi_k$ is the inclusion probability of unit $k$.

• $S$ is the sample, and $U$ is the population.

This ensures that the total weighted sum of auxiliary variables in the sample matches the total sum in the population.

23.5.1 Cube Method for Balanced Sampling

The Cube Method is a widely used approach for balanced sampling, consisting of two phases:

1. Flight Phase: Ensures initial balance on auxiliary variables.

2. Landing Phase: Adjusts the sample to meet constraints while keeping randomness.

library(sampling)

# Cube method functions
samplecube()       # Standard cube method
fastflightcube()   # Optimized flight phase
landingcube()      # Landing phase method

23.5.2 Balanced Sampling with Stratification

Stratification attempts to replicate the population structure in the sample by preserving the original multivariate histogram.

library(survey)
data("api")

# Stratified design with proportional allocation
srs_design <- svydesign(data = apistrat,
                        weights = ~pw, 
                        fpc = ~fpc, 
                        strata = ~stype,
                        id = ~1)

An additional method for balanced stratification is:

balancedstratification()

This method ensures that within each stratum, the sample retains the original proportions of auxiliary variables.

23.5.3 Balanced Sampling in Cluster Sampling

Cluster sampling involves selecting entire groups (clusters) instead of individual units. A balanced approach ensures that the sampled clusters preserve the overall distribution of auxiliary variables.

library(survey)
data("api")

# Cluster sampling design
srs_design <- svydesign(data = apiclus1,
                        weights = ~pw, 
                        fpc = ~fpc, 
                        id = ~dnum)

For explicitly balanced cluster sampling:

balancedcluster()

This method ensures that the cluster-level characteristics of the sample match those of the population.

23.5.4 Balanced Sampling in Two-Stage Sampling

Two-stage sampling first selects primary units (e.g., schools, cities) and then samples within them. A balanced approach ensures representative selection at both stages.

library(survey)
data("api")

# Two-stage sampling design
srs_design <- svydesign(data = apiclus2, 
                        fpc = ~fpc1 + fpc2, 
                        id = ~dnum + snum)

For explicitly balanced two-stage sampling:

balancedtwostage()

This method ensures that auxiliary variables remain balanced across both selection stages, reducing variability while maintaining randomness.