Cluster Sampling in R

Cluster sampling, a widely used technique in statistics and data analysis, offers a practical solution for sampling from large populations. By dividing the population into clusters and selecting a subset of these clusters for analysis, researchers can efficiently collect representative samples while reducing logistical challenges. Here we delve into the principles and applications of cluster sampling and demonstrate how to implement it using R Programming Language.

What is Cluster Sampling?

Cluster sampling is a sampling technique used in statistics and research methodology where the population is divided into groups or clusters, and then a random sample of these clusters is selected for analysis. Instead of individually sampling each element of the population, cluster sampling involves selecting entire groups or clusters and then sampling within those clusters.

How to Perform Cluster Sampling

1. Define the Population: Identify the target population that you want to study. This could be households, schools, hospitals, etc.
2. Define the Clusters: Divide the population into clusters. Clusters are naturally occurring groups within the population, such as geographical regions (e.g., cities, states), organizational units (e.g., schools, hospitals), or any other identifiable group.
3. Randomly Select Clusters: Use a random sampling technique to select a subset of clusters from the population. Ensure that every cluster has an equal chance of being selected to avoid bias.
4. Collect Data from Selected Clusters: Once the clusters are selected, collect data from all units within each selected cluster or from the sampled units within each cluster. This could involve conducting surveys, interviews, observations, or accessing existing records and databases.

Types of Cluster Sampling

1.Single-Stage Cluster Sampling – Entire clusters are randomly selected, and all elements within those clusters are sampled.

Suppose a marketing research firm wants to estimate the satisfaction level of customers in a large city’s supermarkets. Instead of surveying all customers in every supermarket (which could be impractical), they opt for single-stage cluster sampling. Randomly selecting three supermarkets (Cluster A, Cluster B, and Cluster C) from the city’s list of supermarkets and surveying all customers in each selected supermarket to estimate overall customer satisfaction.

R

# Sample Data Generation
set.seed(123)
population <- data.frame(
  Supermarket = paste("Supermarket", 1:1000, sep = "_"),
  CustomerSatisfaction = rnorm(1000, mean = 75, sd = 10)
)

# Single-Stage Cluster Sampling
selected_supermarkets <- sample(population$Supermarket, size = 10, replace = FALSE)
sampled_data <- population[population$Supermarket %in% selected_supermarkets, ]

# Display Sampled Data
head(sampled_data)

Output:

        Supermarket CustomerSatisfaction
203 Supermarket_203             72.34855
225 Supermarket_225             71.36343
255 Supermarket_255             90.98509
354 Supermarket_354             76.16637
457 Supermarket_457             86.10277
554 Supermarket_554             77.49825

2. Two-Stage Cluster Sampling – Two levels of random sampling are involved. First, clusters are randomly selected from the population, and then a sample of elements is randomly selected from within each chosen cluster.

A government agency aims to understand the employment status of households in a particular region. Since surveying every household is time-consuming and costly, they employ two-stage cluster sampling. Randomly selecting three neighborhoods (Cluster X, Cluster Y, and Cluster Z) from a region and then, within each neighborhood, randomly selecting and surveying households to gather employment data.

R

# Sample Data Generation
set.seed(123)
region <- data.frame(
  Neighborhood = paste("Neighborhood", 1:500, sep = "_"),
  AverageIncome = rnorm(500, mean = 50000, sd = 10000)
)
households <- data.frame(
  Neighborhood = rep(sample(region$Neighborhood, size = 500, replace = TRUE), 
                     each = 20),
  HouseholdID = rep(1:20, times = 500),
  EmploymentStatus = sample(c("Employed", "Unemployed"), size = 10000, replace = TRUE)
)

# Two-Stage Cluster Sampling
selected_neighborhoods <- sample(region$Neighborhood, size = 5, replace = FALSE)
sampled_households <- households[households$Neighborhood %in% selected_neighborhoods, ]

# Display Sampled Data
head(sampled_households)

Output:

         Neighborhood HouseholdID EmploymentStatus
1981 Neighborhood_302           1       Unemployed
1982 Neighborhood_302           2         Employed
1983 Neighborhood_302           3         Employed
1984 Neighborhood_302           4         Employed
1985 Neighborhood_302           5       Unemployed
1986 Neighborhood_302           6       Unemployed

3. Multistage Cluster Sampling – Involves more than two stages of sampling, with clusters being selected at multiple levels or stages.

A national health institute wants to assess vaccination rates across the country. Given the vast geographic and demographic diversity, they employ multi-stage cluster sampling. Randomly selecting three states (Cluster P, Cluster Q, and Cluster R) from the country, then within each state, randomly selecting two counties, and finally, within each county, randomly selecting and surveying vaccination centers to gather vaccination rate data.

R

# Sample Data Generation
set.seed(123)
states <- data.frame(
  State = paste("State", 1:50, sep = "_"),
  Population = sample(1000000:5000000, 50, replace = TRUE)
)
counties <- data.frame(
  State = rep(sample(states$State, size = 50, replace = TRUE), each = 20),
  County = rep(paste("County", 1:20, sep = "_"), times = 50),
  VaccinationRate = rnorm(1000, mean = 70, sd = 5)
)

# Multi-Stage Cluster Sampling
selected_states <- sample(states$State, size = 3, replace = FALSE)
selected_counties <- sample(counties$County[counties$State %in% selected_states], 
                            size = 5, replace = FALSE)
sampled_vaccination_centers <- counties[counties$County %in% selected_counties, ]

# Display Sampled Data
head(sampled_vaccination_centers)

Output:

      State    County VaccinationRate
8  State_32  County_8        70.37428
11 State_32 County_11        66.86024
13 State_32 County_13        70.81309
15 State_32 County_15        67.68222
19 State_32 County_19        70.91839
28 State_46  County_8        68.84869

Cluster Sampling in R on Iris Dataset

R

# Set seed for reproducibility
set.seed(123)

# Load the iris dataset
data(iris)

# Randomly choose 2 species out of the available ones for the first stage
selected_clusters <- sample(unique(iris$Species), size = 2, replace = FALSE)

# Define sample as all observations belonging to one of the selected species
cluster_sample <- iris[iris$Species %in% selected_clusters, ]

# View the selected species for the first stage
cat("\nSelected species for the first stage (Clusters):\n")
print(selected_clusters)

# Randomly choose 1 observation from each selected species for the second stage
observations_per_species <- 1
sampled_observations <- lapply(selected_clusters, function(species) {
  species_observations <- rownames(iris[iris$Species == species, ])
  sampled_observation <- sample(species_observations, 
                                size = observations_per_species, replace = FALSE)
})

# Combine the sampled observations into one data frame
cluster_sample <- iris[sampled_observations[[1]], ]
for (i in 2:length(sampled_observations)) {
  cluster_sample <- rbind(cluster_sample, iris[sampled_observations[[i]], ])
}

# View the selected observations for the second stage
cat("\nSelected observations for the second stage (Individual elements):\n")
print(rownames(cluster_sample))

Output:

Selected species for the first stage (Clusters):

[1] virginica setosa   
Levels: setosa versicolor virginica

Selected observations for the second stage (Individual elements):
[1] "114" "3"  

It indicates the selected species for the first stage of the two-stage cluster sampling process.

• The printed list shows the names of the randomly chosen species: “virginica” and “setosa”.
• Selected observations for the second stage of the two-stage cluster sampling process, where one observation is randomly chosen from each selected species.
• Printed list shows the row numbers of the individual observations that were sampled within each selected species. In this case, row 114 corresponds to a species of “virginica”, and row 3 corresponds to a species of “setosa”.

Applications

1. Educational Studies:- Sampling schools or classrooms as clusters to study student performance.
2. Health Surveys:- Sampling medical facilities as clusters to assess patient demographics and health outcomes.
3. Market Research:- Sampling cities or neighborhoods as clusters to analyze consumer behavior and preferences.

Advantages

1. Reduces resources required for data collection and analysis, especially in geographically dispersed populations.
2. Easier to implement compared to other methods, suitable for large-scale surveys or studies.
3. Convenient when the population naturally divides into clusters, facilitating access to sampling units.
4. Focuses resources on a smaller number of clusters, improving sampling and data collection efficiency.

Disadvantages

1. Introduces additional variability due to similarities within clusters, leading to higher sampling errors.
2. May reduce precision compared to other methods, especially with heterogeneous clusters.
3. Risk of bias if clusters are not representative or vary in size, affecting sample representativeness.
4. Requires specialized statistical techniques to account for clustering and obtain unbiased estimates.

Conclusion

Cluster sampling in R offers a practical and efficient approach to sampling when dealing with large and geographically dispersed populations. By selecting entire clusters rather than individual elements, it provides a cost-effective and logistically convenient method for data collection and analysis. While it simplifies the sampling process, researchers must be mindful of its potential drawbacks, such as increased variability and the need for specialized analysis techniques. Overall, cluster sampling in R remains a valuable tool for researchers seeking to obtain representative samples from diverse populations.