Cluster Random Sampling

About cluster Random Sampling

Cluster random sampling is a probability sampling technique used when a complete list of individuals (sampling frame) is unavailable, but clusters (groups) can be identified.

When to Use

• Suitable when populations lack individual lists (e.g., students across Jakarta universities).
• Example: Surveying opinions of UI students vs. all Jakarta students—UI has a list, Jakarta does not.

Procedure

1. Identify clusters: Define groups (e.g., universities, hospitals) containing individuals.
2. Select clusters randomly: Choose a subset of clusters (e.g., 4 out of 50 universities).
3. Sample within clusters: Randomly select individuals from chosen clusters using simple, systematic, or stratified sampling.

Advantages

1. Reduces effort compared to sampling all individuals across a population.
2. Feasible when individual data is inaccessible (e.g., only cluster lists exist).

Parameter Estimation

• Population size $N$ split into $M$ clusters, sizes $N_1,N_2,\dots ,N_M$.
• Sample $m$ clusters, sizes $n_1,n_2,\dots ,n_m$, total sample $n=n_1+n_2+\cdots +n_m$.
• Total estimator: $\hat{X}=\frac{M}{m}{\sum }_{i=1}^m\frac{N_i}{n_i}{\sum }_{j=1}^{n_i}{x}_{ij}$.
• Mean estimator: $\bar{\hat{X}}=\frac{\hat{X}}{N}$.
• Unbiased: $E(\hat{X})=X$.

Variance Analysis

• Variance: $V(\hat{X})=M^2\left(\frac{M-m}{M}\frac{S_b^2}{m}\right)+\frac{M}{m}{\sum }_{i=1}^M{N}_i^2\left(\frac{N_i-n_i}{N_i}\frac{S_i^2}{n_i}\right)$.
• $S_b^2$: Between-cluster variance.
• $S_i^2$: Within-cluster variance.
• Estimator: $\hat{V}(\hat{X})$ uses sample variances $s_b^2$ and $s_i^2$, unbiased for $V(\hat{X})$.

Sample Allocation

• Optimum allocation: Minimize variance given cost $c=c_{1}m+c_{2}m\bar{n}$.
• Result: $\bar{n}=\sqrt{\frac{c_1{S}_{2i}^2}{c_2{S}_{1b}^2}}$, $m=\frac{c}{c_1+c_2\bar{n}}$.