Cluster Sampling

Cluster sampling partitions the population into $M$ clusters (primary sampling units), selects a random subset of $m$ clusters, typically via simple random sampling of clusters, and observes units within the selected clusters, either all of them (one-stage cluster sampling) or a further subsample (two-stage or multistage sampling). Precision relative to SRS of the same total sample size is governed by the design effect, $\text{DEFF}=1+({\bar{n}}_c-1)\rho$, where ${\bar{n}}_c$ is the average cluster size and $\rho$ is the intraclass correlation coefficient measuring within-cluster homogeneity; cluster sampling is less efficient than SRS whenever $\rho >0$ (units within a cluster are more alike than two random population units), which is the typical case, but is usually far cheaper per unit sampled because it concentrates fieldwork geographically or organizationally.

• Cluster sampling's efficiency loss is driven entirely by the intraclass correlation ρ; when clusters are internally diverse (ρ near 0, like randomly assembled groups) cluster sampling loses little precision relative to SRS, but when clusters are internally homogeneous (ρ closer to 1, like households or neighborhoods that share characteristics) the design effect can be substantial.
• Multistage sampling generalizes cluster sampling to two or more levels of clustering, for example selecting a sample of school districts, then schools within districts, then classrooms within schools, trading further cost savings for a compounding of the design effect at each stage.
• Cluster sampling should not be confused with stratified sampling despite both dividing the population into groups first, stratified sampling samples from every group to reduce variance, while cluster sampling samples only a subset of groups to reduce cost, and typically increases variance rather than reducing it.