Multistage Sample vs Cluster Sample

A multistage sample selects in layers. A random mechanism picks large units first, a second random mechanism picks smaller units inside only the ones already chosen, and so on until the last stage reaches individuals. What matters is the list you never have to build. You need a list of the large units, and then a list of individuals only inside the handful you selected.

Suppose a state has 900 school districts, each holding 20 schools, each school 500 students, so 9,000,000 students in all. Randomly select 30 districts, then 4 schools inside each selected district, then 25 students inside each selected school. That is $30 \times 4 \times 25 = 3000$ students, and rosters were needed for only $30 \times 4 = 120$ schools. A given student's chance of selection is the product across the stages, $\frac{30}{900} \times \frac{4}{20} \times \frac{25}{500} = \frac{1}{3000}$, matching 3,000 students out of 9,000,000.

The usual error is "this is a cluster sample." A cluster sample picks groups at random and then measures everyone inside the chosen ones, and it stops there. Take all 500 students in each of the 120 schools and that is what you have. Sampling 25 of them instead adds a stage. Nor is it a simple random sample, even though every student's chance is the same $1/3000$: no sample can hold students from more than 30 districts, so most groups of 3,000 are impossible.

The standard error formulas in this course assume a simple random sample, so they do not transfer to a multistage design. Students inside one school also resemble each other more than two students drawn from the state at random, which usually makes a multistage sample less precise than an SRS of the same size. The course names four random sampling methods, simple random, stratified, cluster and systematic, so describe a multistage design stage by stage rather than reaching for a fifth label.

Clusters are non-overlapping groups that together cover the population, and a good cluster holds the same mix the population does, which leaves the clusters resembling one another. The randomness acts on the clusters, not on people: take a simple random sample of clusters, then collect data from every individual inside the ones drawn. Each cluster is measured completely or not at all.

Use the same school of 1,200 students that the stratified plan uses, this time split into 60 advisory groups of 20 that each mix all four grades. Draw 4 advisories at random and survey all 20 students in each: $4 \times 20 = 80$ students out of 4 groups, with the other 56 advisories contributing nobody. Every student still has a $4/60 = 1/15$ chance of being in the sample, the same rate the stratified plan gives.

That equal chance is where the misreading starts. "Every student had the same chance of being picked, so this is a simple random sample." It is not. An SRS requires every set of 80 students to be equally likely, and here only $\binom{60}{4} = 487{,}635$ samples can occur at all. Two students in the same advisory are either both in or both out, and no sample ever spans five different advisories. Equal chance for each individual is a weaker condition than the SRS condition.

The price of clustering is variability, not bias. Selection is random, so the method has no systematic lean, but neighbors resemble neighbors, so 80 students from 4 advisories usually carry less information than 80 drawn one at a time. More clusters therefore beat bigger clusters at a fixed total, and clusters built to be alike inside, one advisory per grade for instance, are the worst case. What is a cluster sample works through that cost.

Cluster sampling is Unit 1, topic 1.11 Random Sampling. Read the two divisions off a scenario and the naming question answers itself.