Multistage 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.