Multi-Stage Sample

A multi-stage sample is one in which sampling is done sequentially across two or more hierarchical levels, such as first at the county level, second at the census track level, third at the block level, fourth at the household level, and ultimately at the within-household level.

Many probability sampling methods can be classified as single-stage sampling versus multi-stage sampling. Single-stage samples include simple random sampling, systematic random sampling, and stratified random sampling. In single-stage samples, the elements in the target population are assembled into a sampling frame; one of these techniques is used to directly select a sample of elements. In contrast, in multi-stage sampling, the sample is selected in stages, often taking into account the hierarchical (nested) structure of the population. The target population of elements is divided into first-stage units, often referred to as primary sampling units (PSUs), which are the ones sampled first. The selected first-stage sampling units are then divided into smaller second-stage sampling units, often referred to as secondary sampling units (SSUs), which are sampled second. This process continues until the actual elements, also referred to as the ultimate sampling units, are reached.

For example, to obtain a national sample of elementary public school students, one can divide the target population of students into elementary schools in the United States, which are used as first-stage sampling units (i.e. the PSUs). Sample schools are selected at the first stage of sampling. A sampling frame (list) of students is then assembled for each selected school. At the second stage of sampling, a sample of students is selected from each selected school. This design is a two-stage sample.

In another example, to obtain a national sample of housing units, one can divide the target population of housing units into counties, which are used as the first-stage sampling units (i.e. the PSUs). A sample of counties is then selected. Within each selected county, the target population of housing units is divided into census tracts. A sample of census tracts is drawn from within each selected county. The census tracts would be considered the SSUs. Within each selected census tract, the target population is divided into census blocks. A sample of census blocks is drawn from each selected census tract. The census blocks would be considered the third-stage sampling units. Within each selected census block, a sampling frame (list) of all housing units is assembled. A sample of housing units is then sampled from each of the selected census blocks. The housing units would be considered the fourth-stage sampling units. This design is a four-stage sample.

In both examples, the hierarchical structure of each population was used. Also note that there is a size ordering in the second example—there are more census blocks in the United States than there are census tracts, and there are more census tracts than counties. One must use an appropriate method of selection at each stage of sampling: simple random sampling, systematic random sampling, unequal probability sampling, or probability proportional to size sampling. Also, one can incorporate stratified sampling procedures to select a stratified multi-stage sample. In the previous examples, one would at a minimum want to stratify the first-stage sampling units, elementary schools and counties, by the four census regions.

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