In many surveys, the population can be divided into a set of groups of units. One approach is then to treat these groups as strata and select a separate sample from each group. An alternative approach is to treat these groups as clusters, in which case only a sample of them is included in the survey.
If all the units in the selected clusters are included in the sample, the method is regarded as cluster sampling. These two approaches have already been discussed. If only a sample of units is selected from each selected cluster, the method is known as two-stage sampling.
Frequently, a hierarchy of clusters is used. Initially, a population is divided into several large clusters of elements, called primary sampling units (PSU).
Following this, several primary sampling units are selected at random, and from those selected primary sampling units, some smaller units, called secondary sampling units (SSU), are selected again at random.
The process continues until, finally, units are selected within the ultimate units. As the selection of the sample is made in more than one stage, it is known as the multi-stage sampling.
In multi-stage sampling, the primary sampling unit (PSU) is generally a compact geographical area, say, for instance.
The second stage unit (SSU), may, in that case, be a local unit, such as a village in rural areas block of contiguous housing units) in urban areas.
The next stage of selection may involve the selection of households or housing units. If no further stage is needed, then the elements in this stage are called the ultimate sampling unite (USU), and the second stage units may be described as penultimate sampling unite (PUSU).
The multi-stage sampling is, in essence, a process of sub-sampling.
Once the primary sampling units are selected at the first stage, the subsequent stages are nothing but a process of sub-sampling from the units chosen at the immediate previous stages.
Thus the second stage sample of units is chosen from the first stage sample of units. The third stage sample of units are chosen from the second stage sample of units, and so on.
In multi-stage sampling, the fewer the stages of selection, the better and more efficient the sample would be.
A sampling at each stage, down to the penultimate stage, can be done by a variety of procedures e.g., simple random sampling, sampling with probability proportional to size (PPS), systematic sampling, and systematic sampling with PPS.
At the ultimate stage, USUs within selected PUSU can also be selected by different methods; one can select a simple random sample or a stratified random sample of USUs, or one can divide the PUSU into clusters of households and select a number of clusters.
Generally, PSUs, SSUs, and other intermediate stage units down to PUSUs are selected with probability proportional to size, and USUs are selected by simple random sampling or systematic sampling with equal probability.
In some instances, however, the USUs are sampled in the form of clusters of contiguous units.