Sampling from large populations
Two-stage sampling is a sampling scheme that is related to cluster sampling, but is of most use for large populations when the individuals are very widely separated in some sense. For example, many polls are conducted to obtain national information about voting intentions or consumer purchases, and there is a high cost associated with travelling between different regions.
In two-stage sampling, the population is separated into groups of adjacent individuals called primary sampling units. These primary sampling units are typically large — for example a country might be split into 20 or more regions. A small number of these are selected according to some sampling scheme, then individuals are sub-sampled within each selected primary unit.
Costs are reduced by limiting sampling to a small number of primary units. For example, if individuals are only sampled from within say 5 regions, travelling and accommodation costs will be considerably reduced.
The diagram below is a small-scale illustration of two-stage sampling of 16 values from a population of 128 individuals.
Click Take sample to select all 16 primary sampling units and a sample of size 1 from each. This is identical to a stratified random sample.
Change the control Primary units sampled to 2 primary units. Click Take sample to randomly select two of the primary units, with all individuals in each unit being sampled. Since complete primary units are sampled, this extreme is identical to cluster sampling.
Finally, change the control to select either 4 or 8 primary units. Taking a sample now involves a random selection of primary units, followed by a random selection of individuals within each. Typically there are large numbers of individuals and primary units so random sampling is needed for both primary units and individuals within them.
Cost and accuracy (advanced)
In the example above, there was a considerable cost involved with travel between the primary units, so the total cost is reduced when fewer primary units are sampled. Unfortunately, the accuracy of the resulting estimate is usually lower in this situation.
The number of primary units to sample is therefore a trade-off between accuracy and cost. The details are beyond the scope of CAST.
In the diagram below, the vertical bands separate 16 primary units, each of which contains 8 individuals whose values are shown in a jittered dot plot.
Initially, all primary units are similar — their means are nearly the same. Click Take sample several times to observe the sampling distribution of the mean (in the pink band to the right of the population dot plots). When the number of primary units sampled is changed, the distribution of the mean remains the same. The most cost-effective sampling scheme is therefore to sample the smallest number of primary units possible.
Change the slider at the bottom to half-way between Similar and Different. Now some primary units are fairly consistently higher than others. Observe that reducing the number of primary units sampled now has a much greater effect on the standard deviation (and hence accuracy) of the mean.
As with cluster and stratified sampling, formulae for the standard deviation of the resulting estimates are beyond the scope of an introductory course.