In single-stage sampling, the members in the target population are first collected into a sampling frame. Then, either the simple random sampling or the systematic random sampling is used to draw a random sample from the sampling frame. In multi-stage sampling, sample selection is carried out in stages, using smaller and smaller sampling units at each stage. It takes the nested structure of the population or an area into account. The target population is divided into naturally-occurring clusters or non-overlapping groups (i.e. strata). From these clusters and strata, a simple random sampling is selected. Cluster and stratified sampling are two examples of multistage sampling.
Cluster sampling is a type of sampling which involves. Two steps are involved in cluster sampling:
Divide the population into groups (or clusters).
Randomly select one or more clusters and every element within the selected cluster is sampled.
Stratified sampling is useful when we want to improve the precision in a heterogeneous target population. In other words, this sampling approach is useful when we want to study subpopulations (male/female, geographic regions, racial groups, etc.). Two steps are involved:
1. The target population is divided into non-overlapping groups called strata. Strata should be created in such a way to minimize the within stratum variance and to maximize the between stratum variance.
2. Sampling is performed separately within each stratum. For instance, select a sample random sample from each stratum.
Auxiliary information (variables) can be used to set up homogenous strata. Usually, variables related to the outcome variable (Y) are used as auxiliary information. A more precise estimate can be obtained within each stratum.
It can also be used when auxiliary information (variables) that are used to set up homogenous strata are available. There were essentially three reasons for stratifying: 1) homogeneity - to produce reliable estimates for the subpopulations, 2) to improve the sampling efficiency and thereby improving the reliability of overall estimates of the target population, and 3) to ensure that different parts of the population are appropriately represented in the sample (Gonzalez and Miles 2001).
For instance, the TIMSS uses a two-stage sampling procedure to ensure a nationally representative sample of students (Gonzalez and Miles 2001). The first stage selects and assigns a comprehensive national list of all eligible schools to predetermined strata. Using a systematic probability-proportional-to-size (PPS) technique, approximately 150 schools were randomly selected from all secondary schools in each participating country. The probability of selection for a school was proportional to the number of eighth grade students in the school. Stratification by region and urbanization was used to ensure that urban and rural schools in all states were represented. At the second sampling stage, one or two eligible classrooms of eighth grade students within each sampled school were randomly selected (Gonzalez and Miles 2001).
Here are a few ways to allocate sampling units across strata in stratified sampling:
Two important points to consider when allocating the sample among strata:
1. Simplicity: Can be easily presented in a table (e.g. proportional stratified sampling)
2. Precision: the chosen allocation technique should provide the smallest sampling variance (or
sampling error)
The allocation rules specify how the sampling units are allocated among the strata. It is influenced by within-stratum variance, data collection costs in each strata, and the number of sampling units in each stratum.
Multistage random sampling is a combination of cluster and stratified sampling. Under this approach, a simple random sample is selected within each chosen cluster. Stratification may be done at each selection stage. It involves the following stages:
An area (e.g. country) is divided into smaller regions (e.g. state).
A random sample of these regions is collected.
Each selected region is further divided into even smaller geographic subdivisions (e.g. counties).
A random sample of smaller geographic subdivisions is taken from within each of the selected regions.
Each selected geographic subdivisions is further divided into even smaller geographic subdivisions (e.g. counties).
A random sample of even smaller geographic subdivisions (e.g. Census blocks) is taken from within each of the selected geographic subdivisions in Step 3.
Stratified sampling can be done at stages 2, 4, and 6. In other words, the random sample can be stratified by demographic characteristics (e.g. sex, race, per capita income, consumption spending, unemployment rate, and so forth). This process will continue until the sizes of these geographic subdivisions are deemed too small for the purpose of the study.
It is used when members of the target population are too large or too scattered (e.g. all the church members or voters in America). This makes it impossible to obtain a complete list of the members of the target population and draw a simple random sampling from the list.
Cluster and stratified sampling are substantially different. In cluster sampling, we select the clusters and all members of that cluster. In stratified sampling, a random sample is drawn from all the strata.
Gonzalez, E. J. and J. A. Miles. 2001. TIMSS 1999 User Guide for the International Database. Chestnut Hill, MA: International Study Center, Lynch School of Education, Boston College.