# Properties of Complex Samples

A complex sample can differ from a simple random sample in many ways. In a simple random sample, individual sampling units are selected at random with equal probability and without replacement (WOR) directly from the entire population. By contrast, a given complex sample can have some or all of the following features:

Stratification. Stratified sampling involves selecting samples independently within non-overlapping subgroups of the population, or strata. For example, strata may be socioeconomic groups, job categories, age groups, or ethnic groups. With stratification, you can ensure adequate sample sizes for subgroups of interest, improve the precision of overall estimates, and use different sampling methods from stratum to stratum.

Clustering. Cluster sampling involves the selection of groups of sampling units, or clusters. For example, clusters may be schools, hospitals, or geographical areas, and sampling units may be students, patients, or citizens. Clustering is common in multistage designs and area (geographic) samples.

Multiple stages. In multistage sampling, you select a first-stage sample based on clusters. Then you create a second-stage sample by drawing subsamples from the selected clusters. If the second-stage sample is based on subclusters, you can then add a third stage to the sample. For example, in the first stage of a survey, a sample of cities could be drawn. Then, from the selected cities, households could be sampled. Finally, from the selected households, individuals could be polled. The Sampling and Analysis Preparation wizards allow you to specify three stages in a design.

Nonrandom sampling. When selection at random is difficult to obtain, units can be sampled systematically (at a fixed interval) or sequentially.

Unequal selection probabilities. When sampling clusters that contain unequal numbers of units, you can use probability-proportional-to-size (PPS) sampling to make a cluster's selection probability equal to the proportion of units it contains. PPS sampling can also use more general weighting schemes to select units.

Unrestricted sampling. Unrestricted sampling selects units with replacement (WR). Thus, an individual unit can be selected for the sample more than once.

Sampling weights. Sampling weights are automatically computed while drawing a complex sample and ideally correspond to the "frequency" that each sampling unit represents in the target population. Therefore, the sum of the weights over the sample should estimate the population size. Complex Samples analysis procedures require sampling weights in order to properly analyze a complex sample. Note that these weights should be used entirely within the Complex Samples option and should not be used with other analytical procedures via the Weight Cases procedure, which treats weights as case replications.