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Two-stage Sampling

Basic Results-Estimating equations are developed for a basic two stage cluster sampling model. The superpopulation model has parameters equal to the PSU means. The assumptions is made that the number of SSUs are the same in the population for each PSU, and that an equal number of SSUs are selected for each selected PSU.

Constraints-A more detailed development is given of the unbiased constraints needed for estimating PSU means in two stage sampling.

Unequal SSUs-General expressions are given for the variance, and for the partitioned matrices that correspond to the sample and remaining portions in a two stage random permutation superpopulation model. Accompaning documents develop the expressions and define the notation used for two stage sampling.

Projection of SSUs-In a two stage cluster sample where there are equal numbers of SSUs per PSU and we are interested in estimating the PSU means, we project the basic SSU superpopulation random variables (with random variables per selection) onto a subspace that has one random variable per selection.

Selection of all PSUs-We consider a finite population of clusters (PSUs), where each PSU contains the same number (M) of units (SSUs). We select a simple random without replacement sample of all PSUs, and a simple random without replacement sample of m SSUs in each selected PSU. This problem is the simplest problem underlying problems like that for a factorial experiment, or direct adjustment of sample estimates from a population.