A Model for a Nested Study

We present notation that can be used to express a simple model for a nested study. The model is based on the assumption that response has been reported for subjects selected via a two stage cluster sample. At the first stage, a simple random sample of clusters (which we will also refer to as primary sampling units (PSU)) is selected. At the second stage, a simple random sample of subjects (which we will also refer to as second stage sampling units (SSU)) in the selected clusters will be selected. We index the selection of the clusters in the sample by 1,..., i n = , and the selection of the subjects in a selected cluster as 1,..., j m = . We assume that there is a large and equal number of SSUs in each PSU, and a large number of PSUs in the population. We define the population, write a model for a random response, and then develop estimates of parameters in the model that minimize the sum of squared error.