Clustered Populations

Many populations are clustered, with units in a population partitioned into a large number of groups (called clusters), of which only some can represented in the sample. For example, a population of people may be grouped into households, city blocks or geographic areas, while a population of employees may be grouped into employers. The appropriate model is then usually the clustered population model. In its simplest form, values of the variable of interest are assumed to have the same expected value and variance for every unit, and the values for units in the same cluster are assumed to have a constant correlation, ?. Values for units from different clusters are assumed to be uncorrelated. The best linear unbiased predictor of a population total is derived for this model. In general, the best linear unbiased predictor depends on the value of ?. This is inconvenient, because it means that the same set of weights cannot be used for different variables of interest. In practice, however, the value of ? is usually between 0 and 0.2 and the optimal weights do not depend much on ? in this case, particularly if clusters do not vary much in size. So a single set of weights can be used for all variables of interest without much loss of efficiency. The appropriate sample design when the clustered population model applies is usually two-stage sampling. Sample design questions include: which and how many clusters to select; and how many units to select from each selected cluster. Optimal designs are developed for fixed sample size and for fixed cost, where cost is modelled as depending on the number of units and the number of clusters in sample.