Going beyond two stages of sampling is straightforward as long as the invariance and independence properties hold. However, the notation for general r-stage sampling is cumbersome. Three types of sampling units are described below. The term element sampling means that the third-stage sampling units are the populations element. A typical example would be to designate schools as PSUs, classes as SSUs, and the students in each class as elements. Let us now introduce the notation:

1. The N elements of P are partitioned into PSUs U1, U2 . . . , Ui, . . . UNI . The set of PSUs is denoted by UI = {1, . . . , i, . . . , NI}. With Ni being the size of Ui we have N =

∑ i∈UI Ni.