Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 126

next hierarchical level. We may select all of these units, or only some of them
according to a random criterion.
From the above framework, it is worth noting that stratified sampling is only a
particular case of multi-stage sampling, in which we define the selection criteria for
any aggregation level of a hierarchical partition of the frame. In fact, the procedure
followed is to choose all the groups, and perform an SRS within each of them.
Cluster sampling is another widely used strategy. It is based on a reverse
criterion: select a random sample of the groups, and census all the units within
each group. Such a choice can be justified if we do not have a list of units at the last
aggregation level (individuals, establishments, or farms), but we do have a list of
municipalities or census tracks. Another conceivable motivation regards the spatial
distribution of the population. If it is widely scattered over a study region, there
could be prohibitive observation costs due to the travel expenses of

the

interviewers.
In many situations, it is efficient to first select a random sample of groups, and
then a random sample of the units in each group. In this case, the groups are called
primary sampling units (PSUs), and their sample selection is called the first-stage
sample . Therefore the units selected in the second-stage are called secondary
sampling units (SSUs).
Using the same motivation as cluster sampling, it is often difficult to set up a
frame that lists all the individuals in which there is interest. However, the popula-
tion can be naturally partitioned into PSUs, for which a frame either exists or can be
produced at low additional costs. When a sample of PSUs has been drawn (the first-
stage sample), one can list a frame for each PSU in this sample. Then, a sample of
SSUs can be drawn from each selected PSU (the second-stage sample) that are the
elements under investigation (Mandallaz 2008 ). This sequence provides a good
alternative for establishing a detailed map of irregular polygons over a country
(which is a long and very expensive task, see Chap. 5 ) by setting up a rough map of
aggregated polygons. The detailed map is only interpreted and digitized in the
second stage, and then only for the selected aggregates: this procedure saves
considerable resources.
From a methodological point of view, the basic requirement to simplify such a
complex structure of subsequent random selections is that samples for any level
should be drawn independently from each other (S¨rndal et al. 1992 , p. 134). This
strong constraint is clearly essential for deriving the first-order inclusion probabil-
ities as a simple product of the probabilities of each stage, and for decomposing the
variance of the HT estimator into a sum of variances relevant to each sampling
stage.
The two-stage sampling design can be defined as follows. The population U is
partitioned into N 1 PSUs { U 1 , U 2 ,

... , U i , ... , U N 1 }, the number of SSUs in each PSU

is N i and clearly X

N 1

N i ¼ N . In the first stage ,a n 1 -sized sample ( s 1 ) of the N 1 PSUs

i ¼ 1

is selected according to a design p 1 (.). Then, for each i 2 s 1 a sample ( s i ) of units is

drawn from U i according to a design p 2 i (. j p 1 ).