Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 121

It is worthwhile noting that Eqs. ( 6.12 ) and ( 6.13 ) are simply the sum of the HT
estimators within each stratum. This is because the random selections between
strata are assumed to be independent. From a practical point of view, this means
that it is trivial to extend these results to designs that use different selection criteria
for drawing the units within each stratum. To obtain the HT point estimator and its
variance estimator, it is enough to sum the estimator of the total in each stratum and
their variance estimator.
While setting up the stratification, we may often want to consider all the
available prior information in the frame, to control as much as possible the
randomness of the sample, and have more confidence on the quality of results.
This choice, if excessively followed, may risk generating a large number of strata
that are often poorly represented in the population (and even more poorly
represented in the sample).
The problems arising from such a fragmentation of the population is often
reflected in the following unfavorable issues:

1. If we allocate less than a fixed threshold (say T h ) to a generic stratum h, we
typically have n h ¼ T h with a consequent fictitious increase of the sample size

(see Chap. 8 ).
2. If the effective number of observed units r h s in a stratum is less than 2 because of
non-responses, it is no longer possible to estimate the variability and accuracy of
the sample. If r h ¼

0, it is not even possible to produce point estimates.
3. It is difficult to manage panel rotations (see Sect. 6.8 ) or more general sample
coordination (between multiple surveys, see Sect. 6.8 ) in an under-represented
stratum. In extreme situations, we may be obliged to always select the same units
or only sample from a few strata, because there is no unit in the frame belonging
to the same subpopulation.

In these cases, little can be done unless we accept solutions that are not
methodologically desirable, but that introduce as few biases as possible. A practical
solution is the posterior aggregation of similar strata.
The strata function in the sampling package selects stratified random sam-
ples. In this example, we have used it to select n ¼ 100 units without replacement,

25 units for each of the 4 quadrants that divide the study region. The four quadrants are
indicated in the output as: 11,12,21, and 22. The outcome of this function should
be managed by the getdata utility, which extracts the sample data from the
population frame. The HT estimates are obtained with a similar sequence of
R commands to those used for SRS. The only exception is that in the svydesign
function (which defines the design), we should add the option
strata ¼ ~strataid (to define the stratifying codes) and the option deff ¼ T
(if we want deff ( 6.6 ) to be included in the output). This sample is displayed in Fig. 6.4 .

> framepop < - cbind(framepop,strataid ¼ ceiling(framepop$xc*2)*10
+ +ceiling(framepop$yc*2))
> popstr < - table(framepop$strataid)
> popstr