Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 143

where

t y

b 2 c 2 t y

ˈ ¼ ; ð 6 : 28 Þ

2

þʴ 1

and where c is the desired level of precision for the estimation of the total, b t y is
the bias of t y depending on the difference between the estimated ʴ

and the true δ , and

c is the desired level of precision.
Bee et al. ( 2011 ) first considered the problem of estimating a ratio of two
unknown totals representing the same variable in two different periods, which is
very common in conjunctural surveys aimed at estimating variations. The optimal
n is still given by Eq. ( 6.27 ), but now

ˈ is defined as

R

t C , t 1 þ

2

c 2 R 2 b 2

ð t S , t 1 Þ

ˈ ¼ ; ð 6 : 29 Þ

R 2 1 þ 2 R ˁ t S , t t S , t 1

where R is the ratio of the same totals at times t and t ˁ t S , t t S , t 1 is the linear

correlation coefficient of the HT estimators in stratum S at times t and t

1, and

1.
Finally, Bee et al. ( 2011 ) also dealt with non-sampling errors caused by total
nonresponse rates in a multipurpose and univariate approach. It is interesting that
the optimal sample size is again given by Eq. ( 6.27 ), but ʨ is given by Eq. ( 6.28 )if
we are estimating a total, or Eq. ( 6.29 ) for a ratio, minus a quantity that depends on
the non-response probabilities ʸ k . Thus, in both cases the sample sizes are func-

tionally identical to that in Benedetti et al. ( 2010 ), which turn out to be a general
result for these kinds of analyses.

Conclusions
In this chapter, we have reviewed the main basic sampling designs. We have
described the HT estimator and its sampling error for each of them.
However, no solution is always valid and usable. Each scheme has advan-
tages and disadvantages, but generally a combination can achieve stable and
acceptable results in any field of statistical research.
The sample design is the most important stage of a survey, because any
deficiencies cannot generally be compensated for during data editing and
analysis. The classical designs for selecting random samples such as SRS,
stratification, and multistage cluster sampling were all developed to minimize
the survey cost, while controlling the uncertainty associated with the estimates.
Agricultural data are typically collected by directly observing the spatial
units. Therefore, cluster sampling and sample coordination procedures are
required to reduce travel costs and identify the units to be surveyed.

(continued)