Sampling Designs - Sampling Spatial Units for Agricultural Surveys - page 118

total SE DEff

yobs 92083.112 970.903 0.2834

as.factor(q1obs)1 297.621 46.890 1.2067

as.factor(q1obs)2 311.988 46.502 1.1567

as.factor(q1obs)3 363.524 49.555 1.2227

6.5 Stratified Sampling

A traditional approach for dealing with multivariate auxiliary variables during
sample design is to use a stratification scheme. Then, the population units are
classified in a stratum according to the values of their auxiliary variables (Benedetti
and Piersimoni 2012 ; Vogel 1995 ). Thus, an SRS or π ps is selected within each

stratum.
Assume that the population U can be partitioned into H groups according to
some known criteria. Let { U 1 , U 2 ,

... , U h , ... , U H } be this set of groups such that

H

r . As a result, the set of groups (called strata)
are exhaustive and non-overlapping. Let { N 1 , N 2 ,

[

1 U h ¼

U and U h \ U r ¼ ∅, 8 h 6 ¼

h ¼

... , N h , ... , N H } be the number of

units of the population belonging to each stratum, with X

H

N h ¼ N . Such a partition

h ¼1

implies that the following basic choices have been, or should be, established:

1. The set of stratifying covariates has been selected.
2. A method of fixing the required number of strata H has been determined.
3. The criteria used to stratify the population have been defined. If we use discrete
auxiliaries, we need to define the list of codes or code combinations to be used;
while for continuous variables, strata boundaries or limits should be carefully
evaluated.
4. The allocation of sample units to the strata has been determined (see Sect. 8.3 ).

In a sampling strategy, the population is stratified for three main reasons:
administrative purposes, defining the planned analysis domains, and improving
the efficiency of the estimates. At this point, we will postpone our description of
the third topic. The set of the stratifying covariates is usually chosen for the first two
reasons. The first is essentially related to the organization of the data collection
process, and with legal and administrative aspects such as the availability of the
frame only at a local level. The second is related to the data dissemination of the
survey, in particular to estimations for unplanned domains (see Chap. 11 ). This can
create several difficulties because the sample size may not be defined within each
domain. The best way to avoid these difficulties is to fix the sample size in each
estimation domain, introducing an auxiliary variable to the set of stratifying
covariates. The codes of this auxiliary variable identify the estimation domains in