Agriculture Reference
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difficult to know if these simplistic assumptions are more realistic than any other
alternative hypothesis.
Thus, the design weights must be adjusted on the basis of data relating to the
observed and not observed units in the selected sample
s
, using the unbiased
estimator
t
NR
,
y
¼
X
k2r
d
k
=
y
k
. The problem is that the
r
k
s are unknown and
must be replaced with some suitable estimates,
ð
r
k
Þ
r
k
s. Standard statistical modelling
techniques (such as logistic regression) can be used to estimate the response
propensities using auxiliary covariates that are available for both respondents and
non-respondents. However, one of the simplest and most widely used methods is
the response homogeneity group (RHG). This technique multiplies the weight (
d
k
)
of the observed units within a given stratum
k
^
h
by a factor
g
k,RHG
, which is the
ratio between the estimated population size and its expected number of respondent
units within stratum
h
. If we assume that all the units are eligible, the RHG
estimator is (S
¨
rndal et al.
1992
)
2
X
k2s
,
h
d
k
X
k2r
X
k2r
g
k
,
RHG
d
k
y
k
¼
X
k2r
w
k
,
RHG
y
k
; ð
^
k
¼
t
RHG
,
y
¼
X
k2s
,
h
RE
k
d
k
:
Þ
10
23
where
RE
k
is an indicator vector equal to 1 if unit
k
was observed, and 0 otherwise.
Another widely used approach to estimate
^
r
k
is through the within stratum
X
k2h
RE
k
=
respondent
n
h
,S¨rndal
et al.
1992
, p. 581, Lehtonen and Pahkinen
2004
, p. 116). From a modeling point
of view, Eq. (
10.23
) is justified by the underlying hypothesis that the variability of
the response probability (
r
k
) within the stratum
h
is irrelevant, and thus can be
estimated by the rate
rate observed in the sample (i.e.,
^
r
k
¼
g
k
,
RHG
.
It is worth noting that this partition is based on the hypothesis of homogeneity of
the
r
k
, and so does not necessarily respect a stratification used to select the sample
(which has the aim of building strata that are homogeneous with respect to y).
However, the sample design is often based on a very detailed partition of the
population. Therefore, this approach is practical, and it is typical to assume that
nonresponses are homogeneous within the same design stratum.
Moreover, we should consider that a group might be homogenous according to
two different hypotheses. If the
r
k
are homogenous within the group, then
Eq. (
10.23
) is nothing more than a post-stratified estimator that corrects the
sampling weights, in the sense that the population total for any variable is correctly
estimated (Lumley
2010
). However, if a survey variable y has small variabilities
within the stratum, the population total for this outcome will be correctly estimated,
even if
r
k
are not homogenous. It does not matter if the weights for each unit are
wrong, because the values of y are approximately the same within the group
(Lumley
2010
). Clearly, there is no empirical evidence that we can easily achieve
a low variability of responses using a few available auxiliary variables. It is evident
that the post-stratified estimates (i.e., the RHG estimator) are less biased than the
HT estimates.
^
r
k
¼
1
=
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