Agriculture Reference
In-Depth Information
regarding the variable of interest y, which was not included in the sample selection
criterion.
The finite population is often not considered to be an element of primary interest
when making an inference using statistical models.
p
(
s
) is viewed as a nuisance
parameter that must be considered but not exploited in the data analysis. Neverthe-
less, there is a clear recognition that we must take account of the sampling design
s
features when model fitting using survey data (see Chap.
12
). However, the model
is not the target of the analysis when we are using auxiliary information for finite
population inference. It is only a tool for adding information to the sampling
weights, which would otherwise only depend on the inclusion probabilities.
These are the basic principles that underlie the
model-assisted
approach to survey
estimation. In this approach, a statistical model is introduced without many restric-
tions, and all the properties of the resulting estimators are evaluated within a purely
design-based
inference or, more formally, with respect to
p
(
s
).
Within this framework, the pure
design-based
HT estimator has been
supplemented by alternative
design-based
estimators that take advantage of addi-
tional information. Examples include the ratio, difference, and post-stratified esti-
mators (see Fuller
2002
for a review).
Assume that a continuous covariate x is available in the frame for all units of the
population, and that we are interested in estimating the total of a survey variable y.
Focusing on the sample units, we can estimate both the totals to obtain
t
HT
,
y
and
different from its estimated counterpart
t
HT
,
x
, and it is intuitive to consider if the
observed difference in x can also be found in y.
We should correct the HT estimator depending on whether we evaluate this
difference in an additive or multiplicative way. If we assume that the proportion-
ality of the error on x is also likely to be detected on y, the correction to be applied
to the HT estimator is straightforward
'
X
k2s
^
k
¼
X
k2s
g
k
,
RAT
d
k
y
k
¼
X
k2s
w
k
,
RAT
y
k
;
t
x
X
k2s
^
k
t
RAT
,
y
¼
ð
10
:
1
Þ
t
x
X
k2s
^
k
where
g
k
,
RAT
¼
is a constant correction factor applied to the direct weights
d
k
¼
=π
k
to obtain the final weights
w
k
,
RAT
that makes explicit use of the auxiliary
information. Equation (
10.1
) is called the
ratio
estimator.
If we assume that the error on x can additively influence y, the correction to be
applied to the HT estimator is
1
X
k2s
^
k
¼
X
k2s
d
k
y
k
t
DIF
,
y
¼
t
x
þ
t
x
þ
ð
x
k
Þ;
ð
10
:
2
Þ
where
^
k
are the expanded differences between the two variables x and y. Equation
(
10.2
) is referred to as the
difference
estimator.
Search WWH ::
Custom Search