Agriculture Reference
In-Depth Information
X
m
a
¼1
ʵ
ah
ʵ
ah
0
¼
h
0
0
8
h
6
¼
:
ð
10
:
31
Þ
If
m
is balanced and also satisfies
X
m
a
¼1
ʵ
ah
¼
0
8
h
;
ð
10
:
32
Þ
it is said to be in
full orthogonal balance
.
It can be shown that, if the estimator of the sample variance is calculated on a set
of
m balanced half
-samples (also called
balanced repeated replications
(BRR)), it
coincides with that obtained considering all the possible
replications
. The
balanced
half
-sample technique of variance estimation can be applied to a set of splits that
respect the requirement in Eq. (
10.31
). For each of the
replicates
, we can estimate
the total using
2
X
H
1
ʴ
ah
y
h
1
y
h
2
π
h
2
t
a
¼
ð
Þ
π
h
1
þ
1
ʴ
ah
;
ð
10
:
33
Þ
h
¼
where
y
h
1
and
y
h
2
are the observed values of the target variable
y
, and
π
h
1
and
π
h
2
are the first-order inclusion probabilities of the units
h
1 and
h
2, respectively.
The BRR variance estimator can then be calculated using (S¨rndal et al.
1992
,
p. 432)
X
m
2
t
a
t
HT
ð
Þ
V
BRR
tðÞ
¼
a
¼1
:
ð
10
:
34
Þ
m
Note that, when the set is in
full orthogonal balance
, the mean of Eq. (
10.33
)is
equivalent to the HT estimator of the total. For further details on the technical
problem of finding a small set of
balanced half
-samples, see Wolter (
2007
).
When the parameter to be estimated is not a total,
t
HT
can be replaced by
ʸ
HT
and
t
a
by
ʸ
a
(the estimator of
based on data from the
a-
th subsample). In this more
general case, several alternatives to Eq. (
10.34
) have been proposed in the litera-
ture. However, simulation studies have demonstrated that the results are not
substantially different (S¨rndal et al.
1992
, p. 437). The BRR variance estimator
is exactly unbiased, and is exactly the same as using all the possible
replicates
for
the population total. These properties only hold approximately for other estimated
population summaries. Slightly different estimates can be obtained from different
sets of
replicates
in
full orthogonal balance
(Lumley
2010
).
When the design cannot be considered stratified, it is more difficult to split a
sample in such a way that the units do not overlap, thus the use of BRR method is
not recommended (Lumley
2010
).
ʸ
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