Model Quality Report in Business Statistics - Harvard ...

where
y
R ˆ =
h
sh
x
sh
( tˆ
)
( 1−
f ) h 2
2 2
( S − 2Rˆ
S Rˆ
S )
2
ˆ N
h
V
1 y,
rat
= ∑ ysh
h xys
+
h h xs
(4.7)
h
n
h
h
, (that is, the stratified version of equation 6.11 in Cochran). An
alternative variance estimator (see equations 6.12 and 6.13) is
( tˆ
)
2
( 1−
f ) x
h U h 2
2 2
( S − 2Rˆ
S Rˆ
S )
2
ˆ
N
h
V
2 y,
rat
= ∑ ysh
h xys
+
h h xs
(4.8)
h
n x
h
h
2
sh
Note: (4.4) can be written
∑∑
t ˆ = w y where
y,rat
h
sh
h
k
txh
w
h
= .
x
∑
s
h
k
4.3 What does SUDAAN do
For stratified random sampling, the variance formula used is
2 1
where S
zs
= ∑( z − z )
h
k sh
2
( 1 f )
Vˆ 2
= ∑ − n S
(4.9)
nh
−1
sh
variance formula corresponds to the design option DESIGN = STRWOR.
h
h
h
zsh
, and z
k
is the “appropriate linearised value”. Note that the
So, if we want to estimate the variance of the usual expansion estimator (see (4.1)), we use
DESCRIPT. The “linearised value” is
z = w y , and so long as
k
h
k
w
N
h
h
= (that is, the
nh
sampling weight), the variance formula in (4.9) gives the correct variance estimator of (4.3).
What about the variance estimator for the ratio estimator defined in (4.4) Can SUDAAN be
tricked by defining the weight to be
itself will be correct, but the variance formula in (4.9) with
This is not the variance given in (4.7) or (4.8).
w
h
=
t
∑
xh
sh
x
k
2
( − f ) xU
h 2
The answer is no. The ratio estimator
2
sh
z
t
xh
k
= wh
yk
= yk
will give
∑ x
s k
h
ˆ
N
2 h
1
h
V = ∑
S
ys
(4.10)
h
n x
h
h
The answer is to use the RATIO procedure. In general, we can estimate the ratio for any
subgroup d as
Rˆ
d
=
∑∑
h
∑∑
h
s
s
h
h
δ ( d)
w y
hk
hk
h
δ ( d)
w x
h
k
k
(4.11)
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