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STAT 344 Sample Surveys
LAB 6
Chao Xiong
February 15, 2011
Chao Xiong
STAT 344 Sample Surveys LAB 6
Outline
1
Stratification
2
Questions?
Slides will be available on the website:
http://www.stat.ubc.ca/~alex.xio/
My email address: alex.xio@stat.ubc.ca 0
Chao Xiong
STAT 344 Sample Surveys LAB 6
Information about strata and samples
1
2
Strata
Sizes
N1 N 2
Means
Y1 Y2
Variances S12 S22
Samples
Sizes
n1 n2
Means
y1 y2
Variances s12 s22
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3
4
Ng
Yg
Sg2
Ng
YG
SG2
ng
yg
sg2
ng
yG
sG2
STAT 344 Sample Surveys LAB 6
a) Compute the stratified mean estimator ys t and its estimated
variance.
G
X
¯st =
Yˆ
Wg y¯g
1
where
Wg = Ng /N
and
v (Yˆ¯st ) =
G
X
Wg2 (1 − fg )/ng · sg2
1
Chao Xiong
STAT 344 Sample Surveys LAB 6
b) Construct 95% confidence limits for population total
¯st =
Yˆ = N Yˆ
G
X
Ng y¯g
i=1
Its estimated variance
¯st )
v (Yˆ ) = N 2 v (Yˆ
95% C.I
(Yˆ − 1.96 ∗ s.e(Yˆ ), Yˆ + 1.96 ∗ s.e(Yˆ )
Chao Xiong
STAT 344 Sample Surveys LAB 6
c) Test the hypothesis that the average response value of Stratum
II is twice of that of Stratum I.
H0 : Y¯2 = 2Y¯1 v .s. H1 : Y¯2 6= 2Y¯1
For each stratum,
v (¯
yg ) = (1 − fg )/ng · sg2
v (¯
y2 − 2¯
y1 ) = (1 − f2 )/n2 · s22 + 4 · (1 − f2 )/n2 · s22
Under the null hypothesis,
y¯2 − 2¯
y1
∼ N(0, 1)
S.E .(¯
y2 − 2¯
y1 )
Reject the null if
y¯2 −2¯
y1
S.E .(¯
y2 −2¯
y1 )
> Z1.96 or
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y¯2 −2¯
y1
S.E .(¯
y2 −2¯
y1 )
< −Z1.96
STAT 344 Sample Surveys LAB 6
Alternatively, you can use two sample t-test to get more accurate
approximation for small samples
t=
y¯2 − 2¯
y
q 1
s12 · n11 +
where
s12 =
1
n2
∼ t(n1 + n2 − 2)
4(n1 − 1)s12 + (n2 − 1)s22
n1 + n2 − 2
Chao Xiong
STAT 344 Sample Surveys LAB 6
d) Proportional sample size allocation
ng = Wg n
In practise, you will round these numbers to integers. Under
proportional allocation,
X
X
XX
Yˆ¯st =
Wg y¯g =
ng y¯g /n =
ygi /n
G
and
i
X
¯st ) = 1 − f
Wg Sg2
Vp (Yˆ
n
G
the corresponding estimate
X
¯st ) = 1 − f
Vp (Yˆ
Wg sg2
n
G
Compared to SRSOR,
Vs
S2
=P
Vp
Wg Sg2
Chao Xiong
STAT 344 Sample Surveys LAB 6
the conclusion is unless there is not meaningful difference between
strata (which make the 2nd term in the approximation nearly 0),
the stratified srsor with proportional sample size allocation results
in more accurate estimate of the population mean through the use
of stratified mean.
Chao Xiong
STAT 344 Sample Surveys LAB 6
Neyman allocation
¯st ) is minimized if
If we already know Sg , then the variance of V( Yˆ
we set
ng
Wg Sg
=P
Wg Sg
n
the minimized variance is
1 X
1 X
VN = (
Wg Sg )2 −
Wg Sg2
n
N
Note: the above are all theoretic results. In practice, you should
first compute ng under different allocation methods, and then
¯st and Vp (Yˆ¯st ) using
around them into integers. Next, calculate Yˆ
v (Yˆ¯st ) =
G
X
Wg2 (1 − fg )/ng · sg2
1
See the example in the books.
Chao Xiong
STAT 344 Sample Surveys LAB 6
Conclusion
In terms of variance of the stratified mean under stratified srsor,
proportional allocation is less efficient than Neyman allocation.
The advantage increases when the differences between Sg
increases.
The stratified mean with proportional sample size allocation has
lower variance than the sample mean under straight srsor when
V (Yg ) are different.
In some applications, there are a few units with huge sizes that
they form a stratum with huge stratum variance. The optimal
allocation may suggest to have more units sampled than the
stratum size. We should watch out for such situations and adjust
appropriately.
Chao Xiong
STAT 344 Sample Surveys LAB 6
e) Sample size calculation
the variance should not exceed a given value V
|Yˆ¯st − Y¯ | should not exceed a given value e except for a
probability of α
P(
¯st − Y¯ |
|Yˆ
e
≤
)=1−α
ˆ
¯st )
S.E .(Y¯st )
S.E .(Yˆ
we have
¯st ) ≤
v (Yˆ
e2
1.962
¯st − Y¯ |/Y¯ should not exceed a given value
the relative error |Yˆ
a except for a probability of α
Similarly we get
¯
ˆ ) ≤ ( aY )2
v (Y¯
st
1.96
Chao Xiong
STAT 344 Sample Surveys LAB 6
Questions?
Chao Xiong
STAT 344 Sample Surveys LAB 6