A. Illustrated calculations
We illustrate the construction of replicates for variance estimation with a
simple example. Table A.1 contains the original weights and the naive replicate weights for a simple random sample of original size six with two missing
values and two donors per missing value. Observations two and four act as
donors for missing observation three and observations four and five act as
donors for missing observation six. The weights for six naive jackknife replicates are given in the table, where the weights are constructed as if the sample
were complete. The two imputed values are treated as two observations from
a primary sampling unit. We assume the finite population correction can be
ignored. Then ck = 5/6. The naive replicate weights for the respondents are
given in Table A.2. We have
L (
)2
∑
(k)
αi1 − αi = (0.0333, 0.0533, 0.0533, 0.0333)
k=1
for i = 1, 2, 4, 5, respectively. The sum of squares multiplied by 5/6 is 0.1444
∑
while αi2 = 0.2773. The use of the naive replicates severely underestimates
the multiplier for σi2 . Only for observation one, the observation not used as
a donor, is the sum of squares of the naive replicates equal to αi2 .
< Table A.1 around here >
< Table A.2 around here >
∗
, the quadratic equation for b2 is, by
Using ϕ2 and the definition of wij
(24),
[0.2 − 0.2b2 − 0.3333]2 + [0.4 + 0.2b2 − 0.3333]2
− [0.2 − 0.3333]2 − [0.4 − 0.3333]2 = 0.1331 − 0.0533,
1
where 0.1331 = c2−1 α22 , and c−1
2 ϕ2 = 0.0533. The simplified quadratic equation is
0.08b22 + 0.0788b2 − 0.0798 = 0
and b2 = 0.62. The quadratic equation for b4 is
[0.2 − 0.2b4 − 0.3333]2 + [0.2 + 0.2b4 − 0.1666]2
− [0.2 − 0.3333]2 − [0.2 − 0.1666]2 = (6/5)(0.3333)2 − 0.0533.
The simplified equation is
0.08b24 + 0.0654b4 − 0.0798 = 0
and b4 = 0.67. The final jackknife replicates are given in Table A.3 and the
∑
. ∑
respondent weights in Table A.4. Note that i Qi = 0.278 = i αi2 where
(
)2
∑
(k)
Qi = Lk=1 ck αi1 − αi .
< Table A.3 around here >
< Table A.4 around here >
2
Table A.1: Jackknife Illustration
Obs
1
2
3
Naive
Weight 1
2
0.166 0.0 0.2
0.166 0.2 0.0
0.166 0.2 0.2
0.000 0.0 0.0
0.166 0.2 0.2
0.166 0.2 0.2
0.166 0.2 0.2
0.000 0.0 0.0
Donor
4
5
6
2
4
4
5
replicate Weights
3
4
5
6
0.2 0.2 0.2 0.2
0.2 0.2 0.2 0.2
0.0 0.2 0.2 0.2
0.0 0.0 0.0 0.0
0.2 0.0 0.2 0.2
0.2 0.2 0.0 0.2
0.2 0.2 0.2 0.0
0.0 0.0 0.0 0.0
Table A.2: Illustration - Naive Weights for Respondents
Obs
1
2
4
5
αi
0.166̄
0.333̄
0.333̄
0.166̄
(1)
α1i
0.0
0.4
0.4
0.2
Naive
(2)
α1i
0.2
0.2
0.4
0.2
replicate Weights
(3)
(4)
(5)
α1i α1i α1i
0.2 0.2 0.2
0.2 0.4 0.4
0.4 0.2 0.4
0.2 0.2 0.0
(6)
α1i
0.2
0.4
0.2
0.2
Table A.3: Jackknife Weights for Fractional Imputation
Obs Donor
1
2
3
2
4
4
5
6
4
5
Weight
0.166
0.166
0.166
0.000
0.166
0.166
0.166
0.000
1
0.0
0.2
0.2
0.0
0.2
0.2
0.2
0.0
Final Replication Weights
2
3
4
5
0.200 0.2 0.200 0.2
0.000 0.2 0.200 0.2
0.076 0.0 0.200 0.2
0.124 0.0 0.000 0.0
0.200 0.2 0.000 0.2
0.200 0.2 0.200 0.0
0.200 0.2 0.066 0.2
0.000 0.0 0.134 0.0
3
6
0.2
0.2
0.2
0.0
0.2
0.2
0.0
0.0
Table A.4: Illustration - Final Weights for Respondents
Obs
1
2
4
5
Qi =
αi
0.166̄
0.333̄
0.333̄
0.166̄
(1)
αi
0.0
0.4
0.4
0.2
Final
(2)
αi
0.200
0.076
0.524
0.200
replicate Weights
(3)
(4)
(5)
αi
αi
αi
0.2 0.200 0.2
0.2 0.400 0.4
0.4 0.066 0.4
0.2 0.334 0.0
(
)2
(k)
c
α
−
α
.
i
k
k=1
i
∑L
4
Qi
(6)
αi
0.2
0.4
0.2
0.2
0.0278
0.0847
0.1157
0.0502