D. Simulation Study
To test our theory, we performed a limited simulation study. In the simulation, two finite populations of size N = 10, 000 were generated. In population
A, the population values were generated by
Xi ∼ χ2 (1)
Yi = 2 + 0.5Xi + ei
where ei are independently generated by the standard normal distribution.
In population B, we have the same Xi values but the Yi values are generated
by
Yi = 1 + 0.3(Xi − 0.5)2 + ei
and ei are independently generated by the standard normal distribution.
From each of the finite populations, two sets of B = 10, 000 Monte Carlo
samples of size n = 100 and n = 400 were independently generated by simple
random sampling. From each sample, we also generated a response indicator
variable Ri from a Bernoulli distribution with the response rate p = 0.65.
The Yi is observed if and only if Ri = 1. The Xi are observed on the entire
sample.
We used a nearest neighbor imputation where the Euclidian distance
based on the value of Xi ’s was used to identify the nearest neighbor. For each
nonrespondent, we used two nearest neighbor imputation (NNI) methods.
The first NNI method used M1 = 1 for point estimation and the second
NNI method used M2 = 2 donors for point estimation. Both methods used
M2 = 2 donors for variance estimation. The parameters are θ1 = mean of Y
and θ2 = proportion of Y ≤ 2.0.
1
< Table D.1 around here >
Table D.1 contains the means and variances of the point estimators based
on the Monte Carlo samples. The simulation results show that the point estimators based on the NNI method have modest bias and the bias is significant
for population B. The bias of the NNI point estimator can be explained by the
“end effect” where the nearest neighbor associated with the largest x-value
will be imputed from the donor with smaller x-value and so the expected
value of the imputed value is smaller than the expected value of the original.
The bias is larger for M1 = 2 than for M1 = 1 because the end effect is more
serious if we use more donors since the donors with smaller x-values will be
used. However, the NNI method with M1 = 2 is more efficient than the NNI
method with M1 = 1 because it reduces the variance due to imputation by
using two donors instead of one donor.
< Table D.2 around here >
For variance estimation, we used M = 2 nearest neighbors to create the
replication weights. Table D.2 shows the relative biases and t-statistics of
the variance estimator. The t-statistics are the statistics used to test the
hypothesis of zero bias for the variance estimators. The simulation results in
Table D.2 show that the variance estimators are approximately unbiased for
both parameters in both the case with M1 = 1 and the case with M2 = 2.
2
Table D.1: Monte Carlo means and variances of the complete sample
estimators and the imputed estimators, based on 10,000 samples
n Model Parameter
Method
Mean Variance
Complete sample 2.501 0.01478
θ1
Imputed (M1 = 1) 2.500 0.02384
A
Imputed (M1 = 2) 2.500 0.02184
Complete sample 0.350 0.002274
100
θ2
Imputed (M1 = 1) 0.350 0.004097
Imputed (M1 = 2) 0.350 0.00375
Complete sample 1.678 0.07846
B
θ1
Imputed (M1 = 1) 1.661 0.09548
Imputed (M1 = 2) 1.654 0.09104
Complete sample 0.725 0.001975
θ2
Imputed (M1 = 1) 0.724 0.003272
Imputed (M1 = 2) 0.725 0.003002
Complete sample 2.500 0.007646
θ1
Imputed (M1 = 1) 2.498 0.011861
A
Imputed (M1 = 2) 2.497 0.011115
Complete sample 0.351 0.001156
200
θ2
Imputed (M1 = 1) 0.351 0.002027
Imputed (M1 = 2) 0.351 0.001864
Complete sample 1.675 0.03936
θ1
Imputed (M1 = 1) 1.663 0.04514
B
Imputed (M1 = 2) 1.658 0.04386
Complete sample 0.725 0.001003
θ2
Imputed (M1 = 1) 0.725 0.001630
Imputed (M1 = 2) 0.725 0.001508
3
Table D.2: Monte Carlo relative biases and t-statistics of the variance
estimators, based on 10,000 samples
n Model Parameter Imputation Relative t-statistic
Method
Bias (%)
θ1
M1 = 1
1.82
1.29
A
M1 = 2
2.52
1.78
θ2
M1 = 1
0.43
0.29
100
M1 = 2
0.33
0.23
θ1
M1 = 1
1.44
0.96
B
M1 = 2
-1.46
-0.95
θ2
M1 = 1
1.39
0.98
M1 = 2
2.13
1.51
θ1
M1 = 1
1.95
1.37
A
M1 = 2
0.54
0.38
200
θ2
M1 = 1
1.66
1.14
M1 = 2
1.16
0.80
θ1
M1 = 1
1.82
1.32
B
M1 = 2
-1.99
-1.44
θ2
M1 = 1
1.19
0.82
M1 = 2
1.02
0.72
4