VARIANCE ESTIMATORS FOR VARIABLES THAT HAVE BOTH OBSERVED AND IMPUTED VAL

Sandra A.West, Diem-Tran Kratzke, and Kenneth W. Robertson, Bureau of Labor Statistics

Sandra A. West, 2 Massachusetts Ave. N.E., Washington, D.C. 20212

KEY WORDS: Imputation, Mean, Regression, Hot Deck,

Multiple Imputation.

1. Introduction

We will present the results of theoretical and empirical

nvestigations of different variance estimators in the

presence of imputed and observed values in this paper. It

s assumed that all the missing data are imputed by the

ame method. Imputation methods considered include

mean, hot deck, regression, regression plus residual, and

multiple imputation. Variance estimators considered

nclude the standard, two versions of the jackknife, and

andom groups.

The data are employment from the Bureau of Labor

Statistics' Universe Data Base (UDB). The UDB is a

ampling frame of business establishments that is

constructed from the State's Unemployment Insurance

UI) micro data file. The information used to maintain this

ile is obtained from quarterly UI reports which each

employer is required to submit. Although the filing of the

UI report is mandatory, there are always some late,

ncomplete, or missing reports. In previous studies, a

ingle imputation procedure was developed that worked

well for all industries within each State. For this study, the

ecommended imputation method and several alternatives

will be considered. The actual data for non-repondents

were never obtained. Thus non-response had to be

imulated using the patterns of non-response observed on

he files. For the most part, it was assumed that the nonespondents were missing at random. In addition, a fixed

non-response rate was simulated in order to see the effect

on the variance estimators when a large part of the sample

s imputed.

In Section 2, we describe the data sets used and the

design of the empirical investigations. The notation and

evaluation criteria that are used to compare the various

methods are presented in Section 3. Descriptions of the

mputation methods and their properties are presented in

Section 4 and Section 5, respectively. In Section 6,

alternative variance estimators to the standard estimator

are considered. The results of the empirical investigations

are showed in Section 7, along with observations and

conclusions. Future research is discussed in Section 8.

2. Data and Design of Empirical Investigation

Two months of UDB data were used for this study:

December 1991 and January 1992. A unit (establishment)

more homogenous as we move from 2- to 3-d

stratification.

We obtained data from Michigan in these indu

digit SIC code is in parenthesis): Agricultural 

(07), Lumber and Wood Products (24), Trans

Equipment (37), Trucking and Warehousin

Transportation Services (47), General Merchandi

(53), Apparel and Accessory Stores (56), Misc

Retail (59), Nondepository Credit Institutio

Miscellaneous Repair Services (76), Me

Organizations (86), and Private Households (88).

Intuitively, an establishment's employment 

correlated with its own past employment and 

employment of similar establishments. If estab

are placed into strata based on characteristics r

employment, then the more homogenous the strat

higher the correlation will be. Within each 2-d

chosen, we stratified the data further by (1

SIC/county and (2) 3-digit SIC/size class.

Usually a measure of size is created f

establishment based on its most recent reported

employment. This was done in our study. Siz

were formed as follows:

Size Class 1 - Employment < 50

Size Class 2 - 50 ≤ Employment < 250

Size Class 3 - Employment ≥ 250

After some initial results, we increased the n

size classes, as most units fell in the original Size

The original Size Class 1 was sub-divided as follo

Size Class 1a - Employment < 5

Size Class 1b - 5 ≤ Employment < 10

Size Class 1c - 10 ≤ Employment < 20

Size Class 1d - 20 ≤ Employment < 50

For our study we used two non-response pat

the first we simulated the pattern of non

observed in the data as much as possible. If a p

industry had x% of imputed employment, then

response rate of x% was used. It was assumed

missing data mechanism was ignorable, and a ra

of units were chosen to represent the set 

respondents. The second non-response pattern 

that each industry had observed a 25% non-respon

For the empirical study, we allowed only co

single units from private industries. Continuous 

units that existed on the file during the previous
used data from the Model set to determine parameters

which were then applied to the units in the test set.

3. Notation and Evaluation Criteria

Notation

For a given 2-digit SIC let

Ej t, denote the employment for unit j in month t,

$

Ej t, denote the predicted employment for unit j in month t,

Bt denote the set of units that have reported employment

for months t and month t-1,

nrt denote the percentage of units in month t that have

imputed employment values,

NRt denote the set of non-respondents that were obtained

by randomly selecting the percentage nrt of units from

the set Bt (Test set.),

BRt denote the set of units in Bt - NRt (Model set.),

NNRt denote the number of elements in NRt

NBRt denote the number of elements in BRt .

Also let

Vt denote the variance of the employment variable for

establishments in Bt ; that is, the "true" variance,

$Vtmi , , denote the estimator of Vt using variance method m

and imputation method i, where i = 0 denotes no

imputation and the variance estimator is based only on

the respondents.

The following notation will be used for the different

methods of computing the variance:

m = 1 - standard method, denoted by SD

m = 2 - jackknife A, denoted by JA,

m = 3 - jackknife B, denoted by JB,

m = 4 - random groups, denoted by RG.

The following notation will be used for the different

methods of imputation:

i = 1 - stratum mean,

i = 2 - carry over,

i = 3 - hot deck nearest neighbor,

i = 4 - recommended regression,

i = 5 - as in i=4 plus residual,

i = 6 - as in i =4 plus multiple residuals.

Evaluation Criteria

Letting εmi tmi t , ,, V V $ = − denote the error for variance

method m and imputattion method i, then the Percent

Relative Absolute Error will be used:

RAE V mi mi t , , = 100 | | ε .

Note that the imputations were done by 3-digit

SIC/county or 3-digit SIC/size class, but the variances

were computed over the entire 2-digit SIC.

4. Imputation Methods

method of imputation would not be desirable b

adversely affects the distribution of the sample 

skewing the distribution toward the mean. For 

stratification, month t, employment is imputed as f

$

Ek t, = E NBR j t

j BR

t

t

,

∈

∑ , for all k NRt ∈

Thus $Ek t, is equal to the average of the 

employment of all respondents in the stratum.

Carry-Over

Under the carry over method, each non-res

employment is imputed using its own histor

predicted value is therefore independent of size 

industry. It is computed as follows:

$

Ek t, = Ekt s , − , for all k NRt ∈ .

where s ≥ 1 and t s − denotes the last time in 

employment value was reported for the establishm

the paper only s=1 is used.)

Hot Deck-Nearest Neighbor

For any fixed stratification, month t, let k deno

respondent and c denote a respondent such that

E E E E for all j c t, −1 −≤− ∈ k t, −1 j t, −1 k t, −1 B

then $Ek t, = Ec t, .

For any particular non-respondent, this metho

the respondent that appears closest to the non-re

in an ordered list, and substitutes the respondent's

employment value for the non-respondent's.

Regression Model

A common method for imputing missing valu

least squares regression (Afifi and Elaskoff, 19

several papers on estimators for total employme

1982/1983, and West, et al, 1989), it was discov

the most promising models for employment w

proportional regression models. These models sp

the expected employment for establishment j in 

given the vector of E-values (employment in m

reported by units in set BRt):

E EEE E t t t t nt − −−− − 1 11 21 31 1 = ,,, , , , ,...,

is proportional to the establishment j's previous

employment, Ej t, −1. That is,

EE E e E (| ) jt t t jt , , −− − 11 1 = = β

where β is some constant depending on t.

It was further assumed that the E s' are con

uncorrelated. That is,

cov( | ) , E E E e

R j t if j = l S

ν
The model can be rewritten as:

E E jt jt jt , ,, = + β ε −1

where E{εj t, } = 0,

E{ε εj t, l t, } = νj t if j l

otherwise

R , = S

T0

In the previous studies, it was found that the model:

E E jt jt jt , ,, = + β ε −1 with νj t, = σ2Ej t, −1

worked reasonably well for employment data. Thus the

predicted employment value at time t is:

$ $ E E kt kt , , = β −1 , for all k NRt ∈ .

where $β = , ,

∈

−

∈

∑ ∑ E E j t

j BR

j t

t t j BR

1 .

Adding Residuals to the Regression Model

The regression method could be thought of as imputing

or missing employment by using the mean of the predicted

Et distribution, conditional on the predictors Et−1. As a

esult, the distribution of the imputed values has a smaller

variance than the distribution of the true values, even if the

assumptions of the model are valid. A simple strategy of

adjusting for this problem is to add random errors to the

predictive means, that is, drawing residualsrk with mean

zero to add to $Ek t, .

In the earlier studies, the residuals were chosen in three

ways. For this study the residuals will be chosen from a

normal distribution with mean zero and variance obtained

rom the model. Thus the predicted employment value at

month t is imputed as:

$ $ EEs kt kt k , , = + β δ −1 , for all k NRt ∈ .

where δk is a random number from a N(0,1) distribution

and s2 is equal to the mean square error of the regression.

A slight modification of the previous method was

obtained by drawing five random numbers and using the

average value for the added residual. That is,

$ $ EEs kt kt , , = + β δ −1 where δ δ = =

∑ k

k 1

5

5.

5. Effects of Imputation on Standard Variance

Estimator

Consider the population variance for a given 2-digit SIC

at month t:

V E E NBR NNR t jt

j B

t t

t

=− + ∈

∑c ha f ,

2

(5.1)

Assuming that the missing data are missing at

consider the effects of using imputation method

First consider overall mean imputation, that is, 

one stratum. In this situation, formula (5.2) becom

$ $ $ $ V E E E E NBR t jt kt ,, , ,

j BR k NR

t

t t

1 1

2 2

= −+ −

L

N

M O

Q

P ∈ ∈

∑ e j e jb ∑

where $ ( $ )( ) E E E NBR NNR j t, ,

j BR

k t

k NR

t t

t t

=+ + ∈ ∈

∑ ∑ .

This method creates a spike in the em

distribution, since all the missing values are assi

same value, the mean of the respondents, 

$

E E NBR for all k NR kt jt t , ,

j BR

t

t

= ∈ ∈

∑ . The second

(5.3) becomes zero since $ $ E E k t, = resulting

following variance estimator:

$ $ V E E NBR NNR ,, ,

NBR

NBR t jt

j BR

t t

t

t t

11

2

= − += ∈ + ∑ e jb g b

b

where S E E NBR j t

j BR

t

t

2

2

=− − 1 ∈

∑ ,

$ e jb g c h .

Since S2, which is $Vt, , 0 1 , is an unbiased estimat

E Vt ( $ ) , , 11 = NBR

NBR NNR V t

t t

t

−

+

a f1

a f

and hence,

E V

V

t

t

( $ ) , , 1 1 = NBR

NBR NNR

t

t t

−

+

a f1

a f is approximately equ

expected response rate.

Note that the relative bias is approximately 

minus the expected non-response rate:

EV V

V

t t

t

( $ ) , , 1 1 − = − +

+

NNR

NBR NNR

t

t t

a f1

a f .

Next consider the case of mean imputatio

strata; this method produces a series of spike

employment distribution at the means of the im

strata. Let Eh denote the mean of the respon

stratum h which has NNRt h, missing values, 

variance estimator can be written as:

$

V E E NNR E E NB t jt p th h p ,, , , R h

H

j BRt

1 1

2 2

1

= −+ −

L

N

M O

Q

P ∈ =

∑ c h c hb ∑

where H is the number of strata and,
E [ ] NBR E NNR E NBR NNR ( )

E NNR E NNR NNR NNR

p tr t w h t t

wh th h

h

H

t t th

h

H

=+ +

= = = =

∑ ∑

,

,, ,

, where

, since .

1 1

And hence the variance estimator can be written as:

$

.

, ,

,

V

NBR

NBR NNR

S

NNR

NBR NNR

S

S E E NBR

S NNR E E NNR

t

t

t t

p

t

t t

h

p jp

j BR

t

h th h p

h

H

t

t

11

2 2

2 2

2 2

1

1 1

1

1

= −

+

+ −

+

= − L

N

M O

Q

P −

=−−

∈

=

∑

∑

b g

b g

b g

b g

c h b g

c hb g

where , 

Thus, the relative bias of $Vt, , 11 is approximately:

EV V

V

t t

t

( $ ) , , 1 1 − ≈ − + − L

N

M O

Q

P

NNR

NBR NNR

E S

V

t

t t

h

t

b g

b g 1

2 ( )

where ES V h t ( ) 2 is the proportion of the variance explained

by the imputation strata.

Similar results are obtained for imputation methods 2-4.

For example, the formula for method 4 has the proportion

of the variance explained by the regression. The predicted

egression method curtails the spread of the employment

distribution.

The random regression methods 5 and 6 for imputation

adjust the employment distribution for the missing cases

and retain the residual variability exhibited in the

espondents' data. (In all these cases it is assumed that

espondents always respond over conceptually repeated

applications and non-respondents never do.)

In summary, the deterministic imputation methods

methods 1-4) distort the distribution and attenuate the

variance, whereas the stochastic imputation methods

methods 5-6) yield approximately unbiased estimates of

he distribution and the variance. In general for means, all

he methods lead to at least approximately unbiased

estimators.

6. Alternative Variance Estimators

In the empirical study three alternative estimators for

he variance were considered: Two jackknife versions and

a random groups method.

First consider the random groups method. Each unit

was randomly assigned into a group g, where there are G

andom groups. (In this paper, G=20 was used). The

andom group estimator is defined as:

$ $ V V t i t ig , , , ,,

g

4 4

1

= =

∑

G

G

$ $ ( $ VV V t i ti t i , , , , , ,() 21 2 = − G G -1) ⋅

where $Vt i , , 1 is the standard estimator in (5.3), and

$ $ V V t i t ig , , ,( ) , , ,( )

g

2 2

1

⋅

=

= ∑

G

G.

To compute the jackknife B estimator, the jac

estimator of the variance of the mean was multipli

population size. Let $Eg denote the mean estimat

population mean computed with only units in gro

$

E( ) g denote the mean estimator of the populati

computed without units in group g, then the jac

estimator is defined as:

$

Vt i , , 3 =NB E E t g

g

% % ) − − ( )⋅

=

∑e j2

1

1

G

G(G

where % $ ( ) $ E GE G E gg g = −−1 ( )

and % % . E EG (.) g

g

G

= =

∑

1

7. Results / Conclusions

Tables 1 and 2 show the errors in computing 

using the standard variance estimator. Notation:

Vt = VAR, NBR NNR t t + = N, $

Vt i , , 1 3+ = REGi, i=1

$

Vt, , 11 = MEAN, $Vt, , 1 2 = CARRY, and $Vt, , 1 3 = NEA

Table 1.

Percent Relative Absolute Error incurred in S

Variance Estimator due to Imputation

Stratified by 3 digit SIC/county. Non-response 

observed (OB) which is 3%-8% and fixed rate of 

Nonresponse Rate: As observed on file=OB

SIC VAR N REG1 REG2 REG3 MEAN CA

7 256.15 1614 0.84 0.84 0.86 0.97

24 757.13 761 0.03 0.02 0.03 0.19

37 40954.39 503 0.10 0.10 0.10 0.45

42 1300.66 1836 2.64 2.65 2.66 1.56

47 1006.08 785 0.13 0.10 0.11 0.92

53 7711.62 262 0.01 0.01 0.01 0.44

56 3903.65 1622 1.10 1.09 1.11 2.42

59 2659.32 6099 3.39 3.38 3.39 65.24

61 15265.53 302 0.00 0.00 0.00 0.07

76 131.41 1459 0.97 1.03 0.96 2.95

86 921.87 2871 5.67 5.67 5.67 22.09

88 8.17 1495 1.47 1.59 1.59 4.53

Nonresponse Rate: 25%

SIC VAR N REG1 REG2 REG3 MEAN CA

7 255.67 1562 6.50 6.34 6.45 15.96

24 610.17 690 3.36 3.17 3.16 5.44

37 42829.52 470 0.12 0.12 0.12 1.44

42 1313.77 1816 0.62 0.55 0.59 19.52

47 1024.35 756 2.60 2.63 2.71 19.11

53 7964.20 223 2.27 2.28 2.28 5.84

56 4130.02 1530 3.23 3.27 3.29 30.20 1
esponse rates, because certain observations could not be

used due to the requirements of certain imputation

procedures.

Table 2.

Absolute Percent Errors incurred in Standard

Variance Estimator due to Imputation

Stratified by 3 digit SIC/size classe (3 size classes). Nonesponse rates: as observed (OB) and 25%.

Nonresponse Rate: As observed on file=OB

SIC VAR N REG1 REG2 REG3 MEAN CARRY NEAR

7 252.43 1628 0.25 0.24 0.25 0.70 0.08 0.02

24 773.47 788 0.66 0.45 0.59 2.20 0.43 2.16

37 40728.95 506 0.05 0.05 0.05 0.35 0.16 0.24

42 1297.34 1841 2.78 2.77 2.79 3.63 3.53 3.13

47 1004.85 786 0.03 0.01 0.06 0.23 0.29 0.02

53 7495.46 270 0.01 0.01 0.00 0.04 0.01 0.01

56 3869.22 1637 0.74 0.79 0.75 2.06 0.79 0.62

59 927.41 6115 0.08 0.10 0.10 0.00 0.24 1.53

61 15265.53 302 0.00 0.00 0.00 0.02 0.00 0.00

76 130.67 1469 0.62 0.48 0.59 3.59 1.34 0.51

86 719.30 2879 0.01 0.03 0.02 0.13 0.05 0.04

88 8.16 1498 1.59 2.08 1.72 4.66 1.59 1.84

Nonresponse Rate: 25%

SIC VAR N REG1 REG2 REG3 MEAN CARRY NEAR

7 251.39 1626 6.79 5.85 7.06 8.93 3.29 17.42

24 607.01 787 2.39 1.92 2.22 3.57 1.07 1.65

37 40907.09 503 0.42 0.42 0.42 7.14 0.83 0.11

42 1297.34 1841 2.68 2.48 2.58 2.08 1.89 3.05

47 921.11 785 2.07 1.83 1.95 3.41 3.10 3.17

53 6701.98 267 2.47 2.47 2.47 5.13 3.84 1.15

56 3521.63 1634 1.88 1.83 1.89 3.78 13.85 1.11

59 2635.43 6116 1.52 1.49 1.50 2.02 0.20 1.32

61 15409.17 299 1.77 1.76 1.62 64.54 1.73 62.53

76 130.67 1469 1.52 1.84 1.67 12.79 5.32 1.61

86 895.19 2878 0.62 0.51 0.60 0.99 0.73 1.41

88 8.16 1498 1.35 2.82 1.72 17.40 1.35 0.37

Observations from Table 1

for OB%: REG1-3 and CARRY do well; both MEAN

and NEAR can produce very large errors.

for 25%: REG1-3 and CARRY do well, however there is

a large error for REG1-3 and for CARRY. Both

MEAN and NEAR can produce very large errors.

Observations from Table 2

for OB%: REG1-3, for the most part, produce the

smallest errors; however all the methods do fairly well.

There are no large errors for MEAN and NEAR as in

Table 1.

for 25%: REG1-3 do the best, there are no large error as

in Table 1. CARRY, MEAN, and NEAR can produce

large errors.

 As one would expect, the errors, for the most part, are

arger with 25% than with OB%.

County vs. Size Class Stratification

for OB%: Size class stratification produced smaller errors

than county stratification, with the biggest

improvements in the MEAN and NEAR methods The

Note that outliers in an imputation cell fo

county are more likely to occur than in an imput

formed by size class. Thus, it is not surprising th

errors were produced in the variances when the im

was done by county.

In summary, if the standard variance formula

then the imputation method that least dist

population variance is one of the regression typ

simplest regression type which is the single mode

residual added should be used, and stratification s

by 3-digit SIC/size class. This method is ro

different response rates, and resulting error mea

relatively small.

Table 3 shows the errors in computing the 

using different variance methods. The stratifica

done by 3-digit SIC/6 size classes, and only the 2

response rate was considered. Also, only the r

model with no residual added was consid

regression types. For m=1,2,3,4, the following 

used: $Vt, , 1 0 = RespV, $Vt m, ,4 = Rm (REG1), $Vt,m

(MEAN), $Vt m, ,2 = Cm (CARRY), $Vt m, ,3 = NNm (N

Table 3.

Absolute Percent Errors incurred in 4 V

Estimators due to Imputation

Stratified by 3 digit SIC/size classe (6 size classe

response rate: 25%.

SIC VAR N R1 R2 R3 R4 M1 M2 M3

7 251 1623 6.97 6.97 18.29 9.06 6.24 6.49 12.09

24 608 785 2.44 2.32 23.12 7.32 6.43 6.61 49.84

37 40907 503 0.41 0.28 29.74 1.74 7.23 8.93 23.19

42 1298 1840 2.75 2.76 51.74 1.79 3.06 2.96 30.69

47 921 785 2.09 1.95 0.37 6.39 2.10 2.11 0.29

53 6702 267 2.50 2.54 18.84 6.77 5.37 5.59 24.68

56 3522 1634 1.88 2.14 30.60 5.25 4.08 3.90 44.49

59 2635 6116 1.54 1.52 21.94 1.97 1.66 1.61 17.79

61 15360 300 1.77 1.60 15.67 7.16 64.41 64.29 71.73

76 131 1468 2.04 1.85 38.18 6.47 4.44 4.51 8.48

86 859 2877 0.08 0.33 42.50 6.59 0.97 0.54 50.49

88 8 1498 3.12 3.42 10.91 1.35 4.78 5.01 25.23

SIC VAR N C1 C2 C3 C4 NN1 NN2 NN3

7 251 1623 3.18 2.87 23.13 10.50 17.52 17.40 0.11

24 608 785 1.07 1.23 39.19 0.04 1.56 1.74 51.07

37 40907 503 0.83 2.47 26.42 19.79 0.11 1.49 35.85

42 1298 1840 1.89 2.03 43.04 0.34 3.31 3.17 32.40

47 921 785 3.10 3.15 4.95 3.59 3.27 3.28 8.94

53 6702 267 3.84 4.05 24.54 6.76 1.18 1.40 26.45

56 3522 1634 13.85 13.62 62.77 19.97 1.09 0.73 28.39

59 2635 6116 0.20 0.25 21.51 0.64 1.32 1.27 23.16

61 15360 300 1.73 1.32 20.46 10.94 62.53 62.42 66.07

76 131 1468 5.33 5.26 33.72 7.32 4.18 4.03 52.24

86 859 2877 1.30 1.74 71.50 5.53 0.09 0.52 44.18

88 8 1498 1.38 1.60 28.78 1.41 0.87 0.58 32.96
2. For each imputation method, the standard variance

method and the jackknife A method produced the smallest

errors of the four variance methods for most of the SICs.

Occasionally, the random group method and less

requently the jackknife B method resulted in the smallest

errors of the four variance methods, but it produced too

many very large errors to be reliable. For the two

promising variance methods, standard and jackknife A, the

minimum and maximum errors across the SICs are listed in

he following table for the four imputation methods.

Standard Jackknife A

Min.

Error

Max.

Error

Min.

Error

Max.

Error

REG1 .08 6.96 .28 6.97

MEAN .97 64.40 .54 64.29

CARRY .20 13.85 .25 13.62

NEAR .09 62.53 .52 62.42

It is clear from the above table that REG1 imputation

method with standard variance method has the smallest

minimum errors, and the smallest maximum errors.

3. Consider the 16 possibilities from the four imputation

methods and the four variance methods; the combination

hat resulted in the smallest and largest errors out of the 16

are given in the next table for each SIC.

SIC

Min.

Error

Imputation /

Variance

Method

Max.

Error

Imputation /

Variance

Method

7 .09 MEAN / RG 23.13 CARRY / JB

24 .04 CARRY / RG 51.06 NEAR / JB

37 .10 NEAR / SD 35.85 NEAR / JB

42 .34 CARRY / RG 51.74 REG1 / JB

47 .29 MEAN / JB 8.94 NEAR / JB

53 1.2 NEAR / SD 26.45 NEAR / JB

56 .7 NEAR / JA 62.77 CARRY / JB

59 .2 CARRY / SD 23.16 NEAR / JB

61 1.3 CARRY / JA 71.72 MEAN / JB

76 1.8 REG1 / JA 52.23 NEAR / JB

86 .08 REG1 / SD 71.50 CARRY / JB

88 .6 NEAR / JA 32.96 NEAR / JB

Clearly jackknife B is not a good method for computing

variances, regardless of imputation methods. However,

both REG1 and MEAN produced the largest error once,

as opposed to CARRY and NEAR which produced the

maximum error three and seven times respectively.

4. In Table 3, the last column indicates the error in the

variance if only the respondents' values are used to

compute the sample variance estimate, based on a sample

clear that even MEAN and NEAR are better 

imputing. MEAN and NEAR have a slight

maximum value than no imputation, but they ha

large errors.

Our recommendation for use in the Universe D

is the standard variance estimator along w

recommended REG1 method for imputation. F

base where data are imputed by using either strat

the carry over method or hot deck nearest neigh

results indicate that using the standard variance 

is as good or better than using either of the 

methods or random groups. Although jackknife A

did well, the difference did not warrant its use 

simplicity of the standard estimator. In oth

complex situations, other variance estimators 

considered, such as the jackknife variation sugg

Rao and Shao (1992).

8. Future Research

The next step will be to randomly select samp

the population, and consider variance estima

various statistics, such as means, totals, and r

coefficients, when some of the data have been 

Imputation methods could include the popular me

particular the regression type methods. Robust

estimators will be developed for variance estim

total when the imputation is done by regress

addition, the effect on the variance estimator of u

or more imputation methods on the same data se

investigated.

References

Efron, B. (1982), The Jackknife, the Bootstrap a

Resampling Plans, SIAM, PA.

Rao, J. N. K., and Shao, J. (1992), "Jackknife

estimation with survey data under h

imputation," Biometrika 79, 811-822.

Royall, R. M. and Cumberland, W. G., (1978), "

Estimation in Finite Population Sampling", Jo

the American Statistical Association, vol. 73.

Rubin, D., (1987), Multiple Imputation for Non

in Surveys, John Wiley and Sons Inc., NY.

West, S. A., (1982), "Linear Models for Mo

Employment Data", Bureau of Labor Statistics

West, S., Butani, S., Witt, M., Adkins, C.,

"Alternate Imputation Methods for Employme

ASA Proceedings of the Section in Survey 

Methods.

West, S., Kratzke, D., and Robertson, K.

"Alternative Imputation Procedures For Ite

response from New Establishments in the U