Alternative Imputation Procedures For Item Non-response from

 New Establishments in the Universe

S. West, D. Kratzke, and K. Robertson, Bureau of Labor Statistics

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

KEY WORDS: Regression, Bayesian, Multiple

Imputation, Births, Hot Deck

1. Introduction

 In this paper the results of an empirical investigation of

different imputation methods for item non-response from

new establishments are presented. The imputation is for

employment data given that wage data are known. This

investigation began in connection with a revision project

for the Bureau of Labor Statistics (BLS) program that

maintains the Universe Data Base (UDB). The data base

stores information received from the state Quarterly

Unemployment Insurance Address (QUI) files. The QUI

files represent a comprehensive list of all business

establishments that are covered under the unemployment

insurance system in the States. Each employer is required

to submit a QUI report which contains, among other

things, information on monthly employment for the

quarter, total quarterly wages, a standard industrial

classification (SIC) code and a county code for the

establishment. Although the filing of the report is

mandatory, there are always some reports that are filed

late, delinquent, or with partial data in each quarter. In the

case of partial data it is usually the employment data that

are missing. In previous papers imputation methods for

employment and wage data were considered for continuous

units which are units that were present in the previous

quarter. The situation of missing data from new

establishments was not considered. This paper deals with

the latter situation.

 The goal of this project was to develop a single

imputation procedure for new establishments that have

reported total quarterly wages but not employment that

would work reasonably well for all SIC groups within each

state. The methods tested included regression modeling

and distribution modeling with maximum likelihood

estimators for the parameters, multiple imputation, as well

as standard procedures such as hot deck, mean, and

median.

 The data used in this study are discussed in Section 2.

Section 3 presents the notation used in this paper and the

evaluation criteria that are used to compare the various

imputation methods. Section 4 provides a description of

the standard procedures such as mean, median, and several

hot deck procedures. In Section 5, eight regression models

for imputing employment given wages are presented. One

problem with a "best" regression-based prediction method

is that all imputed values will fall on the estimated

regression line and therefore, will lead to biases in

estimates that involve the residual variance for

nonrespondents. Simple methods that attend

to this problem draw random residuals which are added to

the model predictions. Details of such methods are given

in Section 6. In Section 7, imputations are created under

an explicit Bayesian model and multiple imputations are

developed in Section 8. In a multiple imputation context,

several imputed values would be created for each missing

value, where ideally, uncertainty due to the imputation

procedure would be reflected. Section 9 describes the

current method. Section 10 compares the results from the

various imputation methods and summarizes the findings

of this study.

2. Data

 Six quarters of UDB data were available for this study,

from quarter 1 of 1990 to quarter 2 of 1991. A unit

(establishment) is classified as a birth unit if it can not be

matched to any other unit in previous quarters by a number

of criteria. To assure that we did not mistakenly label a

unit as a birth, when perhaps it was inactive for a few

quarters, we decided to use units in quarter 1 of 1991. The

units in this quarter that are classified as birth units are not

matched to units in any quarter of 1990.

 Data from Michigan and California were obtained for the

following industries: Special Trade Contractors, Chemical

and Allied Products, Transportation Equipment, Trucking

and Warehousing, Apparel and Accessory Stores,

Miscellaneous Retail, Nondepository Institution, Personal

Services, Membership Organizations, and Private

Households. Additional industries from Michigan include:

Agricultural Services, Lumber and Wood Products,

Industrial Machinery and Equipment, Real Estate, and

Miscellaneous Repair Services.

 Intuitively, an establishment's total wages are highly

correlated with its total employment at any given point in

time. The more homogenous the strata, the higher the

correlation will be. Several stratifications were tried.

Within each 2-digit SIC chosen, we stratified the data

further by:

(1) 3-digit SIC; (2) 3-digit SIC/size class; (3) 4-digit

SIC/size class; (4) 4-digit SIC/county

 Usually a measure of size is created for each

establishment based on its most recent, nonmissing

monthly employment. But since the target of this study is

to impute employment for new units, we can not create a

measure of size for these units based on employment. For

our imputation procedures, size classes were formed by

breaking units into wage classes at the 25th, 50th, 75th,

and 95th percentile points of quarterly wages.

 In order to validate our procedures, we only selected

birth units that reported both non-zero employment and

wages. Thus, the minimum quarterly employment (sum of

monthly employment in the quarter) an establishment
could have is 3; so that in all the imputation procedures, 3

was set as the lower bound for quarterly employment. We

simulated the pattern of non-response observed in the data

as much as possible. If a particular industry has x% of

imputed employment for birth units, then a response rate of

(1-x)% was used. It was assumed that the missing data

mechanism is ignorable, and random sets of units were

chosen to represent the set of nonrespondents. Most of the

imputation procedures imputed for total quarterly

employment based on total quarterly wages.

3. Notation and Evaluation Criteria

Notation

Ei t, =quarterly employment for establishment i in quarter t,

$

Ei t, =predicted quarterly employment for establishment i in

quarter t,

Wi t, =quarterly wages for establishment i in quarter t.

 The problem is to impute for a new establishment k, that

has Wk t, , but is missing Ek t, . For a given stratified cell, let,

Bt=Set of birth units that have both reported wages and

employment for quarter t,

At=Set of continuous units that have both reported wages

and employment for quarter t,

nrt =percentage of birth units in the t th quarter that have

reported wages but no employment,

NRt=Set of units that were obtained by randomly selecting

the percentage, nrt , from the set Bt ,

BRt=remaining set of units in Bt ,

Mt = the set BRt or set BRt∪ At ,

NNRt=number of units in NRt ,

NMt=number of units in BRt or in BRt∪ At .

 The imputation methods will be applied to units of the

set NRt , The units in set Mt are used to fit different

modelling methods or to obtain imputed values from

standard procedures. The set NRt is called the set of

nonrespondents, or test set, and the set Mt is called the set

of respondents, or the model set.

Evaluation Criteria

 Let e kt kt kt ,,, E E $ = - denote the error in the imputed

value for establishment k. The following error measures

for each stratum will be used.

Percent Relative Error:

RE E k t

k NR

k t

t t k NR

= ∈ ∈100 ∑ ∑ ε , ,

Percent Relative Absolute Error:

RAE E k t

k NR

k t

t t k NR

= ∈ ∈100 | | ∑ ∑ , , ε .

The corresponding mean errors were also computed.

 Errors were computed for each imputed value and then

error measures were computed for each stratum, and then

added across strata for total error for each 2-digit SIC.

Note that RE represents a macro level statistic that

indicates the effect that the imputation procedure has on

total quarterly employment for each 2-digit SIC, while

RAE is a micro level statistic that indicates the effect of

imputation on each unit's quarterly employment.

4. Standard Methods

Mean and Median

 The mean imputation method is a common method of

imputation in many surveys, especially for those surveys

with a high response rate. If the response rate is low, then

this method of imputation would not be desirable because

it adversely affects the distribution of the sample units by

skewing the distribution toward the mean. The mean

imputation method was applied as follows.

 For any fixed SIC group, size class, and quarter t :

$

Ek t, = E NM i t

i M

t

t

,

∈

∑

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

employment of all respondents in the stratum. In the

Tables, these estimates are referred to as Mean-3 or Mean4, depending on whether the imputation was done at the 3-

or 4-digit SIC level. $

Ek t, equaling the median of the total

quarterly employment of all respondents in the stratum was

also tried. These estimates are referred to as Med-3 or

Med-4, again depending on whether the imputation was

done at the 3- or 4-digit SIC level.

Mean and Median - Variations

 The Mean Ratio method, denoted by MeanR, was

calculated in the following manner. For any fixed SIC

group, size class, and quarter t, the mean for total wages,

Wt , and the mean for total employment, Et , was calculated

over Mt. The imputed employment is then:

$

Ek t, = ( ) E Wt t Wk t,

(It will be seen that this is the same basic procedure as

using Regression Model 2, which will be discussed in the

next section.)

 The Median Ratio method, denoted by MedR, is similar

to the preceding one with median replacing mean.

Hot Deck - Nearest Neighbor

 For any fixed SIC group, size class, and quarter t, let k

denote a nonrespondent and c denote a respondent such

that

W W W W for all i M ct kt it kt t ,, ,, − ≤− ∈ , .

then, $Ek t, = Ec t, .

 The Nearest Neighbor hot deck method, denoted by NN,

is desirable because for any particular nonrespondent, it

selects the respondent that appears closest to the

nonrespondent in an ordered list, and substitutes the

respondent's total quarterly employment value for the

nonrespondent's.

Hot Deck - Nearest Neighbor Variations

 Two variations were tried. One, denoted by NNI, used a

linear interpolation in the ordered list. The second,

denoted by NNIR, is identical to NNI, except in the border

cases, when a ratio adjustment was made.

Hot Deck - Random Selection

 For any fixed SIC group, size class, and quarter t,
$

Ek t, = Er t,

1

where Er t,

1 is the employment value of an establishment

randomly selected from Mt . In the tables the estimates are

referred to as RAND.

5. Modeling Employment by Regression

Regression Models

 A common method for imputing missing values is via

least squares regression (Afifi and Elaskoff, 1969). In

several papers on estimators for total employment (West

1982, 1983,) and West, et al (1989), it was discovered that

the most promising models for employment were the

proportional regression models. These models specify that

the expected employment for establishment i in the tth

quarter, given the following vector of E - values for quarter

t-1:

Et t t t tn − −− − − 1 11 12 13 1 = EEE E ,, , , , , ,...,

is proportional to the establishment previous quarter's

employment, Et i -1, . That is,

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

where β is some constant depending on t.

 It was further assumed that the E's are conditionally

uncorrelated. That is,

cov( , | ) , ,

,

E E if i j

otherwise ti t j t t

t i E e − − = = R =

S

T 1 1

ν

0 

 

where νt,i represents the conditional variance of Et i, which

in general will depend on Et i -1, . Choosing a specific

simple function to represent the variance νt,i accurately is

difficult. Fortunately, knowledge of the precise form of νt,i

is not essential, (see Royal, 1978).

 The model can be rewritten as:

Et i, = βEt i -1, + εt,i

where

E{εt,i} = 0,

and

E{εt,i,εt,j} = 

nt i if i j

otherwise

R , =

S

|

T

|0

 In previous papers, νt,i = σ2Et i -1, and νt,i = σ2 were

considered and it was found that the model:

Et i, = βEt i -1, + εt,i with νt,i = σ2Et i -1,

worked reasonably well for employment data.

 A similar model worked well for wages except the data

were first transformed by applying the natural logarithm to

each wage value. Since this model with the above

assumptions worked well with employment and wage

data, it was decided to apply variations of the same model

with employment versus wage data. For the current data

set, the following eight models were considered for total

quarterly employment versus total quarterly wages.

E W jt jt jt , ,, = + b e 1 with e j t, .~N 0 2 c ,s h (1)

E W jt jt jt , ,, = + b e 2 with e j t, .~N w 0 j t

2 ,s , c h (2)

ln ln E W jt jt jt , ,, = + b e 3 d i with e j t, .~N 0 2 c ,s h (3)

ln ln E W jt jt jt , ,, = + b e 4 d i with e j t, .~N wj t 0 2 , ln s , c h (4)

E W jt jt jt , ,, =+ + ab e 5 5 with e j t, .~N 0 2 c ,s h (5)

E W jt jt jt , ,, =+ + ab e 6 6 with e j t, .~N wj t 0 2 ,s , c h (6)

ln ln E W jt jt jt , ,, =+ + ab e 7 7 d i with e j t, .~N 0 2 c ,s h (7)

ln ln E W jt jt jt , ,, =+ + ab e 8 8 d i withe j t, .~N w 0 j t

2 , ln s , c h(8)

 The models will be fit over the set Mt by stratum. The

models were fit for each 3-digit SIC and 4-digit

SIC/county. The more homogenous the strata the better

the models will be.

 Example of fitting model 4:

ln ln E W jt jt jt , ,, = + b e 4 d i with e j t, .~N wj t 0 2 , ln s , c h

and b4 is estimated as:

$

β4 = ln ln , ,

∈ ∈

∑ ∑ E W i t

i

i t

 MMi t t

 For establishment j in NRt , the establishment's predicted

total employment is:

$ exp{ $ ln }. E W jt jt , , = b4

 The regression models 1-8 are denoted by REG1-REG8,

respectively.

Adjustments for Log Models

 Consider models r, for r = 3, 4, 7, 8. If it is assumed that

e j t, is normally distributed, then Ej t, has a lognormal

distribution with

Mean: exp{ b e r jt jt ln( ) . ( ) W Var , , + 5 }

Var: {exp [ ( )] Var j t, e -1} exp{2 b e r jt jt ln( ) ( ) W Var , , + }.

 Therefore, an unbiased estimator of Ei t, is:

 exp{ b e r it it ln( ) . ( ) W Var , , + 5 }.

 As an estimate of Var(ei t, ), the residual mean square

error, MSE, from the regression was used. The predicted

total employment for r = 3 and 4 were computed as:

$

Ei t, = exp{ $ ln( ) . br it W, + 5MSE}

 The log regression models with adjustment are denoted

by REG3ADJ, REG4ADJ, REG7ADJ, and REG8ADJ,

corresponding to the regression models REG3, REG4,

REG7, and REG8 without the adjustment.

6. Adding Residuals to the Regression Models

 The methods discussed in the previous section could be

thought of as imputing for missing total quarterly

employment by using the mean of the predicted Et (or

ln(Et)) distribution, conditional on the predictors, Wt(or

ln(Wt)). As a result, 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, draw

residuals resj t, , with mean zero, to add to $Ek t, (or the

predicted ln(Ek t, ))
 In this project, it was decided to consider this imputation

procedure with the residuals, resj t, equaling:

 1. A randomly selected respondent's residual, using

each of the eight models. These models are denoted by

REG1RES-REG8RES, corresponding to REG1-REG8.

 2. A random normal deviate, from the distribution with

mean 0 and variance MSE. These models are denoted by

REG1NOR-REG8NOR, corresponding to REG1-REG8.

 For example, using model 7 and the first method

described above, a prediction of Ek t, is:

$

Ek t, = exp{ $ $ (ln ) a b 7 7 + + W res kt jt , , },

where resj t, is the residual from a randomly selected

respondent j; that is,

resj t, = [ ln $ $ (ln ) ] E W jt jt , , - - a b 7 7 .

 Using model 6 and the second method described above:

$

Ek t, = $ $ ab d 6 6 + + W s kt k , ,

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

and s

2

 is equal to the MSE.

7. Bayesian Model

 In creating imputed values under an explicit Bayesian

model, three formal tasks can be defined: modeling,

estimation and imputation. The modeling task chooses a

specific model for the data. The estimation task formulates

the posterior distribution of the parameters of that model so

that a random draw can be made from it. The imputation

task takes one random draw from the posterior distribution

of Et , for Et ,Î NRt , denoted by Et,BAY. This is done by

first drawing a parameter from the posterior distribution

obtained in the estimation task and then drawing Et,BAY

from its conditional posterior distribution given the drawn

value of the parameter.

 For the modeling task, consider model 1 and Ej t, having

a N( b1 Wj t, ,s2) distribution. This is the specification for

the conditional density f(Ej t, x Wj t, , q) where q = ( b1,s).

In order to complete the modeling task, the conventional

improper prior for q, Prob(q) proportional to a constant, is

assumed.

 For the estimation task, the posterior distribution of q is

needed. Standard Bayesian calculations show that:

f(s2 x Ej t, ) = s c $ ( ) 0

2

1

2 n 1 - nf( b1, x s2) = N( $

b0 ,s2n)

where

$ $ ( ) s b 0 , ,

2

0

2

=- - å E Wn 1 jt jt

j

e j = MSE

$

b0 ,, ,

2 = å å EW W jt jt

j

j t

j

ν = 1 ∑ 2 Wj t

j

,

where n = number of respondents.

 Since the posterior distribution of q is in terms of

standard distributions, random draws can easily be

computed. The imputation task for this model is as

follows:

1. Estimate s2 by a c n-1

2 random variable, say h, and

let

s 1

2

 = s$

0

2

(n-1)(h)-1

2. Estimate b1 by drawing one independent N(0,1)

variate, say Zo, and let

$ $ . b b sn 00 0 1

5 = + Z0

3. Let no be the number of values that are missing.

Draw no values of Et,BAY as

$

E WZ k t BAY k t k ,, , = + b s 0 1

where the no normal deviates, Zk are drawn

independently.

 The above equation can be rewritten as:

$ $ ( ) [ ]. ,, ,

. ... E W MSE n h Z W Z k t BAY k t k t k =+ - + ,

- b n 0

5 555

0 1

These Bayesian models are denoted by REG1BAYREG8BAY, corresponding to REG1-REG8.

8. Multiple Imputation

 Multiple imputation is the technique that replaces each

missing value with two or more acceptable values from a

distribution of possibilities. The idea was originally

proposed by Rubin. The main advantage of multiple

imputation is that the resultant imputed values will

account for sampling variability associated with the

particular nonresponse model.

 Multiple imputation was obtained from the Bayesian

method by repeating the above three steps. Five

independent imputations were obtained by repeating the

three steps. The average of these five values was taken as

the imputed value. These methods are denoted by

REG1BAYM-REG8BAYM, corresponding to REG1-

REG8.

 Multiple imputation was also obtained for the regression

model with randomly selected residuals and regression

model with randomly generated residuals, N(0, MSE). For

all of the multiple imputation methods, error measures

were computed by using the average of five such repeated

imputations. For randomly selectied residuals, the models

are denoted by REG1RESM-REG8RESM; and for

randomly generated residuals, the models are denoted by

REG1NORM-REG8NORM; corresponding to REG1-

REG8.

9. The Current Method

 The current method is described in Appendix D of the

Exportable ES-202 System. This method will be referred

to as the EXPO method. The EXPO method is stratified

by 4-digit SIC/county/ownership and it uses data from a

year ago to form the ratio for imputing. In our paper,

however, the ownership code will be excluded since only

private ownership was considered in this study.

 A ratio of total quarterly employment to total quarterly

wages of a macro record for the same quarter a year ago is

computed. This ratio is multiplied by the unit's total

quarterly wages to impute for quarterly employment. The
monthly employment is computed by dividing the quarterly

employment by three times a prorate factor which indicates

how many months the establishment is active in the

quarter. For this research, only total quarterly employment

is imputed. Note that this method is similar to using

regression model 2, except with regression model 2 the

ratio is computed at the current time period. That is,

using REG2,

$ $ $ E W where E W kt kt it , , ,,

i J

i t

i J t t

= = Î Îb b 2 2 å å

using EXPO,

$ $ $ , , ,( ) ,( )

() ()

E W where E W kt kt i t

i J

i t

t t i J

= = -

Î

-

- - Î

b b å å 4 4

4 4

where the subscript (t-4) denotes the quarter a year ago.

10. Comparison of Imputation Methods/Conclusions

 At the beginning of the research, it was not clear whether

to use establishments in the set BRt or the set BRt∪ At to

obtain information for imputing employment for

establishments in the set NRt .

 In the first part of the research, model sets with only birth

units were used, excluding those establishments that had

total quarterly wages less than or equal to $110,500 (this

figure was based on 50 employees making minimum wage

of $4.25/hour each). For the States of California and

Michigan, 18 imputation methods were applied to each of

seven SICs (with an additional SIC in California) by

various partitions and, accordingly, the error measures

were computed for each combination. The 18 methods are:

Mean-3, Mean-4, Med-3, Med-4, MeanR, MedR, NN,

NNI, NNIR, RAND, and REG1-REG8. The regression

models were done on 3-digt SIC. Within each state, the

methods were ranked according to the error measures |RE|

and RAE.

 Selecting the best imputation method from the set of 18

methods considered was difficult, because one method of

imputation did not consistently and clearly yield the

smallest error measures. Consequently, in order to

determine the best method of imputing birth total

employment for all the SICs and the two states, the models

were ranked according to several criteria. These criteria

were as follows:

(1) The number of times a method yielded small errors,

i.e., |RE| < 15 and RAE < 55

(2) The number of times a method yielded large errors, i.e,

|RE| > 30 or RAE > 80

(3) The number of times a method ranked in the top 5 (or

the top 10) according to |RE|

(4) The number of times a method ranked in the top 5 (or

the top 10) according to RAE

(5) Total |RE| across all SIC's

(6) Total RAE across all SIC's

 Note that the errors are relative and are summed only

over the nonrespondents. After comparing the scores of

the eighteen methods on the six criteria, eight methods

were eliminated. When the ten remaining methods were

reranked according to |RE| and RAE, Mean-3 and Mean-4

came to the top of the list.

 Next, we included continuous units as well as birth units

in the model set, that is, all establishments in the set At U

BRt . In this preliminary study on all units, the 18 methods

metioned above and the EXPO procedure were done on the

same seven SIC's from Michigan as with the birth units

alone. After the 18 methods were ranked according to |RE|

and RAE and the scores for the six criteria were compared,

the promising methods were: MeanR, REG4-REG8.

 In order to be able to directly compare our procedures

with the current procedure, we decided to try the same

stratification as the current procedure (which is 4-

digit/county), using both continuous and birth units in the

model sets, and including units making $110,500 or more

in the study. Since the standard procedures did not do well

in the preliminary phase using both birth and continuous

units in the model sets, only the distribution modelling was

done in this phase. The following methods were done:

regression models, including adjustment to log models,

regression models with residuals, Bayesian, and multiple

imputation methods. However, due to time limitations, we

only did the Bayesian for regression models 1, 7, and 8.

The multiple imputation was done on the Bayesian method

and on the regressions with residuals. A total of 51

procedures were done on the 12 SIC's from Michigan.

 Based on the six criteria mentioned before, the ten best

methods were REG2, REG2NOR, REG2NORM,

REG6NOR, REG6NORM, REG7, REG7BAY, REG8,

REG8ADJ, and REG8NOR. After comparing these

methods, and noting the variances of |RE| and RAE across

all 12 SIC's, the list was narrowed down to the following

five mehtods: REG6NOR, REG6NORM, REG2,

REG8ADJ, and REG8NOR.

 Since these models did not differ markedly in their

effectiveness, and because of cost considerations and the

simplicity of the calculations under the model, we chose

the REG2 model to be implemented in the ES-202

program. In practice this model can be implemented as a

simple ratio adjustment. Also, this procedure is similar to

the current procedure, except that more recent information

is utilized.

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