Comparing Alternative Median Wage Rate Estimators for the

Occupational Employment Statistics Survey

Carrae Echols, Albert Tou, and Kenneth W. Robertson, all of the U.S. Bureau of Labor Statistics

Key Words: Grouped Data, Median

I. INTRODUCTION

 The Bureau of Labor Statistics’ Occupational

Employment Statistics (OES) program is a joint

federal-state cooperative program that conducts an

annual survey of nonagricultural business

establishments. The program’s objective is to produce

annual estimates of total employment, mean wage rate,

and median wage rate by occupation within industry

group and geographic region. Industry groups are

defined by 3-digit Standard Industrial Classification

(SIC) codes, 2-digit SIC codes, and major industry

divisions. Geographic regions are defined by

metropolitan statistical areas (MSAs), residual areas

within a state that are not part of an MSA (balance of

state), the 50 states plus the District of Columbia,

Guam, Puerto Rico, and the Virgin Islands, and the

Nation as a whole. Because occupational employment

and wage rate estimates are produced at detailed

geographic (MSA) and industrial (3-digit SIC code)

levels, the OES sample design calls for combining up to

three consecutive years of data in order to produce

useful estimates.

 The questionnaire design uses a matrix format that

lists occupational titles down the left margin and

contiguous, non-overlapping wage bands across the top

margin (see the attached sample questionnaire). To

reduce respondent burden, the exact wage values of

individual workers are not collected. The survey,

consequently, uses a “grouped data” procedure to

estimate mean and median occupational wage rates.

Furthermore, since wage rate estimates are calculated

using wage data from three consecutive years, data

from the two previous years must be adjusted to

account for inflation.

 Although the OES survey produces estimates of total

employment, mean wage rate, and median wage rate,

the focus of this paper is on the estimation of median

wage rates. Findings from an empirical study of

alternative median wage rate estimators will be

reported.

II. DESIGN OF THE OES SURVEY

 Sample Design

 To produce MSA-level estimates by 3-digit industry

at the desired level of reliability, the OES survey design

requires a sample size of approximately 1.2 million

establishments. A sample of this magnitude places a

large burden on the responding units. Distributing the

sample across three consecutive years reduces that

burden to a tolerable level. Therefore, approximately

400,000 establishments are sampled each year.

Combining the samples across a 3-year time period

makes it possible to produce reliable MSA-level

estimates on an annual basis, while keeping the

respondent burden to an acceptable level. Overlap

among the samples drawn during any 3-year cycle is

kept to a minimum.

 The sampling frame for the survey is the list of

business establishments that reported their employment

and wage data to the state Unemployment Insurance

(UI) program. The frame for an OES survey conducted

in the 4th quarter of year t is the UI data collected during

the 2nd quarter of year t-1. Although we prefer to use

UI data collected during the 4th quarter of year t-1 as

our frame, UI data from the 2nd quarter are used instead

because that is the most recent time period when frame

employment data are available. Seasonal disparities

that arise between the sampling frame and the data

collection period are partially corrected by a process

called benchmarking. This process ratio adjusts the

weighted sample employment to the universe

employment as of the survey reference month in year t.

 Establishments in the sampling frame are stratified

by state-MSA, 3-digit industry code, and size of

establishment (i.e., employment size class code). The

sample size for a state-MSA/3-digit industry cell is

determined by a Neyman allocation process that

equalizes the expected relative standard errors of

selected “typical” occupations across all state-MSA/3-

digit industry cells. The sample selection process is

based on a permanent random number procedure that

allows the OES survey to minimize sample overlap

with other Bureau surveys. Additionally, procedures

are in place to ensure that any particular establishment

is not surveyed more than once during a three year

cycle.

 The sample allocation process described above

results in a sample size of approximately 400,000

establishments in each of years t, t-1, and t-2. Each

sample unit is assigned a sampling weight that is the

reciprocal of its probability of selection for that year.

When combining samples for two or three years, the

sampling weights are adjusted accordingly.

 Target Population

 The target population is all nonagricultural business

establishments for the 50 states, the District of

Columbia (DC), Guam, (GU), Puerto Rico (PR), and

the Virgin Islands (VI). The reference period of the
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survey is October, November, or December of the

current survey year and is dependent on the

establishment’s 2-digit SIC code.

 Data Collection

 Questionnaires are initially mailed out to almost all

sampled establishments. Some of the larger

establishments, however, are contacted by personal

visit. Two follow-up mailings are sent to nonresponding establishments at three week intervals.

After the follow-up mailings, nonrespondents are

contacted by telephone.

 The OES program currently defines approximately

750 detailed occupations across seven major

occupational groups.

III. ESTIMATING MEDIAN WAGE RATES:

A DESCRIPTION OF THE CURRENT

PROCEDURE AND A CLOSELY RELATED

ALTERNATIVE PROCEDURE

 These two procedures are identical for the first 6

steps below. They differ in the seventh step because

they make differing assumptions concerning the

distribution of the wage-employment data.

 Median occupational wage rate estimates for year t

are calculated as follows:

(1) Calculate inflation factors for the wage rate data

from years t-1 and t-2. The purpose of these factors is

to inflate wage rate data from the past two years up to

the level of the current year (t).

(2) Apply the inflation factors to the wage interval

bounds used during years t-1 and t-2. This is analogous

to applying the inflation factors to the wage data within

those bounds. Note: The wage interval structures for

years t, t-1, and t-2 will, in all likelihood, overlap one

another.

(3) Overlay the wage interval structures of years t, t-1,

and t-2 to create an extended (or universal) interval

structure.

A consequence of this overlaying process is that the 11

wage intervals defined for years t, t-1, and t-2 could

result in the formation of an extended interval structure

composed of 33 wage subintervals.

(4) Use a linear interpolation procedure to determine

the amount of employment within each of the 33 wage

subintervals.

Define

∑∈

=

k t

o q t k o k q

P w p , , , ,

ˆ

where

 wk = combined year sampling weight for

establishment k, ratio adjusted so that the

summed weighted employment matches

population totals, and

 po,k,q = reported employment for occupation o in 

establishment k in wage subinterval q. Note 

that this is within the original 11 interval 

structure.

This is the weighted employment within each wage

interval q, for year t. Define

 

, 1 , , 2 , , 1 3 , , 2

ˆ ˆ ˆ ˆ Po r =a Po q t + a Po q t− + a Po q t−

where

 α1 = the proportion of interval q from year t which 

overlaps in a linear manner with interval r,

 α2 = the proportion of interval q from year t-1 which

overlaps in a linear manner with interval r, and

 α3 = the proportion of interval q from year t-2 which

overlaps in a linear manner with interval r.

 (5) Calculate a cumulative frequency distribution of

estimated occupational employment across all wage

subintervals. Define

∑=

=

'

1

'

ˆ ˆ

r

r

Po r Po r

This is the cumulative frequency of the wage

subintervals up to and including subinterval r’.

 Use the distribution defined in (5) above to identify

the wage subinterval that encloses the median wage

rate.

(6) Define r′′ as the subinterval r′ where the

cumulative frequency Por′

ˆ

 is defined such that

 

( )

 ( ) 







≥

− <

o r o

o r o

P P

P P and

5* ˆ 0.

ˆ

5 * ˆ 0.

ˆ

'

( ' 1)

Step (7A) is a continuation of the current procedure

from Step (6) above.

Step (7B) is a continuation of the alternative procedure

from Step (6) above.

(7A) In the current procedure, a uniform distribution

model is applied to the appropriate wage subinterval to

derive a median wage rate estimate.
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3

 Apply a linear interpolation procedure to that wage

subinterval to determine the median occupational wage

rate estimate. Define

( ) '' 1 ''

"

( '' 1)

,0.5 ''

ˆ

5* ˆ ˆ 0.

ˆ

r r

o r

o o r

o r

lb lb

P

P P

R lb −















 −

= + +

−

where

 0.5 is the 50th percentile,

 lbr is the lower bound of subinterval r, and

 P$ o is the estimated employment for occupation o.

 When using this procedure we assume that the data

are uniformly distributed within the subintervals. Other

percentile estimates can be calculated in an analogous

manner.

 In summary, the method described above is a linear

interpolation procedure that distributes the wageemployment data from three separate interval structures

into one extended interval structure. A second linear

interpolation procedure which assumes a uniform

distribution within the subintervals of the extended

wage interval structure is then used to derive the

median wage rate estimate. This method will be

referred to as the “linear interpolation & linear

interpolation” (LILI) approach. Note that this is the

procedure currently used for the OES survey.

 Note that wage data from survey years t-1 and t-2

are updated (see Robertson and Frugoli) to represent

current wages using 4th quarter-to- 4th quarter wage rate

changes as reported by the Bureau’s Employment Cost

Index (ECI) program. Two key factors are used to

determine wage rate changes. They are

1. the broad ECI occupational groupings that

encompass the wage rate estimate. There are nine

broad ECI occupational groups at the national level,

and

2. the year differential between the current survey year

and the year in which the wage data were collected.

A caveat: Because wage rate changes are determined

by broad occupational groups at the national level

instead of by detailed occupations at the MSA level, it

is likely that these wage rate changes will gauge the

movement of wage rates with varying levels of success

for detailed occupations at the MSA level. For

example, if the wage rate change for the broad

occupational group “administrators and managers” at

the national level is 0.05, then we expect that there is

really a distribution of rates of change for detailed

occupations within this broad group that has a mean

rate of change of 0.05. Similarly, we expect that there

is a geographic distribution of rates of change for this

group that has a mean rate of change of 0.05. Since we

use the average rate of 0.05 across all geographic areas

and all detailed occupations within the broad

occupational group, we expect that there is some level

of error and bias associated with the use of this average.

(7B) In the alternative procedure, which we will refer

to as the “linear interpolation & non-uniform” (LINU)

approach, a “non-uniform distribution” model is

applied to the appropriate wage subinterval to derive a

median wage rate estimate.

 Apply an interpolation procedure to that wage

subinterval to determine the median occupational wage

rate estimate. This interpolation procedure utilizes a

data-derived non-uniform distribution.

• Using the data-derived distribution mentioned

above, locate where the [ ]

th Po Po,(r'' 1)

ˆ ˆ 0.5* − −

person lies, in ascending order based on wage

rates, in interval r’’.

• Next, use the data-derived distribution to identify

the wage rate of that person in interval r’’. This

value is the median wage rate.

IV. DESCRIPTION OF A SECOND

ALTERNATIVE ESTIMATION METHOD

 This section will describe a second alternative

procedure that is considered to replace the current

procedure.

 Binary Search method

 In this method, a binary search procedure is used to

designate a potential median value. The procedure is

calculated as follows:

(1) Calculate inflation factors for the wage rate data

from years t-1 and t-2. The purpose of these factors is

to inflate wage rate data from the past two years up to

the level of the current year (t).

 (2) Apply the inflation factors to the wage interval

bounds used during years t-1 and t-2. This is analogous

to applying the inflation factors to the wage data within

those bounds.

(3) Estimate the employment within each of the 11

wage intervals for each year. Define

∑∈

=

k t

o q t k o k q

P w p , , , ,

ˆ

where

 wk = weight for establishment k, ratio adjusted so

that the summed weighted employment

matches population totals, and
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4

 po,k,q = reported employment for occupation o in

establishment k in wage subinterval q.

Note that this is within the original 11 interval structure.

This value is the weighted employment within each

wage interval q, for year t.

(4) Use an iterative binary search procedure to

designate a test wage value for the median wage rate.

On the first iteration, the test wage value is set at the

midpoint of the range defined by the lower bounds of

intervals 1 through 11. Note that if 50 percent or more

of the employment falls in the 11th interval, we do not

calculate a median wage value. Define

 q’ as the interval in which the test wage value

resides.

 Apply a linear interpolation procedure to that wage

interval to determine the total weighted employment

less than the test wage value. Define

( )

( ) 















−

−

= +

+

−

=

∑ − q t

q q

q

q

q

o test t q t P

lb lb

test value lb

P P ',

' 1 '

'

' 1

1

, , ' 1,

ˆ

.

ˆ 'ˆ

and

, , , , , 1 , , 2

'

ˆ

'

ˆ

'

ˆ

'

ˆP o test = P o test t + P o test t− + P o test t −

where

 test.value = the test wage value being evaluated,

 P o,test,t'

ˆ = the weighted employment less than or

equal to the test wage valuefor year t, and

 P o,test '

ˆ

 = the weighted employment less than or equal

to the test wage value.

 This value, P o,test '

ˆ

, is compared to the value

( ) Po5 * ˆ 0. . If the values are within a designated

tolerance, then the test value is accepted as the

estimated median, otherwise a new potential median

value is calculated which halves the search space in the

appropriate direction. The procedure iterates until a test

wage value within tolerance is found.

V. COMPARISON OF ALTERNATIVE

ESTIMATION METHODS

 In order to conduct the empirical research we

utilized data from the Bureau’s National Compensation

Survey. These data are collected by personal visit, and

are point wage data (the OES survey collects interval

based wage data. Using the NCS data, we calculated

the “True” median wage value for each estimate. We

then placed the NCS data into the OES interval

structures to simulate OES data, and used the three

estimation procedures described in this paper. Two

statistics, the percent relative error (%RE) and the

percent absolute relative error (%ARE), are used to

evaluate the alternative estimation methods. These

statistics are defined as follows:

%RE = 100 * [ Estimate – “True” ] / “True”

%ARE = | %RE |

These statistics were calculated for each estimate, and a

frequency distribution developed. The following tables

present the distribution of these statistics for each

method.

Table I. – Distribution of Percent Relative Errors

Method

Percentile

Error

LILI

(current method)

Binary

Search

LINU

95th 10.68 10.75 7.04

90th 6.85 6.86 5.00

75th 2.97 2.98 2.21

50th 0.49 0.50 0.12

25th -1.50 -1.49 -2.17

10th -3.95 -3.94 -5.30

5

th -5.78 -5.72 -8.87

Mean Error 1.30 1.31 -0.28

Table II. – Distribution of Percent Absolute Relative

Errors

Method

Percentile

Error

LILI

(current method)

Binary

Search

LINU

95th 12.04 12.14 11.69

90th 7.91 7.94 7.68

75th 4.53 4.52 4.38

50th 2.19 2.19 2.18

25th 0.89 0.91 0.90

10th 0.32 0.33 0.33

5

th 0.16 0.16 0.17

Mean Error 3.64 3.68 3.49

 The method which performed the best in each case

has been highlighted.

 The relative error statistic may be used with these

data to determine if the distribution of errors is biased.

The mean of the LINU distribution is slightly negative,

while the means of the other two distributions are

positive. The LILI and Binary Search methods are very

similar. The LINU method does the best from the
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median to the 95th percentile. However, it performs the

poorest from the 25th percentile to the 5th percentile.

This is because the other two methods tend to be

slightly biased towards overestimation with these data.

Most of the larger differences in these distributions are

showing up in the outer parts of the distribution. It is

difficult to choose a best method based on the results in

this table.

 The percent absolute relative error statistic is used to

guage how well the method performed based on the

absolute size of the errors. An examination of Table II

shows that all of the methods are almost identical out to

the 95th percentile. The LINU method performs the

best at the 95th percentile, so that would be the method

of choice based on this table.

 Choosing between these methods is difficult, as they

perform approximately the same when reviewing both

the relative error and the relative absolute error. As

mentioned previously, the primary difference in these

estimators appears to be at the outer portion of the error

distributions. If we chose a new estimator at this point,

we would choose the LINU method. However, as the

reduction in the magnitude of the errors is small, we

will continue to research additional methods before

altering the current procedures. The current procedure,

with its operational simplicity, will be retained.

 One issue that concerns us is the percentage of large

errors associated with all of these methods. Given this,

we attempted to isolate where the bulk of these errors

are occurring. We suspected that most of them would

be occurring in the two higher wage intervals with the

largest width. In order to explore this we divided the

estimates data set into two data sets; one with those

estimates where the median fell in one of the two

widest intervals, and another with all of the other

estimates. We then calculated the absolute relative

error distribution for these data sets. These errors are

presented below for the currently used (LILI) method.

Table III. – Distribution of Percent Absolute Relative

Errors for two data sets

LILI (current method)

Percentile

Error

Median IS in one

of the two widest

intervals

Median IS Not in

one of the two

widest intervals

95th 30.08 7.64

90th 23.64 6.07

75th 14.88 3.74

50th 8.16 1.92

25th 4.17 0.79

10th 1.59 0.29

5

th 0.58 0.14

Mean 10.82 2.66

 A review of Table III indicates that the two widest

intervals do, in fact, have a much worse error

distribution than the rest of the intervals.

VI. CONCLUSIONS AND FUTURE

RESEARCH ACTIVITIES

 In this paper we have presented the results of an

empirical research project. We explored several

alternative grouped-data percentile procedures for use

with the OES survey data. Our primary conclusion

concerns the performance of the current median wage

rate estimation procedure. This procedure appears to be

working reasonably well; it balances a slightly larger

error distribution against significantly lower processing

costs.

 Another conclusion we can draw from this work is

that the errors in the tails of the (error) distribution are

larger than we would like to see. We have

recommended several ways to reduce the size of these

errors. The first recommendation is to add more

intervals. The use of additional intervals would allow

each individual interval to have a smaller width. One

additional wage interval is being implemented for the

1999 survey. Another recommendation is to select the

interval bounds in a manner that distributes the

potential errors evenly across all of the intervals. This

recommendation is also being implemented with the

1999 survey. Another alternative to explore is the use of

multiple sets of wage intervals. Each set of intervals

could be targeted towards the expected wage rates of a

group of occupations; however, there are workload,

burden, and data quality issues associated with this

suggestion that must be explored before any

recommendations can be made.

 In the future we plan to continue our efforts to

improve the OES survey estimates. Research is being

planned that will include both estimator and

questionnaire design issues.

 The opinions expressed in this paper are those of the

authors and do not reflect the opinions or policy of the

Bureau.

Bibliography

“Statistical Issues for the Redesigned Occupational

Employment Statistics Survey” Kenneth W. Robertson

and Pamela L. Frugoli, 1999 ASA proceedings,

forthcoming.