Accepted for publication by Computational Statistics and Data Analysis

AN APPLICATION OF MATHEMATICAL PROGRAMMING

TO SAMPLE ALLOCATION

Richard Valliant

Bureau of Labor Statistics

Room 4915

2 Massachusetts Ave. NE

Washington DC 20212, USA

James E. Gentle

Institute for Computational Sciences and Informatics

George Mason University

Fairfax VA 22030-4444, USA

November 1996
ABSTRACT

The problem of sample allocation in multipurpose surveys is complicated by the

fact that an efficient allocation for some estimates may be inefficient for others. There

may also be precision goals that must be met for certain estimates plus constraints on costs

and minimum sample sizes for strata to permit variance estimation. These requirements

lead to formulating the allocation problem as one of mathematical programming with an

objective function and constraints that are nonlinear in the sample size target variables.

We discuss a flexible approach for a two-stage sample allocation that uses multicriteria

optimization programming. Software was developed to permit survey designers to easily

explore alternative problem formulations and to compare the resulting allocations. The

method is illustrated using a business establishment survey that estimates the costs to

employers of providing wages and benefits to employees and the percentages of

employees that receive certain benefits.

Keywords: Nonlinear optimization; multicriteria programming; two-stage sampling;

variance component estimation.
1

1. INTRODUCTION

Multipurpose surveys produce estimates for many variables and domains. These

multiple products complicate the problem of sample allocation since a given allocation will

not be efficient for all estimates. To reduce costs and produce datasets with richer

analysis possibilities, there may also be pressures to consolidate and use the same sample

for multiple surveys. This may be particularly true in surveys sponsored by governments.

This paper will discuss an application of multicriteria optimization to sample design, using

as an illustration, data drawn from the Employment Cost Index (ECI) and the Employee

Benefits Survey (EBS) conducted by the Bureau of Labor Statistics (BLS). The two

surveys use the same two-stage sample of establishments and occupations to estimate

personnel costs and the percentages of employees receiving various benefits. The

estimates are made for the population as a whole and for domains. We cover several

topics applicable to many sample surveys—estimation of variance components for a multistage design and complex estimator, smoothing of variance component estimates to

eliminate inconsistencies, and the use of constrained nonlinear programming to optimize

the sample allocation. The paper also illustrates some of the practical compromises and

approximations that must be employed in the design of a complex sample.

Kish (1988) noted the variety of purposes for which a given survey may be used

and why purposes may conflict. A number of different variables may be measured and

estimates may be made for diverse domains. Some domains may be straightforward

combinations of design strata, like regions, while others may be crossclasses, like

occupational groups, that cut across design strata. The ECI, for example, collects the cost

to employers of paying for wages and salaries, leave (e.g., holidays, vacations), medical
2

and life insurance, legally required benefits (e.g., social security, unemployment insurance,

workers' compensation), and retirement plans. The EBS collects the numbers of

employees who are eligible for or receive various benefits. Estimates are made of an index

of change in costs between time periods, of the average cost per employee per hour

worked, and of the percentage of employees receiving various benefits within domains,

including industry group (e.g., construction, manufacturing, wholesale, services),

establishment size, class of worker (e.g., professional, technical, and related occupations,

sales occupations, service occupations), and geographic region.

Sample allocation problems, using either variance minimization or cost

minimization, can be formulated using constrained, multicriteria optimization described,

for example, in Narula and Weistroffer (1989), Steuer (1986), or Weistroffer and Narula

(1991). An objective function that is a weighted combination of the relvariances of

different estimators is formed with each weight being the “importance” of each statistic in

the overall survey design. A final component of the objective function is a weight times

the total cost (or sample size). More formally, the objective function to be minimized in

this study is

f = J +

=

∑w w +

c

L

L

1

1

where = 1, , L are the indexes of the estimators, J is the relvariance of estimator 

(i.e., the variance of the estimator divided by the square of its expected value), c is the

total survey cost, which is implicitly a function of sample size, and w1 wL 1

, ,

+

 are weights

assigned to the relvariances and the cost. Because the relvariance is a unitless measure,

we can include estimators in the objective function for variables that are measured on
3

different scales, e.g., wages and proportions. However, since the relvariances are unitless

and the cost is measured in monetary units, they should not simultaneously enter the

objective function. Using a reduced gradient programming algorithm, the weighted

combination is minimized subject to a variety of constraints, including one on total cost or

sample size, minimum and maximum sample size constraints in each stratum, and

relvariance constraints on individual estimators.

Including both relvariances and cost in the objective function has no special

mathematical advantages since, when the relvariances receive nonzero weight, the cost

weight wL+1 should be zero, and vice versa. However, inclusion of cost and relvariances

in f is extremely convenient when using software that flexibly allows weight adjustment.

One of the important byproducts of this research was a system, described in section 5, that

lets a user interactively change the weights in f , re-optimize, and compare results to

previous allocations.

To evaluate the objective function, specific formulas are needed for the

relvariances. The ECI/EBS sample design and estimators, like many others in complex

surveys, are complicated and appropriate variance formulas cannot be taken directly from

a textbook. Sections 2 and 3 sketch the derivation of relvariances needed for this

application. Because a 2-stage sample is used, variance components for each stage are

derived, and their estimation is discussed in section 4. The software for optimization and

the numerical results are covered in sections 5 and 6.

2. SAMPLE DESIGN
4

This section describes the general sample design used in the ECI and EBS

programs. Sample data from this design will be used in estimating variance components

that, in turn, will be used in exploring how to allocate the sample. The design involves

two stages of selectionestablishments at the first stage and occupations at the second.

First, a sample of establishments is selected within each stratum with probabilities

proportional to total employment in each establishment as shown on the frame at a

particular date. Strata are defined by standard industrial classification (SIC) and

employment size. At the second stage, a sample of occupations is selected within each

sample establishment. This is done by selecting a systematic sample of individual

employees from a personnel list in each establishment and enumerating all workers in the

occupations held by the selected employees. If, for example, a janitor is selected in the

systematic sample from an establishment, then all janitors are enumerated. Occupations

with more workers are, thus, more likely to be in the sample. The occupation sampling

procedure is simple to implement in the field but does allow a particular occupation to be

selected or “hit” more than once, introducing some complications into analysis. This point

is discussed further at the end of this section.

In order to proceed, we need some notation. Let h denote a stratum defined by

SIC/size and i an establishment within the stratum. Define

phi = inclusion probability of establishment hi

nh = number of sample establishments in stratum h

p j hi = expected number of times that occupation j is selected within establishment hi

mh

 = number of sample occupations assigned to sample establishment hi, which is the

same for each sample establishment in stratum h,
5

sh = set of sample establishments in stratum h, and

shi = set of sample occupations within sample establishment hi.

The same number mh of sample occupations is assigned to each sample establishment in a

stratum in order to control work loads. Note that the number of unique occupations

obtained in the subsample from an establishment will be less than mh when a particular

occupation is hit more than once.

The quantities phi and p j hi are general. Specifically for this application, if Ehi is

the number of employees in the establishment, then the selection probability of

establishment hi is p hi h hi h = n E E where Eh is the total frame employment in stratum h.

If Ehij is the number of employees in occupation j in establishment hi and no occupation

has Ehij > Ehi mh, then the selection probability of an occupation within the establishment

is p j hi = mh Ehij Ehi . The overall selection probability of unit hij is then

p hij p hip = j hi = nhmh Ehij Eh. In a case where there are one or more occupations with

Ehij > Ehi mh, the term p j hi is the expected number of times that occupation j is selected

given that establishment hi is selected. In that situation, phij is the unconditional

expectation of the number of times that the combination hij is selected.

To make variance calculations tractable, we later assume that establishments are

sampled by stratified simple random sampling without replacement and that occupations

are sampled with probabilities proportional to size but with replacement. The reasoning

behind these assumptions is discussed in the next section.
6

3. THE ECI AND EBS ESTIMATORS

In both the ECI and EBS most published estimates are specific to domains, e.g.,

the average cost per hour per employee for wages and salaries in goods-producing

industries or the percentage of clerical and sales workers who receive paid vacation.

Suppose that De is a domain of establishments defined by grouping strata (e.g.,

manufacturing) and Do is a class of occupations (e.g., clerical and sales). Let yhijk be the

variable measured on worker k in stratum/establishment/occupation hij. For ECI yhijk

might be the worker's average hourly wage; for EBS yhijk = 1 if worker hijk has a

particular characteristic (e.g., receives long-term disability insurance) and 0 if not.

Computations will be facilitated by having an indicator for units that are in a particular

establishment/occupation domain. Since entire occupations and/or establishments are

assigned to a domain or not, let d hij =1 if establishment/occupation hij is in the domain and

0 if not. An estimator of the domain total of y is

T y y hij

j hi

hij

hij

h i s j s

h hi

=

∈ ∈

∑∑ ∑d

g

p

. (1)

where, for Uhij the universe of workers in hij, y y hij hijk k Uhij= ∑ ∈

is the total for occupation

j, and g

j hi is the number of times that occupation j is selected in establishment hi. The

total yhij is collected by the interviewer after the sample is selected for only those

occupations in the sample. The term g

j hi is needed in (1) because of the assumption,

discussed later, that occupations are sampled with replacement within an establishment.
7

Note that the sum over j s ∈ hi in (1) can be replaced by a sum over j ∈Uhi , the

universe of occupations in establishment hi, since g

j hi is 0 for all occupations not in the

sample. The estimated number of employees in the domain is

TE hij E

j hi

hij

hij

h i s

h

j Uhi

=

∈ ∈

∑∑ ∑ d

g

p

.

Because g

j hi = 0 for nonsample occupations, Ehij is needed only for the sample

occupations and is collected at the time of sampling by the interviewer. The proportion of

employees in the domain who have the characteristic (or the mean per employee if y is

quantitative) is then estimated as

m = Ty TE.

To approximate the variance of m, use the usual first-order approximation for a ratio

m − m ≅ (Ty − mTE ) TE

= ⋅

∈ ∈

∑∑ ∑

1

T

z

E

j hi

hij

hij hij

h i s j s h hi

g

p

d (2)

where Ty and TE are finite population totals, m = Ty TE

, and zhij = yhij − mEhij .

3.1 Variance Decomposition

To calculate a variance to be optimized in the allocation of the sample, we use the

concept of anticipated variance introduced by Isaki and Fuller (1982). The anticipated

variance (AV) of m is

ExEp[(m − m)− ExEp (m − m)] = ExEp (m − m) −[ExEp (m − m)]

2 2 2

.
8

where Ex denotes expectation with respect to a superpopulation model and Ep is a

expectation taken with respect to a sample design. When m is design-unbiased (or

approximately so), the AV can be written as

Ex p (m) Ex p (Ty mTE ) TEvar ≅ var −

2

where varp is the design variance. Ex

var

p (m) is a variance anticipated at the time the

sample is constructed and has several practical advantages for sample design. The designbased variance calculation will allow finite population correction factors to enter the

variance in a somewhat easier way than a purely model-based computation would. The

superpopulation approach, on the other hand, permits explicit modeling of characteristics,

which will prove to be an important advantage when estimating variance components.

To evaluate the effects of different sample sizes at the two stages of selection, we

need to write the variance as a sum of components associated with establishments and

occupations within establishments. The standard approach to deriving variance

components is to apply the conditional variance formula

varp (m)= varp Ep (m s1)+ Ep varp (m s1) (3)

where s1 is the vector of all first-stage stratum samples sh. Although it is possible to

compute variance components under certain probability proportional to size (pps) sample

designs, the results often involve joint probabilities of selection and are difficult or

impossible to work with in practice. Särndal, Swensson, and Wretman (1993, ch.4)

discuss the design-based methods. If the strata are based on size, as in the ECI/EBS, and

are numerous and narrow, a reasonable simplification is to assume that all establishments

in a particular stratum have about the same number of employees. In that case, a pps
9

sample selected without replacement is equivalent to a simple random sample selected

without replacement (srswor) in each establishment stratum. The selection probability of

establishment hi is then phi h h = n N . Being able to incorporate a finite population

correction factor is important in this application because the sampling fraction in some

strata is non-negligible. The second-stage sample of occupations within sample

establishments is also pps. It seems less reasonable to assume that the second stage

selection can be well approximated by equal probability sampling since the number of

employees in different occupations varies widely in many companies. In the subsequent

development, we assume that pps sampling with replacement is used to select occupations.

The mechanics of second-stage occupation sampling, that allows an occupation to be hit

more than once, is very similar to with-replacement pps sampling. Interviewers typically

select the second stage samples from alphabetically sorted personnel files whose order is

unrelated to occupation. Consequently, the possibility of having periodicities that can foul

the properties of systematic sampling is small.

3.2 First and Second-stage Variance Components

The first term on the right-hand side of (3) will generate the between-establishment

variance component. Since E( j hi) j hi g = p , phij phipj hi = , and we assume srswor at the

first stage, it follows that E (T T ) N z p y E h hs h

~ − m s1 = ∑ where, ~ ~ z z n hs hi h i sh

= ∑∈ with

~ ~ z z hi hij j Uhi

= ∑ ∈, 

~z z hij = d hij hij , and Uhi being the universe of occupations in establishment

hi. From the usual formula for the variance of a stratified total under srswor we have
10

var

p p ( y E ) ( )

h

h

h h

h

E T T

N

n

− m s = ∑ − f S 1

2

1

2

1 (4)

where S h (zhi zh ) (N )

i U h

h

1

2

2

= − −1 ∑∈

~ ~ with ~ ~ zh

zhi Nh i Uh

= ∑∈ and Uh the universe of

establishments in stratum h.

For the second-stage variance component, we need

var ( ) var

~

p y E

i s hi

p j hi

hij

h j U j hi

T T

z

h hi

− =

















∈ ∈

m ∑∑ ∑ pg

p

s1 2

1

. (5)

Defining p p j hi j hi mh

∗

= to be the 1-draw selection probability of occupation j and using

Result 2.9.1 of Särndal, Swensson, and Wretman (1993, p.51), the variance on the righthand side of (5) becomes

var

~ ~

~

p j hi

hij

j U j hi h

j hi

hij

j hi

hi

j U

z

m

z

z

hi hi

g

p

p

∈ p

∗

∗

∈

∑ ∑

















= −

















1

2

.

Consequently,

E (T T )

N

n

S

m

p p y E

h

h

hi

h i Uh h

var − =

∈

m s1 ∑ ∑

2

2

(6)

where S hi j hi(zhij j hi zhi)

j Uhi

2

2 2

= −

∗ ∗

∈

∑p p

~ ~ . Combining (4) and (6), the design-based variance is

T ( ) ( )

N

n

E p

f S

h

h

h h

h

2

2

1

2

var m = ∑ 1− +

∈

∑ ∑

N

n

S

m

h

h

hi

h i Uh h

2

2

. (7)

A troublesome point is that the term S2hi

2

 is specific to a particular establishment. To use

expression (7) for allocation, a separate variance component would have to be estimated

for every establishment and domain of interest.
11

Use of a reasonable model for the ~zhij 's will help solve this problem. First,

consider a model for the per employee mean, yhij Ehij , for those establishments that do

have employees in a particular occupation. Certain occupations will tend to have larger

than average yhij Ehij 's because of their high pay or years of experience. Establishments

in some SIC/size strata may also tend to offer more (or less) pay or benefits than others.

There is also likely to be residual error associated with a particular

establishment/occupation combination. Considering these factors, we will adopt the

following model

yhij Ehij = m +ah + bj + hij e , ah ~ (0,s za )

2

, b j ~ (0,s zb )

2

, e hij ~ (0,s ze Ehij)

2

(8)

with the errors a, b, and e being independent. Because SIC/size strata are relatively

homogeneous groups, no factor for establishments is included. The model for

zhij Ehij = yhij Ehij − m is then a b e h + j + hij . Since zhij Ehij is a mean, we assume its

variance is inversely related to number of employees. Model (8) is undoubtedly an oversimplification since it does not account for the differences in unionized and non-unionized

occupations, among other complications. The model may also predict the average pay

yhij Ehij to be negative in an unusual case. The model does, however, account for some

of the key determinants of wage levels—specifically occupation and industry.

Next, we need to compute the model expectation of the components S2hi

2

 and S1h

2

.

There is a considerable amount of algebra involved that is sketched in the Appendix. The

final result, expressed in relvariance terms, is
12

Jm≡ m xvar (m)≅ −











 +

− − − ∑ ∑2 2

1

2

2

2 E T N 1

N

n

v T

N

n m

v p y h

h

h h

h y

h

h h

h

h

(9)

where

v h z VhE z Vhj E

j

1 z h

2 2 2

= s a + s b∑ + s e

all 

, v h z V V V 2 hEE z hEE z hEM

2

1

2

2

2

= s a + s b+ s e,

VhE (Ehi Eh ) (N )

i U h

h

= − − ∑∈

~ ~

2

1 , Vhj ( hijEhij hi jEhi j N ) (N )

i U h i U h

h h

= − ′ ′ − ∑∈ ∑′∈

j j

~ ~

2

1 ,

V E (E E ) N

hEE hi hi hi

i U

h

h

1

~

~ ~

= −

∈

∑ , V E E E N

hEE hi hi hij

i U j U

h

h hi

2

2

~

~ ~

= −













∈ ∈

∑ ∑ , and

VhEM (EhiMhi Ehi) N

i U

h

h

= −

∈

∑

~

with jhij being an indicator, defined in the Appendix, for whether an establishment

contains an occupation, Ehij = dhij Ehij , 

~ ~

Ehi Ehij j Uhi= ∑ ∈, Eh Ehi N

i U h

h

= ∑∈

, and Mhi

being the number of occupations in Uhi . The summation over “all j” in v1h means sum

over all occupations defined for the survey.

For an estimator of a total Ty

, rather than the mean Ty TE, expression (9) must be

modified only slightly. The model yhij Ehij = mh +ah + b j + hij e , with mh being the fixed

mean and ah, bj, and e hij being independent random effects with means of 0 and

variances sya syb sye

2 2 2

, , and , leads to the same form of expression as (9) but with

v h ( y h )VhE y Vhj E

j

1 y h

2 2 2 2

= s a + m +s b∑ +s e

all 

~

 and v h ( y h )V V V 2 hEE y hEE y hEM

2 2

1

2

2

2

= s a + m +s b +s e~ ~ .

Note that an alternate model for yhij Ehij would be to use m rather than mh

 as in (8).

This might have led to more stable estimates v1h and v2h; however, we applied some
13

general smoothing procedures to stabilize all variance component estimates, as described

in the next section.

4. VARIANCE COMPONENT ESTIMATION AND SMOOTHING

The variance components in expression (9) were estimated using data from the

ECI for the quarter ending in September, 1992, and from EBS for the year 1992. The

datasets differ somewhat from those used for publications from those surveys and

estimates here will differ from ones that may have been published by BLS. The total

number of strata used here was 360, formed by crossing 72 SIC groups with five size

classes: <50, 50-99, 100-249, 250-999, and 1000+ employees. The SIC groups will be

referred to here as pseudo-SIC's (psic's). Since both the ECI and EBS publish hundreds

of statistics quarterly or annually, we made a selection of some of the more important ones

to use in this study, shown in Table 1. The sample design that produced our test data used

only psic’s as explicit strata. Establishments were sorted within psic’s by geographic

location and size but those characteristics were not explicitly used as strata in the original

design. The 360 strata were, thus, finer subdivisions of the original strata, making the

assumption of srswor of establishments more tenable.

The variance components sza szb sze

2 2 2

, , and in (8) were estimated for each of the

preceding variables/domains using the MIVQUE0 method (Hartley, Rao, and LaMotte

1978) treating ah, bj, and e hij as random effects. The other components of v1h and v2h—

VhE , Vhj , V1hEE

, V2hEE, and VhEM—were estimated using simple method-of-moments

estimators appropriate if observations on establishments were independent and identically

distributed in each stratum. Method-of-moments estimation is reasonable from a design-
14

based point-of-view when size strata are sufficiently narrow that the pps sampling actually

used in the surveys is approximately equivalent to simple random sampling. A

compromise form of VhEM , different from the one defined after expression (9), had to be

devised because Mhi , the number of occupations in an establishment, was not always

available. The method used is sketched in Appendix A.2. Since the contribution of VhEM

to v2h was relatively small, this compromise was of little consequence.

The various parts were combined to estimate v1h and v2h for totals in the ECI and

proportions in the EBS. As Figure 1 illustrates, these components are related to size in

each stratum. The figure shows plots of log(v1h Ty )

2

 and log(v2h Ty )

2

 versus the log of

the average employment size per establishment using total compensation as the variable

and the full population as the domain. Plots for other variables and domains had similar

features. The logs of the relvariance components are generally linearly, or possibly

quadratically, related to the log of the average employment size with the exception of

strata where the components are poorly estimated due to small establishment sample sizes.

In Figure 1 points based on sample sizes of nh > 4 are shown as •'s while points with

samples of nh ≤ 4 are Δ 's.

Estimated variances are themselves subject to variability. To obtain more stable

estimates of variance components, we smoothed the point estimates within each psic h

across the size classes h′ by fitting models of the form

log(vkhh′)= ah +b log(Ehh′)+ khh′

e (k=1,2) (10)
15

where vkhh′ is a variance component and Ehh′

 is the average population employment per

establishment in (psic h/size class h′). Based on plots like Figure 1, for a given variable

like total compensation, a common slope b for all psic's was reasonable while allowing the

intercept ah to differ among psic's accommodated the different levels observed for some

groups. Fitting models with some curvature would have yielded somewhat better fits, but

our major goal was to stabilize component estimates. As long as the relative sizes of

stratum component estimates are reasonable, the sample allocations in section 6 will not

be sensitive to some misspecification in model (10).

Weighted least squares estimates of ah and b were calculated that minimized the

sum of squares ∑h,h′

whh′[log(vkhh′)− ah − blog(Ehh′)]

2

 with whh′ = nhh′

log(Ehh′). The

weights whh′ were designed to be roughly inversely proportional to the variances of

log(vkhh′) observed in scatterplots like Figure 1. Strata for estimating the parameters in

(10) were limited to those with nh ≥10 , a larger cutoff than in Figure 1, that eliminated

poor point estimates for virtually all variables and domains. We then used these parameter

estimates to compute smoothed, predicted variance components to use as inputs to the

optimization algorithm described in the next section. Figure 2 shows the point estimates

and smoothed values of log(v1h Ty )

2

 and log(v2h Ty )

2

 plotted versus the log of the

average employment size for total compensation in psic 61 (Banking, savings and loan,

and other credit organizations) and 83 (social services). In 3 of the 4 panels of Figure 2,

smoothing leads to a larger component in the 1000+ size class where the point estimates
16

were based on small sample sizes. This was a common and desirable effect of the

smoothing in many psic's having a small sample of large establishments.

5. THE APPROACH TO OPTIMIZATION

The ECI and EBS publish many estimates that have varying degrees of importance.

This makes the problem of sample allocation far more complicated and interesting than

Neyman allocation to strata based on a single variable. In addition to the references

mentioned in section 1 on multivariate allocation in surveys, there has been a considerable

amount of related work, including Bethel (1985, 1989), Chromy (1987), Hughes and Rao

(1979), Kokan (1963), Kokan and Khan (1967), and Zayatz and Sigman (1994). These

papers deal almost entirely with single-stage sampling. Notable exceptions are Leaver,

et.al. (1987, 1996) which discuss a complex, multistage allocation problem for the U.S.

Consumer Price Index.

Multicriteria optimization programming is one method for dealing with

multivariate allocation. Our approach will be to minimize a weighted sum of the

relvariances of a number of important statistics subject to various constraints defined

below. Because the statistics from these surveys are of disparate typesproportions of

employees, total dollar costsuse of relvariances puts the estimates on a comparable

scale. To write the optimization problem mathematically, let w be a weight associated

with estimator ( =1,...,L) and J be the anticipated relvariance of the estimator,

defined by (9). The total cost is c and its weight in the objective function is wL+1, which is
17

either 0 or 1. The optimization problem we have formulated for the ECI and EBS is to

find {nh,mh;h = 1, , H} that

minimize f = J +

=

∑w w +

c

L

L

1

1

(11)

subject to

(1) nh,min ≤ nh ≤ Nhfor establishment sample sizes nh,

(2) n n n h

h

= ∑ ≤ 0, a bound on the total number of sample establishments,

(3) mh,min ≤ mh ≤ mh,max , i.e. the number of occupations sampled per

establishment in stratum h is bounded above and below,

(4) 

n m

n

m

h h

h S

h

h S

S

∈

∈

∑

∑

≤ ,max , i.e. the average number of occupations sampled per

establishment is bounded above in a subset S of strata.

(5) J J

1 2

0

1 2

≤ for ∈SE, i.e. the coefficient of variation of an estimator 

is bounded for all estimators in some set SE.

As noted in the Introduction, the cost and relvariances are on different scales and do not

enter f simultaneously. Differential stratum costs were not available for these surveys but

could be accommodated in our approach.

The number of occupations sampled from an establishment is bounded in two ways

with constraints (3) and (4). First, an upper and lower bound on mh

 is used in each

stratum. Second, the average number of occupations sampled per establishment will be

bounded above, across some subsets of strata. This approach allows some flexibility in
18

assigning the number of sample occupations, which may lead to a more efficient

allocation, while still restricting the average burden per establishment.

The weights {w }

L

=1

 in the objective function are based on subjective judgments as

to the relative importance of each estimator in meeting the goals of the surveys. Because

analysts may have different opinions on how the weights should be assigned, we have

designed software for solving the optimization problem that flexibly allows the effects of

modifying the weights to be explored. We used the PV-WAVE AdvantageTM package

sold by Visual NumericsTM running under UnixTM and X-WindowsTM to develop a

program called ALLOCATE with a graphical user interface (GUI) to adjust parameters of

the optimization problem and then to solve the problem. The PV-WAVE interface calls a

C program known as GRG2 (Lasdon and Waren 1978; Windward Technologies Inc.

1994) that solves the nonlinear problem.

Figure 3 shows the main ALLOCATE window that is divided into four sections:

action buttons, allocation table, constraints table, and weight slider bars. The allocation

table in the upper right shows the stratum population and sample sizes for both stages of

sampling. The table also shows two kinds of sample sizes: “trial” and “optimal.” How the

trial values are initialized is optional; two possibilities are the lower bounds of the

variables or an allocation used in the survey for a previous time period. The trial entries in

the table can be modified directly. The constraints table in the lower right shows the

constraint bounds and the values of the constraints corresponding to the trial and optimal

allocations. The bounds in the table can be also edited directly. Weights for the objective

function, i.e., w 's in (11), are assigned by moving slider bars in the lower left-hand part of
19

Figure 3. If the objective function consists of only one component (which itself may be a

sum of relvariances), the slider bars do not appear in the GUI. Action buttons are in the

upper left-hand portion of the window. Clicking any of the first five of these leads to

secondary choices, two of which will be sketched briefly.

The buttons labeled “Determine optimum allocation...” and “Compare trial and

optimum...” and their secondary choices are pictured in Figure 4. After selecting the

former and preparatory to computing an optimum, the user can set some tuning

parameters for GRG2 or select starting values for the algorithm which may be the lower

bounds, the trial allocation, or the previously computed optimum. After an optimum has

been found, the user can compare it with the trial in several ways after pressing “Compare

trial and optimum...” One is by plotting the trial and optimal first-stage stratum sample

sizes versus each other as shown in Figure 4. Strata to which the points correspond can

be identified by clicking them with the mouse. Three of the points in Figure 4 are labeled

as illustrations.

The optimization problem can have a nonlinear objective function and nonlinear

constraints. In the ECI/EBS problem, the objective function f is a function of nh

−1

 and

mh

−1

 while the constraints depend on nh and mh

. The variables (the sample sizes) are

restricted to integer valuesa restriction that is unimportant in most problems. The

sensitivity of using integer solutions that are near to the continuous solution can be

investigated by assigning them as trial allocations and seeing whether the algorithm seeks a

different solution. In the continuous version of this problem (without the integer

restrictions), both the objective function and the constraints are smooth. A variety of

algorithms is available for solving this optimization problem (Moré and Wright, 1993).
20

The one selected here, GRG2, is a reduced gradient programming method that

iteratively searches for a solution. Termination occurs when the improvements in the

objective function are small. Starting values may affect convergence, and it is wise to run

any optimization with several initial allocations in order to check results.

6. NUMERICAL RESULTS

To illustrate the flexibility of this approach to allocation and its superiority to

“rule-of-thumb” allocations commonly used in sampling, we present results for three

ECI/EBS minimization problems. The sample size of establishments in the ECI/EBS

dataset in 1992 was about 4,360 which will be maintained as the constraint n0 in the first

two problems. Keeping the same total sample permits us to analyze whether the current

sample can be profitably redistributed among the strata. The third problem is the

minimization of the number of sample establishments subject to the same set of constraints

used in the first two problems.

Table 2 shows the minimum, maximum, and average numbers of sample

occupations per sample establishment that will be used for each size class. Each row in

the table refers to a size class, and implicitly to all pseudo-SIC's (psic) in the size class.

For the <50 size class, for instance, the number of occupations assigned to each sample

establishment is constrained to be between 2 and 6. The average number of occupations,

mS,max , allocated per establishment across all psic's in the <50 class is constrained to be no

more than 4. The constraints on the average mS,max in each size class are somewhat larger

than the mh = 4, 6, or 8 used in the ECI/EBS in 1992. Thus, the total workload within
21

establishments is allowed to be more than the current amount but extreme increases are

avoided.

Upper bounds on cv’s of different estimates are listed in Tables 3-5. The bounds

are set to be approximately equal to the cv’s achieved for the 1992 ECI/EBS. Table 3 lists

bounds on the cv’s that were used for ECI estimates of total costs for the full population

for six variables. Bounds on cv’s of ECI estimates of total compensation costs for major

occupational groups (MOGs) and industry divisions are listed in Table 4. Listed in Table

5 are the bounds used for cv’s of EBS occupational domain estimates of percentages of

employees receiving certain benefits in establishments of 100+ employees (referred to

subsequently as EBS/100+). The cv’s of EBS estimates of percentages for small

establishments (EBS/<100) are unconstrained. Note that the domains for occupational

groups, like professional and technical workers and administrative support, are

crossclasses that cut across the basic psic/size strata, and we, thus, have no direct control

over the number of sample employees in those domains.

A five component, weighted objective function was used that had as its parts (1)

the relvariance of the estimate of total compensation from the ECI, (2) the sum of the

relvariances of ECI estimates of total compensation for major occupational groups A-K in

Table 4, (3) relvariance of the ECI estimate of total benefit costs, (4) the sum of the

relvariances of the benefit costs of paid leave, insurance, legally required benefits, and

retirement and savings, and (5) the total establishment sample size n nh h

H

= ∑ =1. By

adjusting the weight slider to 0 for component 5, we formulate a problem for minimizing a

function of relvariances; by adjusting the other four sliders to 0 and the last to 1, the

problem becomes one of minimizing the total establishment sample size. This is an
22

extremely handy way for a user to set up a problem because no reprogramming is needed

to switch from relvariance minimization to cost minimization.

Selection of weights is subjective but should be guided by the goals and priorities

of the survey. Total compensation and benefits are key targets in the ECI and, thus,

deserve relatively large weight. In some surveys, sponsors may have difficulties in

specifying goals in a way that easily translates into weights. Even in those cases, a set of

domains can usually be identified as being important and equal weights assigned to each.

The exercise of assigning weights can, in fact, be a useful way of forcing survey designers

to think more clearly about their goals.

We report here on the results from three choices of the vector of weights

w = ( ) w w w w w ′

1 2 3 4 5

, , , , in the objective function. The first was w1 = (1 0 5 1 0 5 0)

′

, . , , . ,

which gives twice the weight to the estimates for total compensation and total benefits as

to the other estimates. The second was w2 = (1, 0,1, 0,0) which eliminates the

components for occupational groups and individual benefits. The third was

w3 = (0,0,0,0,1) for the sample size minimization problem. The user sets each of the

weight vectors with slider bars in Figure 3. Notice, in particular, that total sample size is

excluded from w1 and w2 by setting component 5 to 0. The relvariances are excluded

from w3 by setting w1 w2 w3 w4, , , and to 0. Thus, we do not include sample size and

relvariances in the objective simultaneously even though the general form of the objective

defined in section 5 incorporates both. Though we report a limited number of weight

choices here, one of the major advantages of the software is the facility to easily

manipulate the weights to do sensitivity analysis.
23

The first three rows of Table 6 list the values of the weighted objective function

for the optimal allocations for the weight vectors w1 and w2. Three other allocations are

included in rows 4-6 of the table for comparison. Allocation 4 was actually used in the

ECI/EBS in 1992; allocation 5 is an allocation of establishments in proportion to the total

employment Eh in each stratum; and allocation 6 is in proportion to the total number of

establishments Nh in each stratum. For the latter two allocations, we forced at least two

sample establishments to be assigned to each stratum and then redistributed the remainder

over the other strata in proportion to Eh or Nh —thus, keeping the same nh,min = 2

constraint as in the nonlinear optimization problems. The first two optima (allocation 1

for w1; allocation 2 for w2) are substantial improvements over the other allocations in

terms of objective function value. The rule-of-thumb allocationsproportional to Eh and

proportional to Nhare notably worse than the optima for either weight vector.

Table 7 shows the averages by size class of the optimum values of the secondstage allocations for the three optimal allocations and the 1992 ECI/EBS allocation. The

optimal allocations generally call for more occupations to be sampled per establishment.

One might expect that the second-stage allocations would be closer to the mS,max bounds

in Table 2 since there is no penalty in the optimization to keep those maxima from being

achieved. However, the contribution of the second stage of sampling to the relvariance of

most estimates is small implying that selecting more occupations did not reduce the overall

relvariance much. For example, the median percentage contribution of the second-stage

to the total relvariance across the 66 estimates considered here was only about 8% for

each of the allocations.
24

Allocation 2 is the optimum when total compensation and benefits are equally

weighted in the objective function. Since these are two of the key estimates from the ECI,

it is interesting to see how that allocation compared to the others. The cv’s achieved by

optimum allocation 2 are compared in Figure 5 to those obtained from allocations 1, 4,

and 5 in Table 6. The cv’s for 63 estimates are plotted. Forty-three of those were

constrained in the optimization while the remaining 20 were not. Allocation 2 generally

improves over the others for key estimates—in particular the ECI and EBS/100+. In the

lower right-hand panel of Figure 4, the allocation proportional to Nh does produce lower

cv’s for EBS/<100 estimates because the large number of small establishments leads to

larger samples being allocated to the <100 strata under the proportional-to- Nh rule. On

the whole, allocation 2 not only substantially reduces the value of the objective function

for w2 = (1, 0,1, 0,0) compared to the other allocations, but also reduces the cv’s for

important individual survey estimates.

Because cost is always an issue in survey design, we prefer an allocation that meets

survey goals using the smallest feasible sample size. When the sample size is minimized

( w3) and the same constraints are used as for allocations 1 and 2, we obtain an optimum

of n=3,672 (allocation 3). Thus, the constraints can be satisfied for an establishment

sample that is 84% (3,672/4,360) of the size used for allocations 1 and 2. The cost of

allocation 3 would be somewhat more than 84% of allocations 1 and 2 since the mh

’s in

Table 7 are larger for allocation 3. Figure 6 shows the cv’s for the sample size minimizing

allocation plotted versus those for allocations 1, 2, and the 1992 allocation. There are
25

increases in individual cv’s incurred with allocation 3 as opposed to allocation 2, but

allocation 3 is almost as good as allocation 1 and better than the 1992 allocation.

To summarize, allocations 2 and 3 do well in our example, both in terms of the

objective function and the cv’s for domain estimates. Allocation 2 will have a cost that is

similar or slightly higher than the cost of the 1992 allocation. But, if budget cutting is

important, allocation 3, with an establishment sample 84% of the 1992 size, will produce

domain estimates almost as precise at lower cost.

Ideally, survey design and sample allocation in a periodic survey like ECI/EBS

should be a continuing process. The stability of variance component estimates over time

should be studied, along with the sensitivity of the optimal allocations to changes in

parameter estimates. Predicted relvariances from expression (9) should also be validated

by comparing to relvariances estimated directly by replication or some other method.

7. CONCLUSION

Nonlinear optimization can be a powerful technique in sample allocation in

multipurpose surveys, but a number of factors have probably limited its use. Despite there

being a number of studies in the literature, optimization algorithms themselves may not be

well-known to all survey practitioners. The algorithms for handling nonlinear objective

functions and nonlinear constraints are complex and would require significant

commitments of time and resources to program from scratch. However, commercial

versions of some algorithms are available that can be linked into specially written interface

software as was done here. At the time that the ALLOCATE software was developed,

the GUI building tools were more limited than they are today, which influenced our choice
26

of the proprietary PV-WAVE. Now, however, many other options are available that

other developers might find preferable—particularly Java, PowerBuilder, and C++.

The former two allow cross-platform development.

The functionality and use of our system is spelled out in two Bureau of Labor

Statistics manuals—Allocate User’s Guide and Allocate Programmer’s Guide. These

manuals include many more screenshots and more detailed descriptions of how to use the

system and write the C program that calls the GRG2 optimizer. Both of these are

available from the authors to anyone wishing to develop their own systems.

Optimizers other than GRG2 are also available, many of which are described and

compared by Schittkowski (1985), that use penalty methods or sequential quadratic

programming (SQP) techniques. Leaver, et. al. (1987, 1996), for example, applied an

SQP method developed by Fiacco and McCormick (1968). Although there are

performance differences among algorithms, the particular optimizer used here was less

important than illustrating the usefulness of a flexible allocation system for sample design.

For organizations that conduct a variety of surveys and must periodically redesign to

update and improve existing operations, developing systems to facilitate the use of

optimization algorithms seems well worthwhile.

APPENDIX

A.1 Model expectation of the design variance of m

First, consider Ex (S2hi)

2

. Under model (8), we have

( )

~ ~

~ ~ ~

z E

E E E

hi hij h j hij

j U

h hi j hij

j U

hij hij

j U

hi

hi hi

= + +

= + +

∈

∈ ∈

∑

∑ ∑

a b e

a b e .
27

Using p j hi Ehij Ehi

∗

= ⋅,

~z ~ ~ ~

z E E E E

hij

j hi

hi h hij hi hij

j U

j hij hi j hij

hi j Uhi

p

a d b d b ∗ ′

′∈

′ ′

′∈

− = −











 + −











∑ ∑  .

+ −













′ ′

′∈

e d ∑e hij hij hi hij hij

j U

E E

hi

~

= Aj + Bj + Cj, say. (A.1)

Under model (8), each zhij has expectation 0, so that

Ex (zhij p j hi zhi) x (zhij p j hi zhi)

~ ~ var ∗ ~ ∗ ~ − = −

2

. The variance of the first term in (A.1) is then

var ( ) ( )

~ ~

x Aj = s za d hijEhi − EhiEhid hij + Ehi

2 2 2

2 and p j hi ( j) j U

A

hi

∗

∈ ∑ var = s za Ehi(Ehi Ehi)

2 ~ ~

− .

The variances of Bj and Cj follow similarly. Combining results yields

E (S hi) z Ehi(Ehi Ehi) z (EhiEhi j UEhij) z (EhiMhi Ehi) hi

x 2 s a s b s e

2 2 2 2 2

= − + − + − ∑ ∈

~ ~ ~ ~ ~

. (A.2)

Substituting this expression for the model-expectation of the second term in (7), leads to

the second term on the right-hand side of (9).

Turning to S1h

2

, we have

( )

~ ~ ~ ~ ~ ~

~ ~

z z E E E E N

E E N

hi h h hi h j hij

j U

j hi j

i U j U

h

hij hij

j U

hi j hi j h

i U j U

hi h hi

hi h hi

− = − + −













+ −













∈

′ ′ ′

′∈ ′∈

∈

′ ′ ′ ′

′∈ ′∈

∑ ∑ ∑

∑ ∑ ∑

a b b

e e

= Ai + Bi + Ci, say. (A.2)

Under model (8), var( ) ( )

~ ~

Ai E E

i U z hi h ∑∈ h ∑i∈Uh

= s a −

2

2

. To compute var(Bi), define

jhij

hi j

=







1

0

if establishment contains occupation

if not

.

With that definition, Bi = ∑ j

b j(jhijEhij −∑i′∈Uhjhi′jEhi′j Nh )

~ ~

all 

 from which it follows

that var(Bi) V

i U hj j ∑∈ h

= s b∑

2

all 

 where Vhj was defined at the end of section 3. Recalling

that var(e hij)= s ze Ehij

2

, the variance of Ci is
28

var( ) [{( ) } ]

~ ~

Ci z Nh Nh Ehij N E

j U h hi j

hi i Uh

j Uhi

= − +

∈

−

′ ′ ∑ ∑′∈ ∑ ′∈ ′

s e

2 2

2

and var( ) ( )

~

Ci N E

i U z h h

h ∑′∈

= s e −

2

1 . Combining results for Ai Bi Ci

, , and gives

= sza hE + szb∑ hj + s e

j

V V z Eh

2 2 2

all 

. (A.3)

Finally, from (7), (A.2), and (A.3) we obtain (9).

A.2 Compromise Form of VhEM

A compromise form of VhEM was used because Mhi was not available for all

establishments. Let ~Eh be the mean number of employees per occupation per

establishment in a domain. Suppose that under model (8) we have var( )

~

ehij = sze Eh

2

rather than var(ehij)= sze Ehij

2

, i.e., substitute an overall stratum mean for the

establishment-specific mean. Then 

i U j U

p j hi (Cj) h hi

∗ ∑∈ ∑ ∈

var = sze NhV hEE

Eh

2

2

~

~

 and

VhEM V E = 2hEE h

~

~

. This form of VhEM was estimated by method-of-moments.

REFERENCES

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the Section on Survey Methods Research (American Statistical Association, 1985),

209-212.

, Sample Allocation in Multivariate Surveys, Survey Methodology, 15 (1989), 47-57.

Chromy, J., Design Optimization with Multiple Objectives, Proceedings of the Section on

Survey Methods Research (American Statistical Association, 1987), 194-199.

Fiacco, A.V., and McCormick, G., Nonlinear sequential unconstrained minimization

techniques, (Wiley, New York, 1968).
29

Hartley, H.O., Rao, J.N.K., and LaMotte, L., A Simple Synthesis-based Method of

Variance Component Estimation, Biometrics, 34 (1978), 233-244.

Hughes, E., and Rao, J., Some Problems of Optimal Allocation in Sample Surveys

Involving Inequality Constraints, Communications in Statistics 3⁄4 Theory and

Methods, A8 (15) (1979), 1551-1574.

Isaki, C. and Fuller, W., Survey Design Under the Regression Superpopulation Model,

Journal of the American Statistical Association, 77 (1982), 89-96.

Kish, L., Multipurpose Sample Designs, Survey Methodology, 14 (1988), 19-32.

Kokan, A.R., Optimum Allocation in Multivariate Surveys, Journal of the Royal

Statistical Society A, 126 (1963), 557-565.

Kokan, A.R., and Khan, S., Optimum Allocation in Multivariate Surveys: An Analytical

Solution, Journal of the Royal Statistical Society B, 29 (1967), 115-125.

Lasdon, L. and Waren, A., Generalized Reduced Gradient Software for Linearly and

Nonlinearly Constrained Problems, in: H. Greenberg (Ed.), Design and

Implementation of Optimization Software (Sijthoff and Noordhoff, Alphen aan den

Rijn 1978).

Leaver, S.G., Weber, W.L., Cohen, M.P., and Archer, K.P., Item-Outlet Sample Redesign

for the 1987 U.S. Consumer Price Index Revision, Proceedings of the 46th Session,

Vol. LII, Book 3 (International Statistical Institute, 1987), 173-185.

Leaver, S.G., Johnson, W.H., Baskin, R., Scarlett, S., and Morse, W., Commodities and

Services Sample Redesign for the 1998 Consumer Price Index, Proceedings of the

Section on Survey Methods Research, (American Statistical Association, 1996), to be

published.
30

More, J. J., and Wright, S. J., Optimization Software Guide (SIAM, Philadelphia, 1993).

Narula, S., and Weistroffer, H., Algorithms For Multiple Objective Nonlinear

Programming Problems, in A. Lockett and G. Islei (Eds.), Improving Decision Making

in Organizations, (Springer-Verlag, Berlin, 1989), 434-443.

Särndal, C.-E., Swensson, B., and Wretman, J., Model Assisted Survey Sampling, (New

York: Springer-Verlag, 1993).

Schittkowski, K., NLPQL: a FORTRAN subroutine solving constrained nonlinear

programming problems, Annals of Operations Research, 5(6) (1985), 485-500.

Steuer, R., Multiple Criteria Optimization: Theory, Computation, and Application,

(Wiley, New York,1986).

Weistroffer, H., and Narula, S., The Current State of Nonlinear Multiple Criteria Decision

Making, in G. Fandel and H. Gehring, (Eds.), Operations Research, (Springer-Verlag,

Berlin, 1991), 109-119.

Windward Technologies Inc., GRG2 User’s Guide: Software for Solving Nonlinear

Optimization Problems with Nonlinear Constraints, (Windward Technologies Inc. and

Optimal Methods, Inc., Meadows TX, 1994).

Zayatz, L., and Sigman, R., Feasibility Study of the Use of Chromy’s Algorithm in Poisson

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on Survey Methods Research (American Statistical Association, 1994), 641-646.
31

Figure Titles

Figure 1. Log of stratum relvariance components for total compensation plotted versus

log of average establishment employment size. The upper panel shows data for firststage relvariance components; the lower panel is for second-stage components.

Figure 2. Log of predicted and point estimates of stratum relvariance components for

total compensation plotted versus log of average establishment employment size for

two SIC groups. Solid circles are point estimates used in smoothing ( nh ≥10 ); dotted

circles are point estimates excluded from estimation ( nh < 10 ).

Figure 3. Main window of the ALLOCATE software.

Figure 4. Two branches a user can take in the ALLOCATE program.

Figure 5. Coefficients of variation for optimal allocation 2 plotted versus cv’s for

alternative allocations.

Figure 6. Coefficients of variation for optimal allocation 3 (n=3,672) plotted versus cv’s

for alternative allocations (n=4,360).
32

Table 1. Important variables and domains from the ECI/EBS.

ECI EBS

Variables Variables

 Total compensation % workers receiving:

 Cost of benefits for Life insurance

 All benefits Medical insurance

 Life insurance Retirement, savings plans

 Legally required benefits Paid sick leave

 Retirement, savings plans Paid vacation

Domains (used for total compensation) Domains

 Full population All occupations

 9 major occupational groups Professional, technical, related

occupations

 7 industries Clerical, sales occupations

 Production, service occupations

 1 size class (100+)

Table 2. Minimum, maximum, and average numbers of occupations to be assigned to

pseudo-SIC's in each size class.

Size Class mh,min mh,max mS,max

< 50 2 6 4

50-99 2 10 8

100-249 4 12 10

250-999 4 12 10

1000+ 6 12 10

Table 3. Bounds on cv's for estimates of different totals of employee costs.

Cost cv bound

Total compensation .03

Total benefits .02

Paid leave .03

Insurance .03

Legally required benefits .03

Retirement and savings .03
33

Table 4. Bounds on cv’s of estimates of total compensation for major occupational

groups (MOG’s) and industry divisions.

Group cv

bound

Group cv

bound

MOG Industry

 A-Professional, technical .30 Construction .10

 B-Executive .25 Manufacturing .05

 C-Sales none Transportation .05

 D-Administrative support .20 Wholesale .10

 E-Precision workers .40 Retail .10

 F-Machine operators none FIRE .10

 G-Transportation .20 Services .10

 H-Handlers .10

 K-Service .20

Table 5. Bounds on cv's of the estimated percentages of employees in different

occupational groups receiving certain benefits for establishments with 100+ employees.

Benefit All

occupations

Professional,

technical

Clerical, sales Production,

service

Medical insurance .015 .015 .015 .015

Life insurance .010 .010 .010 .010

Retirement, savings .020 .020 .020 .020

Vacation .010 .010 .010 .010

Sick leave .020 .020 .020 .020

Table 6. Values of the weighted objective function for three optimal allocations and three

comparison allocations. Optimal allocation 1 is optimal for w1; optimal allocation 2 is

optimal for w2. The total establishment sample size for allocations 1, 2, 4, 5, and 6 is

n=4,360;total size for allocation 3 is n=3,672.

Allocation w1 = (1 0 5 1 0 5 0)

′

, . , , . , w2 = (1, 0,1, 0,0)

1. Optimal allocation 1 0.263 1.025×10−3

2. Optimal allocation 2 1.064 0.426×10−3

3. Optimal allocation to

minimize total sample size

11.875 1.042×10−3

4. ECI/EBS 1992 3.579 1.345×10−3

5. Proportional to Eh9.415 1.108×10−3

6. Proportional to Nh7.484 1.914×10−3
34

Table 7. Average optimum values, across industries, of the second-stage allocation mh

for optimal allocations 1, 2, and 3.

Allocation

Size Class 1992 Optimal 1 Optimal 2 Optimal 3

< 50 4 4.0 4.0 3.9

50-99 6 6.9 7.2 8.0

100-249 6 9.4 9.5 10.0

250-999 8 9.9 9.9 10.0

1000+ 8 9.2 9.3 10.0
35

Figure 1. Log of stratum relvariance components for total compensation plotted

versus log of average establishment employment size. The upper panel shows data for

first-stage relvariance components; the lower panel is for second-stage components.

ln(avg size)

ln(relv1h)

2 4 6 8 10

0

5 10 15 20

nh <= 4

nh > 4

ln(avg size)

ln(relv2h)

2 4 6 8 10

0

5 10

nh <= 4

nh > 4
Figure 2. Log of predicted and point estimates of stratum relvariance components for total compensation plotted versus log of average

establishment employment size for two SIC groups. Solid circles are point estimates used in smoothing ( nh ≥10 ); dotted circles are

point estimates excluded from estimation ( nh < 10 ).

v1h 61

ln(avg size)

ln(relvar component)

3 4 5 6 7 8

4 6 8 10 12 14 16

point estimate

smoothed

v2h 61

ln(avg size)

ln(relvar component)

3 4 5 6 7 8

2 4 6 8

point estimate

smoothed

v1h 83

ln(avg size)

ln(relvar component)

3 4 5 6 7

4 6 8 10

point estimate

smoothed

v2h 83

ln(avg size)

ln(relvar component)

3 4 5 6 7

1 2 3 4 5 6 7

point estimate

smoothed
1

Figure 3. Main window of the ALLOCATE software.

Allocation table

Slider bars Constraints table

Action buttons
2

Figure 4. Two branches a user can take in the ALLOCATE program.
3

1992 allocation

optimal allocation 2

0.0 0.05 0.10 0.15

0.0 0.10 0.20

•

•

•

•

•

•

• •

•

•

••

•

•

o••

ooo

o

o

oo

o

oo

o

o

o

o

o o

ECI

EBS/<100

EBS/100+

optimal allocation 1

optimal allocation 2

0.0 0.05 0.10 0.15

0.0 0.10

•

•

•

•

•

•

•

•

•

•

•

o•

oo

o

o

o

o

o

o

o

o

o

o

o

o

allocation proportional to Eh

optimal allocation 2

0.0 0.05 0.10 0.15

0.0 0.10 0.20

•

•

•

•

•

•

•

• •

•

•

• •

•

•

•• •

oo

o

o

oo

o

o

o

o

o

oo

o

allocation proportional to Nh

optimal allocation 2

0.0 0.05 0.10 0.15

0.0 0.10

•

•

•

•

•

•

• •

•

•

•

•

•

o

•• •

ooo

o

o

o

o

o

oo

o

o

o

o

o

o

o

Figure 5. Coefficients of variation for optimal allocation 2 plotted versus cv's for alternative allocations.
optimal allocation 2

optimal allocation 3

0.0 0.05 0.10 0.15

0.0 0.10

•

•

•

•

•

•

•

•

•

•

•

•••

•

•

o

o

o

o

o

o

o

oo

o

o

o

o

o

ECI

EBS/<100

EBS/100+

optimal allocation 1

optimal allocation 3

0.0 0.05 0.10 0.15

0.0 0.10

•

•

•

•

•

•

•

•

•

••

•

o

o

o

o

o

o

o

o

o o

o

o

o

o

o

o

1992 allocation

optimal allocation 3

0.0 0.05 0.10 0.15

0.0 0.06 0.12

•

•

•

•

•

• •

•

•

•

•

•

•

•••

•

o

oo

o

o

o

o

o

o

oo

o

o

o

o

o

o

Figure 6. Coefficients of variation for allocation 3 (n=3,672)

plotted versus cv's for alternative allocations (n=4,360).