DETERMINING WITHIN-PSU SAMPLE SIZES FOR THE CONSUMER EXPENDITURE SURVEY

Sylvia Johnson-Herring, Sharon Krieger, David Swanson

U.S. Bureau of Labor Statistics

2 Mass. Ave., NE, Room 3650, Washington, DC 20212

Any opinions expressed in this paper are those of the authors, and do not constitute policy of the Bureau of 

Labor Statistics.

Key Words: variance, sample size, sample

allocation, constrained least squares

Introduction

The Consumer Expenditure Survey is a national

household survey conducted by the Bureau of Labor 

Statistics (BLS) to find out how Americans spend 

their money. The survey’s sample design is updated 

approximately every ten years, and at that time many 

decisions need to be made, such as the number of 

geographic areas in which to collect data, and the 

number of households from which to collect data in 

each area. In this paper we describe an automated

method of determining the number of households to 

sample in the selected geographic areas.

Background

The Consumer Expenditure Survey is a national

household survey conducted by the BLS to find out 

how Americans spend their money. Data for the

survey are collected by the Bureau of the Census 

under contract with the BLS. One of the primary uses 

of the data is to provide expenditure weights for the 

Consumer Price Index (CPI).

The Consumer Expenditure Survey consists of two 

separate surveys, the Diary (CED) and Quarterly

Interview (CEQ) surveys. The purpose of the CED is 

to obtain detailed expenditure data on small,

frequently purchased items such as food and apparel. 

The purpose of the CEQ is to obtain detailed

expenditure data on large items such as property,

automobiles, or major appliances, or expenses that 

occur on a regular basis, such as rent, utility bills, or 

insurance premiums. This paper focuses on the CEQ.

Primary Sampling Units

The selection of households for the survey begins 

with the definition and selection of primary sampling 

units (PSUs), which consist of counties (or parts

thereof), groups of counties, or independent cities.

The sample of PSUs currently used in the survey 

consists of 105 geographic areas, of which 87 urban 

geographic areas1 have also been selected by the BLS 

for the CPI. The 105 PSUs are classified into four 

size categories:

• 31 “A” PSUs, which are Metropolitan 

Statistical Areas (MSAs) with a population of 

1.5 million or greater

• 46 “B” PSUs, which are MSAs with a

population less than 1.5 million

• 10 “C” PSUs, which are nonmetropolitan

areas used in the CPI

• 18 “D” PSUs, which are nonmetropolitan

areas not used in the CPI, often referred to as 

“rural” PSUs

The 31 “A” PSUs are self-representing, and the 74 

“B”, “C”, and “D” PSUs are non-self-representing. 

Examples of “A” PSUs are the Boston, Chicago, and 

San Francisco metropolitan areas. Examples of “B” 

PSUs are the Hartford, Connecticut; the Dayton,

Ohio; and the Provo, Utah metropolitan areas. 

Examples of “C” PSUs are the Morristown-Jefferson 

City, Tennessee, and Mount Vernon, Illinois

nonmetropolitan areas. An example of a “D” PSU is 

Caribou-Presque Isle, Maine, a rural area composed 

of parts of Aroostook County. 

For some analyses the “B”, “C”, and “D” PSUs are 

grouped together by their region-size classes. There 

are 12 region-size classes which are created by crossclassifying the four regions of the country (Northeast, 

Midwest, South, and West) with the three size classes 

(B, C, D). When these region-size classes are used, 

they are treated just like the other self-representing 

PSUs. No “C” PSUs were selected in the Northeast 

region, so there are actually 11 region-size classes. 

Hence the CEQ can be thought of as having 42 self-

 

1

The new official terms for “urban” and “rural” areas are 

“CBSA” and “Non-CBSA” areas, but in this memorandum 

we will use the terms “urban” and “rural” to denote them. 

“A core-based statistical area (CBSA) consists of the

county or counties associated with at least one core of 

10,000 or greater population, plus adjacent counties having 

a high degree of social and economic integration with the 

core(s) as measured by commuting ties.” Non-CBSA areas 

are areas outside CBSAs. Non-CBSA areas “display a high 

degree of rurality.”
representing geographic areas, 31 “A” PSUs plus 11 

region-size classes for the smaller PSUs.

Within-PSU Sample Sizes

In the CEQ’s current sample design, usable data are 

collected from 7,760 households in each calendar 

quarter of the year: 4,260 usable interviews in the 

“A” PSUs, and 3,500 usable interviews in the “B,” 

“C,” and “D” PSUs. In order to guarantee that

enough data are collected to satisfy publication

requirements for all 42 geographic areas, the sample 

of 7,760 households is allocated in a way that at least 

120 usable interviews are obtained in each of the 38 

geographic areas used in the CPI, with no minimum 

number of usable interviews required in the 4 “D” 

geographic areas.

Thus the sample allocation problem is to allocate

7,760 households to the 42 geographic areas in a way 

that the following constraints are satisfied:

• 4,260 usable interviews are collected in the 31 

“A” PSUs

• 3,500 usable interviews are collected in the 11 

“B,” “C,” and “D” geographic areas

• 120 or more usable interviews are collected in 

each of the 38 geographic areas used in the CPI

CEQ staff recently re-evaluated the minimum sample 

size requirement of 120 usable interviews to

determine whether it is still an appropriate number. 

One of the results of the re-evaluation was the

development of a new automated method of

allocating the nationwide sample to individual

geographic areas. The new method allowed repeated 

analyses to be conducted easily using different

minimum sample size requirements. The method

involved setting up the sample allocation problem as 

a mathematical optimization problem, and then using 

SAS software to solve the optimization problem. 

“Target” versus “Required” Sample Sizes

In the past there were various interpretations of the 

word “required” in the phrase “minimum required

sample size.” At times the requirement that at least 

120 usable interviews be obtained was interpreted as 

a “target” sample size, meaning that the expected

number of usable interviews should be at least 120,

E( ix ) ≥ 120

while at other times it was interpreted as a “required” 

sample size, meaning that there should be a very high 

probability that at least 120 interviews be obtained,

P{ ix ≥ 120} ≥ 0.95

where ix is the number of usable interviews

collected in geographic area = i.

For example, under the first interpretation (“target” 

sample size), we would have to visit 185 households 

in each quarter of the year in order to collect 120 

usable interviews in the Boston metropolitan area, 

assuming that usable interviews are obtained at 65% 

of the residential addresses in the CEQ’s sample.2

E( ix ) = 185 × 0.65 = 120

However, under the second interpretation (“required” 

sample size) we would have to visit 202 households 

in order to be 95% certain of getting at least 120 

usable interviews, again assuming a 65% survey

participation rate.

P{ ix ≥ 120} = k k

k k

−

=

− 















∑

202

202

120

0.65 (1 0.65)

202 = 0.95

Table 1 shows the difference in the sample size that 

would be needed when using a “target” as opposed to 

a “required” minimum number of usable interviews. 

The number of selected addresses needed to achieve 

a “target” minimum sample size is approximately 

10% less than that needed for a “required” sample 

size.

Table 1.

Minimum Sample Size

(assuming a 65% survey participation rate)

Number of 

sample 

households (n)

Expected number of 

usable interviews 

(=0.65n)

95% Confidence 

Interval

Target sample size (2-sided 95% confidence interval)

62 40 [33, 47]

92 60 [51, 69]

123 80 [70, 90]

154 100 [88, 112]

185 120 [107, 133]

215 140 [126, 154]

Required sample size (1-sided 95% confidence interval)

72 47 [40, +∞)

105 68 [60, +∞)

137 89 [80, +∞)

170 110 [100, +∞)

202 131 [120, +∞)

234 152 [140, +∞)

These estimates were produced using formulas from 

the binomial distribution for the mean and variance 

of the number of usable interviews,

μ = ( )i E x = 0.65 n

( )

2

i s = V x = 0.65(1 − 0.65) n

 

2 Approximate ly 15% of the residential addresses selected 

for the CEQ survey are ineligible for the survey, and 20% 

do not participate in the survey due to refusal or no one 

being home. This leaves 65% of the sample who participate 

in the survey.
and the normal distribution was used to approximate 

the binomial distribution to estimate a 95%

confidence interval on the number of usable

interviews:

1-sided confidence interval: [ μ − 1.64σ , +∞ )

2-sided confidence interval: [ μ − 1.96σ , μ + 1.96σ ]

After some discussion, it was decided that “target” 

sample sizes would be satisfactory since the width of 

the 2-sided confidence intervals are relatively small.

Setting Up the Optimization Problem

As mentioned earlier, the current CEQ sample design 

calls for allocating 7,760 households to the 42

geographic areas in a way that the following

constraints are satisfied:

• 4,260 usable interviews are collected in the 31 

“A” PSUs

• 3,500 usable interviews are collected in the 11 

“B”, “C”, and “D” geographic areas

• 120 or more usable interviews are collected in 

each of the 38 geographic areas used in the CPI

This can be written in mathematical terms as the

following:

• 1 2 31 x + x +L+ x = 4,260

• 32 33 42 x + x +L+ x = 3,500

• ix ≥ 120 for i=1,2,...,38

where ix is the number of usable interviews collected 

in geographic area=i.

The objective of the CEQ’s sample design is to

allocate the nationwide sample of households to 

individual geographic areas in a way that minimizes 

the standard error of the CEQ’s published

expenditure estimates at the national level. Allocating 

the sample proportional to the population of each

geographic area comes very close to achieving that 

goal. Although allocating the sample proportional to 

population does not minimize the nationwide

standard error, it is a very simple sample design that 

is known to achieve “near minimization.” We chose 

to implement this method because of its simplicity

and its “near optimal” properties.

For some of the geographic areas with small

populations (e.g., Anchorage and Honolulu) the

requirement that at least 120 usable interviews be 

collected during each calendar quarter of the year 

conflicts with the objective of allocating the sample 

proportional to population. For example, the

Anchorage metropolitan area has approximately

0.09% of the U.S. population, and allocating the

7,760 usable interviews directly proportional to

population would give Anchorage only enough

addresses to obtain 7 usable interviews. This conflicts 

with the publication requirement that at least 120

usable interviews be collected in each geographic

area.

Since the objective cannot be achieved, we decided to 

allocate the sample as close to population 

proportionality as possible. This involved setting up a 

least squares problem in which the squared difference 

between each geographic area’s proportion of the

population and its proportion of the sample is

computed, and then the sum of those 42 squared 

differences is minimized.

Thus the sample allocation problem is to solve the 

following constrained least squares problem:

Given values of n, pi, p, find values of ni that

Minimize ∑=

















−

42

1

2

i

i i

p

p

n

n

Subject to n1 + n2 + L+ n31 = 4,260

n32 + n33 +L+ n42 = 3,500

ni ≥120 for i=1,2,...,38

ni ≥ 0 for i=39,...,42

where

ni = number of housing units assigned to geographic area=i

 n = number of housing units nationwide (n = 7,760)

pi = population of geographic area=i

 p = population in all geographic areas(p = 1 2 42 p + p +L+ p )

Solving the Optimization Problem

The sample allocation problem described above can 

be seen to have both equality and inequality

constraints. This creates a practical problem because 

optimization problems with equality constraints are 

usually solved with different techniques than those 

with inequality constraints.

Least squares problems with equality constraints are 

usually solved with linear algebra and linear

regression theory, while problems with inequality

constraints are usually solved with iterative search

techniques.

Fortunately, the SAS procedure PROC NLP

(NonLinear Programming) can handle both kinds of 

constraints simultaneously. An example using PROC 

NLP to solve the problem above is given at the end of 

this paper.
Estimating the CEQ’s Standard Error

The CEQ’s variance resulting from the sample

allocation process described above was estimated

using the following formula:

i i

i

i

i

i

i

i

i

p n

p

V x

p

p

x

p

p

V x V

42 2

1

2

42

1

2

42

1

( ) ( )

s

∑ ∑ ∑ = = =

















= 















=

































=

where

i

x = sample mean of geographic area = i

x = sample mean of the nation,

i

i

i i

i i

i

i

i

i i

x

p

p

p

p x

p

p x

x ∑

∑

∑

∑

=

=

=

=

















= = =

42

1

42

1

42

1

42

1

2 s

= expenditure variance of a randomly selected 

household

The CEQ’s variance under the proposed sample

allocation method is estimated by substituting the

values of ni obtained from the optimization problem 

(the output of “PROC NLP”) into the formula

∑=

















=

42

1

2

2

( )

i i

i

p n

p

V x

s

.

Then the standard error is computed by taking the 

square root of the variance.

SE = ∑=

















42

1

2

2

i i

i

p n

p s

This allows comparisons to be made to the current 

method of sample allocation. The value of σ does not 

have to be known because the change in standard 

error is the number in which we are interested, and 

when the ratio of two estimates of the standard error 

is taken (to compare the standard error of using, say, 

80 as the minimum sample size compared to 120) the 

value of σ in the numerator and σ in the denominator 

cancel out with each other.

CEQ’s Standard Error with Different Minimum 

Sample Size Requirements

After allocating the CEQ’s nationwide sample to

individual geographic areas using PROC NLP, the 

percent change in the CEQ’s standard error was

computed for various minimum target sample sizes. 

Standard errors were computed at the All Items, All 

U.S. level. The baseline used in the comparison was 

the current sample allocation. The current minimum 

target sample size is around 120, but for technical 

reasons it is not exactly equal to 120. The results of 

the comparisons are shown in Table 2.

Table 2.

The Effect of Minimum Target Sample Sizes 

on CEQ’s Standard Error

Minimum Target 

Sample for each PSU 

Change in CEQ’s 

Standard Error (%)

 0 −4.16

 10 −4.16

 20 −4.15

 30 −4.10

 40 −4.04

 50 −3.96

 60 −3.88

 70 −3.74

 80 -3.54

 90 −3.21

100 −2.72

110 −2.04

120 -1.14

130 +0.06

140 +1.45

150 +3.28

160 +5.63

170 +10.07

180 +14.41

From this table it can be seen that the CEQ’s standard 

error is minimized when the sample is allocated

directly proportional to population, i.e., when 0 is the 

minimum number of usable interviews required in 

each geographic area. Reducing the target number of 

usable interviews from 120 to 0 would reduce CEQ’s 

standard error by 4.16%. The other extreme is where 

the sample is divided equally among all the

geographic areas, i.e., 180 usable interviews per area. 

Increasing the target number of usable interviews 

from 120 to 180 would increase CEQ’s standard error 

by 14.41%. 

Reducing the minimum target number of usable

interviews from 120 to 80 per geographic area would 

reduce the CEQ’s standard error by 3.54%. The

Changes in CEQ's Standard Error 

with Different Minimum Sample Sizes

-10

-5

0

5

10

15

20

0 50 100 150 200

Minimum Sample Size

% Change in Standard Error

Proposed Sample 80
above graph of Table 2 shows that nearly all of the 

reduction in the CEQ’s standard error is achieved by 

reducing the minimum target sample size to 80, and 

that little further reduction is achieved by reducing 

the minimum target sample size below 80.

A More Detailed Analysis

Four additional sample allocation methodologies

were examined to study the effect that the minimum 

target sample size of 80 would have on the standard 

error of the CEQ and CPI estimates. They are:

Method 1. The number of households allocated to 

each geographic area is as close to population

proportionality as possible, but with the added

constraint that a minimum number of usable

interviews be collected in each of the 38 geographic 

areas used in the CPI.

Method 2. The number of households allocated to 

each geographic area is as close to population

proportionality as possible, but with the added

constraints that (1) a minimum number of usable

interviews be collected in each of the 38 geographic 

areas used in the CPI, (2) the number of households 

in each of the four “D” geographic areas is less than 

or equal to 100.

Method 3. The number of households allocated to 

each geographic area is as close to population

proportionality as possible, but with the added

constraints that (1) a minimum number of usable

interviews be collected in each of the 38 geographic 

areas used in the CPI, and (2) the number of

households in each of the four “D” geographic areas 

is exactly equal to 100.

Method 4. The number of households allocated to 

each geographic area is as close to population

proportionality as possible, but with the added

constraints that (1) a minimum number of usable

interviews be collected in each of the 38 geographic 

areas used in the CPI, and (2) the total number of 

households in the four “D” geographic areas is equal 

to 400.

The percent changes in standard error relative to the 

current allocation were computed for the four

proposed methods of sample allocation at the Total 

U.S. (CE population) and Urban only (CPI

population) levels. The CEQ’s actual 2000 sample 

allocation is used as the basis of comparison.

Table 3.

The Effect on CEQ’s Standard Error when 

Minimum Target Sample Size is 80

Proposed Method

Total U.S.

(CE 

population)

Urban only 

(CPI 

population)

Method 1 −5.56 −1.21

Method 2 −1.00 −4.00

Method 3 −0.95 −3.86

Method 4 -3.54 -3.86

Table 3 shows that the reduction in standard error for 

the CPI population closely matches the reduction in 

standard error for the CE population when using

Method 4 to allocate the sample. The standard error 

is reduced for both surveys by approximately the

same amount.

Table 4.

The Current and Proposed Sample Allocations for “A” 

PSUs in the West Region, and its Effect on CEQ’s 

Standard Error, with Minimum Target Sample Size 80

PSU Population

Current 

Sample 

Size

Proposed 

Sample 

Size

Change 

in SE 

(%)

A419 Los Angeles 8,863,164 231 290 −10.81

A420 Greater L.A. 5,668,365 152 187 −9.88

A422 SanFrancisco 6,253,311 158 206 −12.44

A423 Seattle 2,970,328 119 100 +9.08

A424 San Diego 2,498,016 104 85 +10.78

A425 Portland 1,793,476 130 80 +27.48

A426 Honolulu 836,231 112 80 +18.32

A427 Anchorage 226,338 125 80 +25.00

A429 Phoenix 2,238,480 132 80 +28.45

A433 Denver 1,980,140 121 80 +22.98

Total U.S. 240,218,238 7,760 7,760 −3.54

Table 4 shows the current and proposed sample sizes 

for “A” PSUs in the West region after using PROC 

NLP to allocate the sample using Method 4. The 

table also shows the effect of using this method of 

sample allocation on CEQ’s standard error. PSUs

with populations under 4 million will have their

sample sizes reduced, while PSUs with populations 

over 4 million will have their sample sizes increased. 

This shows that reducing the minimum target number 

of usable interviews collected quarterly in each urban 

geographic area to 80 would reduce the standard

error in the larger “A” PSUs and increase the

standard error in the smaller “A” PSUs. Using

Method 4 reduces the standard error for the

nationwide sample by approximately 3.54%. The

standard errors increase in some geographic areas and 

decrease in others, but the standard error for the

nation as a whole decreases by 3.54%.
Conclusion

Finding a way to allocate the CEQ’s nationwide

sample of households to individual geographic areas 

is facilitated by automating the sample allocation

process. Estimates of standard error can be computed 

from the output of PROC NLP and analyzed to find 

an allocation method which minimizes the

nationwide standard error. Our research shows that 

allocating the CEQ’s sample using Method 4 reduces 

the standard error of the published expenditure

estimates at the national level by about 3.54%. This 

automated process to select an optimal sample

allocation method can be effective in future sample 

redesigns.

Appendix

Automating the Sample Allocation Process

Below is the optimization problem for the sample 

allocation using Method 4, along with a SAS

program that solves it.

Given values of n, pi, p, find values of ni that

Minimize ∑=

















−

42

1

2

i

i i

p

p

n

n

Subject to n1 + n2 +L+ n38 = 7,360

n39 + n40 + n41 + n42 = 400

ni ≥ 80 for i=1,2,...,38

ni ≥ 0 for i=39,...,42

Where

ni = number of housing units assigned to geographic area=i

 n = number of housing units nationwide (n = 7,760)

pi = population of geographic area=i

 p = population in all geographic areas(p = 1 2 42 p + p +L+ p )

***********************************************

* COMPUTE THE SQUARED DIFFERENCE BETWEEN EACH *

* AREA’S PROPORTION OF THE POPULATION & ITS *

* PROPORTION OF THE SAMPLE. *

**********************************************;

%MACRO MAC1;

SUM_POP = SUM(OF POP1-POP42);

%DO I=1 %TO 42;

 SQR&I = ((N&I/7760) - (POP&I/SUM_POP))**2;

%END;

%MEND MAC1;

*************************************************

* SOLVE A CONSTRAINED LEAST SQUARES PROBLEM TO *

* FIND THE NUMBER OF HOUSING UNITS IN EACH PSU *

* THAT MINIMIZES THE SUM OF SQUARED DIFFERENCES *

************************************************;

PROC NLP DATA=POP_DATA(KEEP=POP1-POP42) NOPRINT

 OUT=RESULTS(KEEP=N1-N42)

 /* CONVERGENCE CRITERIA */

 GCONV=1E-15 FCONV2=1E-15 MAXITER=100000;

 /* DECISION VARIABLES */

 DECVAR N1-N42;

 /* COMPUTE THE SQUARED DIFFERENCES */

 %MAC1; 

 /* SUM THE SQUARED DIFFERENCES */

 F1=SUM(OF SQR1-SQR42);

 /* FUNCTION TO BE MINIMIZED */

 MIN F1;

 /* PROBLEM CONSTRAINTS */

 BOUNDS N1-N38>=80, N39-N42>=0;

 NLINCON F2=7360, F3=400;

 F2=SUM(OF N1-N38);

 F3=SUM(OF N39-N42);

RUN;