SAMPLE REDESIGN FOR THE INTRODUCTION OF THE TELELPHONE POINT OF PURCHASE SURVEY FRAMES IN

THE COMMODITIES AND SERVICES COMPONENT OF THE U.S. CONSUMER PRICE INDEX

Sylvia G. Leaver, William H. Johnson, Owen J. Shoemaker, Thomas S. Benson

United States Bureau of Labor Statistics, 2 Massachusetts Avenue, N.E., Rm. 3655, Washington, D.C. 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: Sample design; Optimization;

Components of Variance

This paper describes the methods used to allocate

data collection resources for the most recent redesign

of the sample for the commodity and services (C&S)

component of the U.S. Consumer Price Index. These

methods rely on models relating price change sampling

variance and data collection costs to design variables

which are the number of items to price and outlets to

visit per item group in each sample city. With these

models, the optimal allocation of data collection

resources to minimize sampling variance of price

change, subject to budgetary and operational

constraints, can be found using nonlinear

programming techniques. This work represents an

expansion of models developed for the 1987 and 1996

C&S sample redesigns. Models for sampling variance

and costs are given, and solutions to the design

problem posed under varying assumptions are

discussed. A closing section characterizes the changes

in sample allocation from previous designs.

Background

For a full discussion of the Consumer Price Index

(CPI), we refer the reader to Chapter 17 of The BLS

Handbook of Methods (1997). See also Leaver and

Valliant (1995) for a more detailed description of the

C&S sample design, selection, and estimation

procedures

The current CPI is a Laspeyres aggregation of a

combination of Laspeyres- and geometrically-averaged

sub-indexes computed for the breadth of consumerpurchased commodities and services. The C&S

component, which represents 72.5% of the expenditure

weight of the CPI, is computed from measurements of

price change on a sample of commodities and services,

collected from selected outlets in sample cities across

the United States. Consumer items are grouped into

strata, the most finely defined item classes for which a

price index is computed. Let IX(I,t,0) denote the month

t index for a collection of strata, termed an item

aggregate I, where month 0 represents the index base

or reference period. Then

( , ,0) 100(( ( ))/( ( ))) 0IX I t CW I CW I t = ⋅



















 ∑













= ∑

∈ i∈I

t

i I

t t RI (i)IX (i) / RI (i)IX (i) 0

where CWt,(I), is an estimate of expenditures (called a

cost weight) on a collection of items during period t,

computed as a weighted average of individual indexes

over all item strata in the item aggregate I, where

RI (i) t

is the expenditure-weighted relative importance

of item i at time t.

In this application, we were concerned with the

short term or d-month percentage price change:

An index area is the most basic geographic area

for which a price index is computed on a monthly,

bimonthly, or semiannual basis. There are two types of

index areas: self-representing areas, such as New

York, which were selected with certainty; and non-selfrepresenting areas, whose sample comprises two or

more primary sampling units (PSU’s) selected

according to a probability sample. The 1998 revised

U.S. All Cities CPI is a weighted average of 38 index

area CPI’s; 31 from self-representing and 7 from nonself-representing areas. For purposes of variance

estimation and operational manageability, the sample

for each index area is segmented into two or more

subsets, called replicate panels.

Each item stratum is composed of one or more

narrowly defined classes called entry level items

(ELI's). An ELI describes the level of specification for

a class of goods with which a data collector enters an

outlet for initial pricing.

In CPI sample selection, ELIs are selected from

each stratum by a systematic probability proportional to

size (pps) procedure, where, with the 1998 revision,

the ELI weights were derived from expenditures

reported in the 1993-1995 Consumer Expenditure

Surveys. ELI selections are independently drawn for

each replicate panel within each PSU.

Sample frames and weights used in outlet selection

are derived from the Telephone Point of Purchase

Survey (TPOPS), a random digit telephone survey

conducted by the U.S. Bureau of the Census for the

























−

−

− = ⋅ 1

( , )

( , ,0 )

( , , ) 100 dd

IX I t

IX I t

PC I t t
BLS. Beginning in 1997 the TPOPS replaced the

Continuing Point of Purchase Survey, a household

survey. The TPOPS survey provides the names and

addresses of outlets and dollar amounts of purchases,

for item classes known as POPS categories. A POPS

category is a class of items which are normally sold in

the same kind of outlet. Each ELI belongs to only one

POPS category. Outlet frames and selection weights

are derived from POPS survey data for each PSU-POPS

category-replicate panel.

In outlet selection, outlets are selected by

systematic pps from frames for each PSU-replicate

panel for POPS categories corresponding to ELIs

selected in item sampling. Selected items are then

priced in sample outlets on a monthly, bimonthly, or

seasonal basis.

History

Hansen, Hurwitz,and Madow (1953), Kish (1965),

and Cochran (1977) present several examples of

sample design optimization via cost and error

modeling. Groves (1990) discusses sample design for

social surveys.

Cost and sampling error models were first

formulated for the C&S sample design for the 1978

CPI Revision (Westat, 1974). Item classes comprised

two categories - food, and other goods and services,

and sample size allocation were made for six PSU

classes. Selection of the sample design implemented in

that revision was based on evaluation of a number of

alternative designs. The 1987 CPI Revision (CPIR)

redesign (Leaver, et al., 1987) expanded on this

approach, refining models for eight item groups and

ten PSU classes. This implementation relied on

detailed use of administrative records and modeled

estimates for cost and variance function estimates.

Solution methods used nonlinear programming

techniques to identify local minimizers of a modeled

relative variance function, under varying assumptions

of annual inflation and price change interval. For

another BLS survey, Valliant and Gentle (1994)

developed a generalized system for constrained

optimization of a two-stage stratified sample design

implemented on a UNIX platform, with a weighted

summed relative variance objective function.

The approach taken in this application generally

follows that taken for the 1987 and 1998 CPIRs

(Leaver, et al., 1996). Sampling variance was

minimized in this application. Data collection cost

models were revised; costs were derived from

administrative records and a time and travel study of

CPI data collection. The size of the nonlinear

programming problem solved was expanded, and

detailed distribution of item-outlet sampling resources

used stratum-level variance estimates not previously

available for sample design allocation.

The Design Problem

The primary objective of the C&S sample redesign

was to determine values for all sample design variables

which would minimize the sampling variance of price

change for the C&S portion of the CPI. Sample design

variables for the C&S component were the number of

ELI's to select in each item stratum and the number of

outlets to select per CPOPS category-replicate panel in

each sample PSU. The number of PSU's, the number

of replicate panels per PSU, and the item stratification

were previously determined (Williams et al., 1993;

Lane, 1996 and Williams, 1996.)

Certain simplifying assumptions were made to

render the problem tractable. Newly revised item

strata were divided into thirteen item groups: four

subgroupings of food at home, food away from home

and alcoholic beverages, household furnishings and

operations, fuels and utilities, apparel, transportation

less motor fuel, motor fuel, medical care, education

and communications, and the combined group of

recreation and other commodities and services. The 87

PSU’s were divided into 15 groups according to size

and number of replicate panels. It was assumed that

the same outlet sample sizes would apply to all PSU's

within the same PSU group. It was also assumed that

the same item sample selection sizes would apply

across all PSU's . This reduced the allocation problem

to one of determining the values of the design variables

{ Ki, i=1,...,13}, the number of ELI selections per

item group-replicate panel within each PSU and

{ Mij , i=1,...,13, j =1,...,15 }, the designated number

of outlet selections per item group-POPS categoryreplicate within each PSU, which would minimize a

modeled price change sampling variance, subject to

additional allocation and cost constraints.

The variance of price change for all C&S items

was modeled as a function of the design variables, as

were total annual data collection costs. Nonlinear

programming methods were then used to determine

optimal values for the design values under various cost,

variance, and sample share constraints. Detailed

descriptions of these activities follow.

The Sampling Variance Function

For the purposes of the allocation problem, we write

the All U.S. City Average C&S price change estimator

as PC(⋅,⋅, t,t − d ) ( , , , )

,

= ∑ ∑ PC i k t t −d

k

i k

RI

i

, where

PC(i, k , t, t −d ) is the estimated price change from time t-d to
t for item group i and index area k, and RIi, k

is the

population-expenditure-weighted relative importance

of item group i in index area k. Deriving a component

form of the variance of this price change estimator,

accounting for the stages of sampling described above, would be

extremely difficult. Rather than this direct route, we have taken

a more indirect, modeling approach described below. Four

sources of variation were modeled: PSU selection, item

selection, outlet selection, and other sources, such as

sampling within the outlet.

The variance function for the CPI revision was

modeled for index areas. Each self-representing PSU

is a single index area. Non-self-representing PSU's

were selected to represent 7 index areas, whose sample

consisted of 2 to 22 PSU's. The variance model

assumes that the total variance of price change for item

group i within index area k can be expressed as a sum

of four components:

s s s s s

i, k psu,i, k eli,i, k outlet,i, k error,i, k

2 2 2 2 2

= + + +

where

s

i, k

2 is the total variance of price change for

item group i in index area k,

s

psu,i,k

2 is the component of variance due to

sampling PSU’s in non-selfrepresenting areas, 0 for selfrepresenting areas,

s

eli,i, k

2 is the component of variance due to

sampling of ELI’s within item strata,

s

outlet,i, k

2 is the component of variance due to

sampling of outlets, and

s

error,i, k

2 is a residual component of variance

attributable to other aspects of the

sampling process, including the final

stage of within-outlet item selection,

called disaggregation .

We assume that the variance of price change of an

individual sampled unit or quote has the same

structure:

s s s

s s

unit i k unit psu i k unit eli i k

unit outlet i k unit error i k

, , , , , , , ,

, , , , , ,

2 2 2

2 2

= +

+ +

, where

s

unit,i, k

2 is the total variance of price

change of an individual sampled

unit or quote for item i in area k,

s

unit, psu,i, k

2 is the component of unit variance

due to sampling PSU’s in non-selfrepresenting areas,

s

unit, eli,i, k

2 is the component of unit variance

due to sampling of ELI's within

item strata,

s

unit,outlet,i,k

2 is the component of unit variance

due to sampling of outlets, and

s

unit, ,i, k

2 is the corresponding residual

component of unit variance.

It follows that each component of s

i, k

2

 can be written

in terms of its corresponding unit variance

components:

)

,

/( 2

, , ,

)

'

,

/( 2

, , ,

/( )) 2

, , ,

(

'

/

2

, , ,

2

,

i k

MiKkHkN i k unit error i kMkHkN unit outlet i k

i

NC iKkHkN k unit item i kN i k unit psu i k

s s

s s s

+ +

= +

where

Nk

is the number of PSU’s in index area k ,

Nk

' is the number of non-self-representing

PSU’s in the index area,

Hk

is the number of replicate panels per PSU

in the index area,

Mi, k

' is the number of unique in-scope outlets

selected per PSU-replicate

NCi

is the percent of strata in item group i

 which are non-certainty strata.

Note that the expected number of quotes per PSUreplicate panel- item group is estimated by the product

of the designated outlet sample size and the number of

item stratum selections, Mij Ki⋅ .

Thus the sampling variance of price change for the

All U.S. City Average C&S index is

2

,

2

,

2

i k

i

i k

RI

k

s TOTAL = ∑ ∑ s

The Cost Function

The total annual cost of the C&S portion of the

CPI includes costs of initiation data collection and

processing, personal visit and telephone pricing, and

pricing data processing, each of which were developed

in terms of outlet and quote related costs. For PSU

group j and item group i, outlet related costs for

initiation are:

)

2

(

,

( , ) 0.25 ijMij b

ijMij a O i

C jH jN iK O Mij CI = ⋅ ⋅ ⋅ + , where

CIO Mij Ki( , ) is the outlet-related initiation cost

for item group i in PSU group j

Nj is the number of PSU’s in group j,

Hj is the number of replicates per

PSU in PSU group j,

CO,i is the initiation cost per outlet for

item group i,
and ( )

2

aij Mij + bijMij is an overlap function used to

predict the number of unique sample outlets,

accounting for the overlap of elements in the outlet

sample within and between item groups for a replicate

panel. The number 0.25 accounts for the rotation or

reselection of one-fourth of the sample each year.

Quote related initiation costs are:

CIQ Mij Ki N jH j SeasIi CQ i Mij Ki NRi = ⋅ ⋅ ⋅ ⋅ ⋅ 25 ,( , ) 0.

where

CIQ Mij Ki( , ) is the quote-related cost of initiation

for item group i in PSU group j,

SeasIiis a seasonal items initiation factor

for item group I,

CQ,i is the initiation cost per quote for

item group i, and

NRi

is the outlet initiation response rate

for item group i.

The costs of ongoing price data collection and

processing were also developed as both outlet and

quote related costs. For PSU group j and item group i,

outlet related costs for ongoing pricing are:

[( ) (1 ) ]

( , ) ( )

, , , , , , , , , ,

2

PV O i PV T i T O i T O i T O i

O ij i ij j j i ij ij ij ij

C C R C R

CP M K MB N H NR a M b M

⋅ + ⋅ − + ⋅

= ⋅ ⋅ ⋅ ⋅ +

where

CPO Mij Ki( , ) is the total outlet-related cost for

ongoing pricing,

CPV ,O,i

is the cost for a personal visit for

pricing per outlet for item group i,

CPV ,T,i

is the travel cost for a personal visit

for pricing per outlet for item group i,

RT,O,i

is the proportion of outlets priced by

 telephone for item group i,

CT,O,i

is the per outlet cost for telephone

collection,

MBij is a factor to adjust for the monthly/

bimonthly mix of outlets and quotes by

PSU and major product group.

Quote related costs for ongoing pricing are:

[ (1 ) ]

( , )

i PV,Q,i T,Q,i T,Q,i T,Q,i

Q ij i ij j j ij i i

SeasR C R C R

CP M K MB N H M K NRQ

⋅ − + ⋅

= ⋅ ⋅ ⋅ ⋅ ⋅

⋅

, where

CPQ Mij Ki( , ) is the total quote-related cost for

ongoing pricing,

CPV ,Q,i

is the per quote cost for a personal

visit for pricing,

RT,Q,i is the proportion of telephone

collected quotes for item group i,

CT,Q,i is the per quote cost for telephone

collection for item group i, and

NRQ,i is the quote level pricing response

rate for item group i.

SeasRi is a seasonal items ongoing pricing

factor for item group i.

The total cost function associated with data

collection and processing for C&S, summed over all

item groups and PSU groups, is then given by:

C CI M K CI M K

CP M K CP M K

Total O ij i Q ij i

i j

O ij i Q ij i

= +

+ +

∑[ ( , ) ( , )

( , ) ( , )]

,

Thus, the sample design problem can be expressed as

the nonlinear programming problem:

Minimize sTotal

2

({ Ki },{ Mij }) subject to:

CTotal ≤ $5,300,000

Ki ≥ Number of item strata in item group i,

Ki ≤ Maximum number of item hits for item group i,

Mij ≥ 2 , i=1,..., 13, j=1,...,15

Average item hits per stratum ≥ 9

Model coefficients

Estimates of components of the cost function were

developed using agency administrative records. Fiscal

year 1996 data were used to obtain a total cost per

outlet to initiate, and then data provided by the field

office produced a per hour cost of initiation. Outlet

unit costs and quote unit costs of initiation, by item

group, were derived by taking these per outlet and per

hour costs and combining them with data obtained

from a data collection time and travel study conducted

in 1987. Travel costs per quote, by item group, were

estimated by using an overall travel cost per outlet and

again comparing it to data from the 1987 time and

travel study.

Pricing costs were figured in a similar manner.

Distinctions between personal visit and telephone

collection of data were made based upon cost

accounting information from the field office and from

an analysis conducted within the Prices Statistical

Methods Division. Outlet initiation survival rates and

quote and outlet retention rates for each item group

were developed from field initiation records and

ongoing pricing records for mid 1993-mid 1997.

“Overlap” functions were modeled to project the

number of unique outlets realized in sample selection

as a function of designated sample size. These were

obtained by modeling the number of unique outlets
obtained in simulations of sampling procedures for

each PSU and item group, using CPOPS and TPOPS

sampling frames for the most recent rotations for each

PSU-item stratum (Johnson et al., 1999).

Components of price change variance were

computed using weighted restricted maximum

likelihood components of variance estimation methods

and C&S price micro-data collected from 1993-97

(Shoemaker and Johnson, 1999). Component

estimates were developed for 6-, and 12-month price

change for the 13 item groups for each index area.

Problem Solution

SAS NLP was used to find a local minimum to the

design problem. Solutions were found using

components of variance estimates for 6-, and 12-month

price change components of variance estimates. For

each item group, the number of item selections was

bounded below by the number of strata in the item

group and above by a ceiling of 133% of the item

group’s pre-1998 revision item stratum hits allocation.

Only minor differences were observed between the

problem solutions found for differing pricing intervals.

The solution found for 6-month price change was

selected because variance component estimates were

considered most stable for this interval.

Item hits were then distributed among item strata

within each item group, with consideration given to

differences in relative importance, stratum level price

change variance estimates, and response rates among

the item strata within each item group, as well as

special problems identified by commodity analysts and

field staff. Similarly, designated outlet sample sizes

were adjusted among the various POPS categories in

item groups to manage variation in expected response

rates and respondent burden.

Although major revisions in the CPI occur every

10 years, incremental revisions are planned with each

year with the staggered rotation of TPOPS categories

in sample PSUs. The table below characterizes the

latest TPOPS rotation sample design, contrasting it

with the design implemented in sample rotations for

the four years prior to the last CPOPS rotation revision,

which occurred in 1996. In general, the sample design

represents a considerable reduction in modeled

sampling variance from that projected under the same

model for these prior allocations. This reduction is

primarily attributable to a 33% increase in the data

collection resource budget; the gains in projected

precision match almost one to one with gains in

allocable budget. In addition, the allocation also

shifted resources in many item groups from sampling

many outlets to fewer outlets, with more item

selections per outlet. This is due primarily to the

larger residual component of price change sampling

variance estimated for most item groups. This

component was regarded as negligible in earlier

estimation (Leaver, et al., 1987).

Acknowledgments

The authors would like to thank Janice Lent,

David Chapman, and Alan Dorfman for their careful

reading of earlier drafts of this paper and their helpful

comments, Glenn Pontanilla and Jane Martinez of the

Office of Field Operations, and Valerie Harris of the

Office of Prices and Living Conditions for their

assistance in survey cost accounting, and Kirk

Hagemeier and Sean McIllece of the Prices Statistical

Methods Division, Robert Cage of the Division of

Consumer Prices and Price Indexes, and Mark Crosby

of the Division of Consumer Prices and Consumption

Studies for their assistance in obtaining production

sampling frames. The authors also would like to thank

Janet Williams and Brian Hedges for their support on

this project.

References

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Methods, Washington. DC: U.S. Government Printing

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Pre-revision and Latest Allocation of C&S Design

Item Group Root Mean 6-

Month PC

Variance

9807-9901

% Change in

Modeled PC SE

from 1994

Rotation to

Latest Revision

% Total

Costs,

1990-94

Rotation

% Total

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Design

% Total

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1999

TPOPS

Design

92-94 CE

Relative

Importance

(relative to total

C&S)

Total, All Items less Rent and

Owners’ Equivalent Rent

 .0977 -15.9 100.0 100.0 100.0 100.0

Food at home (4 groups) .2312

.3430

.6099

.1956

-2.6

 +1.5

-2.7

+0.6

5.3

4.8

4.6

13.2

3.3

2.9

2.3

7.1

Total:15.6

6.8

8.5

6.5

8.8

3.38

3.41

1.78

5.18

Total: 13.75

Food away + Alcoholic

Beverages

.1201 -2.0 5.3 4.8 5.4 8.14

Household Furnishings &

Operations

.4992 -26.4 8.1 12.3 12.1 10.41

Fuels and Utilities .3524 -2.0 4.4 4.2 5.3 6.91

Apparel & Upkeep .8844 -11.1 13.5 11.7 8.2 7.34

Transportation less Motor Fuels .1435 -19.1 14.4 20.33 13.4 20.01

Motor Fuels .2864 -26.7 3.1 2.72 2.9 4.23

Medical Care .1985 -21.3 6.0 6.23 5.0 7.35

Education & Communication .2364 -17.9 5.1 7.15 5.7 7.58

Recreation + Other C&S .1899 -19.7 12.3 15.22 11.4 14.26

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http://www.stats.bls.gov/mlr/cpijw001.htm