Reduction of Sample Areas in the Consumer Price Index and Consumer Expenditure Survey December 2007

Designs 

Lawrence R. Ernst, William H. Johnson, William E. Larson 

Bureau of Labor Statistics, 2 Massachusetts Ave., N.E., Room 1950, Washington, DC 20212-0001

Ernst.Lawrence@bls.gov

Abstract

The Consumer Price Index (CPI) and the Consumer 

Expenditure Survey (CE) are surveys with multistage designs, 

revised every 10 years. The first stage CPI and CE samples 

include a set of areas (PSUs) selected from the set of U.S 

Core Based Statistical Areas. CE additionally selects a set of 

PSUs to represent the rest of the nation. After selecting the 

original sample of PSUs, a reduction was considered for 

budgetary reasons, a reduction implemented in CE only. In 

this paper we describe: the details of the reduction process 

used and alternative approaches; the adjustment of the PSU 

weights resulting from the reduction, which was complicated 

by the use of a maximization of overlap procedure in the 

original selection of the new sample PSUs; and possible 

improvements to the overlap procedure in the next redesign. 

KEYWORDS: PSUs, maximization of overlap, PSU 

weights 

1. Introduction 

Every 10 years the Bureau of Labor Statistics has 

updated the area samples for the CPI (Consumer Price 

Index) and the CE (Consumer Expenditure Survey). 

The CPI uses data from CE in order to produce 

aggregation weights used in constructing higher level 

indexes. The CE derived weights are used in 

aggregating across item strata and index areas. For 

instance, aggregating bananas over all sampled areas in 

order to produce a U.S. level banana index. 

Because of this usage of CE data in the CPI, the 

desire has been to have the same sample areas for both 

the CE and CPI. In addition there has been concern 

about potential bias in the C-CPI-U (chained CPI for 

urban consumers) if the CE and CPI have different 

sample areas. 

CE covers the entire U.S. population within the 50 

states and the District of Columbia. CPI-U covers the 

portion of the area covered by CE that is in CBSAs 

(Core Based Statistical Areas). Thus the CPI area 

sample is normally a subset of the CE area sample. 

There are several steps in determining the area 

sample: 

1. A set of certainty areas has to be determined. 

This is done by means of a population cutoff. The 

cutoffs used or examined in previous area samples 

were 1,200,000, 1,500,000, and 1,800,000. As the CPI 

divides its total sample into 120 units called variance 

PSUs, we examined using total CBSA population 

divided by 120, that is 2,142,306, as a cutoff. 

Although, simulations indicated that with a higher 

cutoff we should expect lower variances resulting from 

fewer certainty areas and more non-certainty areas, 

2,142,306 was chosen in order not to lose a large 

number of the current certainty areas. 

2. Simulations using components of variance of 

price change from current samples were used to 

determine the optimal number of non-certainty PSUs 

to select in each cluster given the set of certainty areas 

and the constraint of 120 variance PSUs total. (There 

are eight clusters, which are the cross product of the 

four census regions and the two types of areas used in 

CPI, metropolitan and micropolitan.) 

3. Each cluster of non-self representing areas was 

partitioned into sampling strata with the number of 

sampling strata equal to the number of PSUs to be 

selected in the corresponding cluster using a 

stratification procedure which attempted to minimize a 

sum of squared differences using several values from 

the 2000 Census and latitude and longitude. The 

number of strata in a cluster was the number of PSUs 

determined to be optimal in step 2. 

4. Overlap maximization procedures were run. 

Since the cost of introducing a new area to the CE/CPI 

sample is large, there is a desire to have as much 

overlap as possible, that is to retain as many of the 

current sample PSUs in the new sample as possible 

while preserving the unconditional selection 

probabilities of the PSUs in the new design. The 

following procedure was used to produce the adjusted 

probabilities of selection: 

The first step was to determine what is meant by 

an overlap PSU. Given the considerable changes in 

definitions of the PSUs it is possible that part of a PSU 

might currently be in the CPI sample but not other 

parts. The preliminary definition was that 30% of the 

counties or 30% of the 2000 population of a PSU 

currently be covered by the CPI sample. This was 

complicated by the fact that counties are composed of 

Minor Civil Divisions (MCDs) in the Northeast region. 

Current CPI PSUs in the Northeast are defined at the 

MCD level, while the new PSUs are defined at the 

county level. It was decided that a county composed 

of MCDs was overlap if at least 5% of its 2000 

population was covered by the current CPI sample. A 

PSU composed of MCDs is considered overlap as long 

Section on Survey Research Methods

2918
as 30% of the counties are overlap and at least one of 

those counties is at least 30% covered by the current 

CPI sample based on number of MCDs. 

The overlap maximization procedure, which is a 

modification of the procedure used in the previous 

redesign, attempts to increase the likelihood of 

selecting PSUs which are overlap. Some changes in 

the program had to be made due to the massive 

redefinition of PSUs. The procedure used operates at 

the level of the intersection of a new stratum with a 

stratum for the 1998 CPI Revision PSU sample. Due 

to redefinitions, there are many cases where only part 

of a new PSU lies within one of these intersections. 

Thus the new PSUs were broken in pieces for the 

purpose of overlap maximization and then the pieces 

were added together to give the total new probability 

of selection of a PSU. See Section 4 of this paper for a 

further description of this procedure. 

5. A controlled selection program was run to 

construct a complete set of patterns in each Census 

region and then select a pattern in each region. Among 

the controls were the requirements that: 

Each state had to receive the number of PSUs it 

was expected to have based on population (rounded up 

or down to one of the two nearest integers). 

In each region the number of overlap PSUs had to 

equal the expected number of overlap PSUs based 

upon the adjusted probabilities of the overlap PSUs 

(rounded up or down). 

In each cluster of sampling strata, the number of 

PSUs selected had to equal the expected number of 

PSUs selected from the cluster (rounded up or down.) 

The unconditional probability of any new PSU 

being selected had to remain unchanged by the 

controlled selection procedure. 

The resulting selected sample of 86 areas, which 

will be referred to as the 86 PSUs design, was to be 

used for the CPI, with the CE using these 86 areas and 

16 additional areas that are outside CBSAs. CE data 

for the new area sample was collected by Census 

starting in 2005. The CPI was originally supposed to 

start introducing the new area sample in 2008, which 

would replace the current area sample, an 87 PSUs 

design. 

CPI’s initial request for funding for introducing 

new areas and a continuous rotation of the housing 

sample was denied. It was decided that the cost of the 

CPI revision should be brought down. CPI does its 

own data collection so it was felt that the greatest 

savings would be realized by reducing the number of 

areas. 

A description of the sample cut procedure is 

presented in Section 2. The remaining sections focus 

on the overlap procedure, principally because it affects 

the changes in the PSU weights due to the sample cuts. 

In Section 3 a linear programming overlap procedure is 

described, which has some advantages over the 

currently used procedure. In Section 4 a slightly 

modified version of the overlap procedure actually 

used in the selection of the PSUs in the 86 PSUs 

design is presented. The changes, which are mainly 

changes in presentation, were principally done to 

simplify comparisons between the procedure actually 

used in selecting the PSUs for the 86 PSUs design and 

the linear programming procedure. The original 

description of the overlap procedure actually used is 

presented in Johnson, Shoemaker, and Rhee (2002). It 

is a generalization of a procedure due to Perkins 

(1970). Finally, in Section 5, the calculations of the 

PSU weights after the sample cut are described. In 

particular, these weights are affected by the use of an 

overlap procedure. 

2. Description of Sample Cut Procedure 

The CPI program decided to reduce the number of 

sample areas in the new area sample from 86 to 75 

while keeping the same overall quote and housing 

units sample sizes by increasing within PSU sample 

sizes. It was believed that eliminating the overhead 

and startup costs and not having to maintain staff in 

these 11 areas would save approximately $1,000,000 

per year. 

As the most expensive areas to collect data in the 

new design are the areas which are not in the current 

CPI and therefore are areas for which we do not 

already have data collection staff and offices in place, 

the only areas considered for being cut were newly 

selected areas. 

In the CPI area sample, PSUs can be A-size (self 

representing), X-size (mostly metropolitan CBSAs 

with a few exceptions), and Y-size (mostly 

micropolitan CBSAs). Given the increasing 

population share of the Y-size areas and their relatively 

small samples, only the X-size areas were considered 

for being dropped. Note also that in the CPI’s staff 

documentation of the sample cut process, several of 

the A PSUs were changed to X PSUs in conjunction 

with the sample cut since it was intended that indexes 

would no longer be published for these areas. 

However, since these PSUs remain certainty in the 75 

PSUs design, we consider them in this paper to remain 

A PSUs after the sample cut. 

In the Northeast region, there was one newly 

selected PSU and it was cut. 

In the Midwest region, all three newly selected 

PSUs were dropped. 

In the South region, there were 10 newly selected 

areas. Three of these areas were dropped, with these 

three areas selected from the 10 with equal probability. 

This is the only region in which some of the newly 

selected PSUs were not cut. 

Section on Survey Research Methods

2919
In the West region, all four newly selected PSUs 

were dropped. 

Thus a new sample of 75 PSUs for the CPI was 

determined. The PSU weights of the X sample PSU 

remaining in the 75 PSUs design were increased to 

reflect that these PSUs are only in sample for the 75 

PSUs design if they are in sample for the 86 PSUs 

design and they are retained in the sample cut. The 

details of the weight adjustments are presented in 

Section 5. 

Due to budget problems, CE needed to save about 

$1.2 million and in 2006 implemented this reduced 

sample by having the Census Bureau, which collects 

the data for CE, drop data collection in the 11 areas 

CPI decided to cut and to adjust the PSU weights in 

the same way CPI intended. 

CPI ultimately decided to return to the originally 

selected 86 PSUs sample, deciding that it could make 

up the estimated $1,000,000 savings in other ways. As 

of this time, Congress has not yet approved CPI 

revision funding to implement a new area sample. 

3. Overlap Procedure of Ernst (1986) 

As mentioned in the Introduction, the overlap 

procedure used in the selection of the CPI PSUs in the 

86 PSUs designs and earlier designs is a generalization 

of a procedure due to Perkins (1970). 

An alternative is an overlap procedure due to 

Ernst (1986), which has at least four major advantages 

over Perkins’ procedure, namely: 

A. It generally yields a higher expected overlap 

than Perkins’ procedure and never yields a lower 

expected overlap. 

B. It is easy to modify this procedure to handle 

the case when the PSUs are redefined in the new 

design. 

C. The procedure does not require that either the 

initial design or the new design be one PSU per 

stratum. In this paper, initial design and new design 

mostly refer to the 87 PSUs and 86 PSUs CPI designs, 

respectively. 

D. The procedure can handle many different types 

of rules for what is considered an overlap, or what is 

considered a partial overlap. (See the last paragraph of 

this section for a discussion of partial overlap.) 

In addition, the Ernst (1986) procedure shares 

with Perkins’ procedure the desirable property that it is 

not required that the PSUs in the initial sample be 

selected independently from stratum to stratum. The 

reason that this property is mentioned is that there are 

some overlap procedures, such as the procedure of 

Causey, Cox, Ernst (1985), that require this 

independence assumption but do not preserve this 

independence from stratum in the new design and 

hence cannot be used in two consecutive redesigns. 

Therefore, these overlap procedures are not described 

here. This issue is discussed in more detail in Ernst 

(1986). 

We proceed to describe the procedure of Ernst 

(1986). Since the CPI is a one PSU per stratum 

design, we restrict the description to this case in order 

to simplify it, although this restriction is not necessary 

and is not used in Ernst (1986). Also, since in the CPI 

the PSUs definitions differ in the 87 and 86 PSUs 

designs, we do not restrict our description to the case 

when the PSU definitions are identical in the old and 

the new designs. 

Note first that each stratum S in the new design 

represents a separate problem, with NkS ,,...,1, k =

denoting the PSUs in S and π k denoting the 

probability of selection of in the new sample. Let 

 be the strata in the initial design having at 

least one PSU that intersects a PSU in S. The general 

idea of the procedure is to select one of the and 

then condition the selection of the new sample PSU 

from S on which PSU from was chosen in the 

initial sample. Furthermore, in general, one specific 

 is not chosen with certainty, but instead a 

probability is assigned to each of the , with the 

Sk

r ,...,II1

i I

i I

i I

i y i I

riy ,,...,1, i = being a set of variables in the 

optimization process. For each = ri ,,...1 let 

,,...,1, ij = ujI i denote the PSUs in let denote 

the joint probability that is the selected initial 

stratum, is the PSU from this stratum selected in 

the initial sample, and is the new PSU selected 

from S. Let be the probability of selection of in 

the initial sample. 

;i I ijk x

i I

ij I

Sk

pij ij I

The ’s are variables in the linear programming 

problem to be described and the only such variables 

besides the ’s. After an optimal set of values is 

obtained by solving the linear programming problem, 

the desired probabilities, which are the probabilities of 

selection of each of the ’s conditioned on the initial 

sample PSUs in , can be expressed in terms of 

the optimal ’s and ’s and the known ’s as 

follows. 

ijk x

i y

Sk

r ,...,II1

i y ijk x pij

, ),Pr(

),,Pr( ),Pr(

iji

ijk

iji

kiji

ijik py

x

II

SII

IIS = =

from which it follows that 

∑

=

=

r

i ij

kij

rjjk

i

i

r p

x IIS

1

1 )....,,Pr( 1

where riI ,,...,1 i ij = is the PSU in stratum that was i I

Section on Survey Research Methods

2920
in the initial sample. 

All that remains is to state the linear programming 

problem that yields the optimal set of ’s and ’s. 

The constraints will be presented first and then the 

objective function. Since new unconditional 

probabilities must be preserved, the probabilities of all 

the three stage events with as the new sample PSU 

in S must sum to 

ijk x i y

Sk

π k , that is 

∑∑

= =

==

r

i

u

j

kijk

i

Nkx

1 1

π ,...,1,

Similarly, since , we also have that iijiji ),Pr( = ypII

∑

=

===

N

k

iijijk i ujriypx

1

, ,...,1,,...,1

The final constraint is 

∑

=

=

r

i

i y

1

1

which arises from the fact that exactly one initial 

stratum is chosen to provide information that is input 

in the process of selecting a new sample PSU from S. 

As for the objective function, it will be of the form 

∑∑∑

===

r

i

u

j

N

k

ijkijk

i

xc

111

where for each is a constant to be 

determined. 

ijk c ,, kji

One possibility is for to be either 0 or 1 

depending on the magnitude of the intersection of 

and ; that is outcome leads to overlap and 

hence if, for example, a certain percentage of 

 is covered by based on either the new design 

population or the number of subdivisions of the PSU, 

such as counties, while otherwise . This 

approach is based on the requirement of a single initial 

sample PSU covering a certain percentage of the new 

sample PSU rather than an alternative of all the initial 

sample PSUs intersecting combined covering this 

percentage. 

ijk c

ij I

Sk ,, kji

cijk =1

Sk ij I

cijk = 0

Sk

A second possibility is for each to be 

determined by considering PSU together with a set 

of initial PSUs with one initial PSU from each initial 

stratum other than that intersects S. This yields a 

set of PSUs one from each stratum . For each 

such set the product of the initial selection probabilities 

of each PSU in this set except for PSU is calculated 

and the product summed over all such sets for which 

the set covers the required percentage of to obtain 

. This is the desired goal of the approach taken in 

the selection of sample PSUs for the 86 PSUs CPI 

design, since for this selection a PSU (except for a 

PSU composed of MCDs) is considered to be overlap 

if 30 % of the 2000 population of the PSU or 30% of 

the counties in the PSU are covered by the 87 PSUs 

sample, regardless of the number of PSUs from the 87 

PSUs design needed to meet this 30% figure. 

However, for the procedure actually implemented, as 

described in Section 4, this overlap goal may actually 

not be met since that procedure attempts to retain the 

intersection of an initial sample PSU and a new PSU 

regardless of how little of the new population of the 

new PSU is in the intersection. 

ijk c

ij I

i I

r ,...,II1

ij I

Sk

ijk c

A third approach is to have the value of be the 

proportion of that is in the intersection of and 

 based on 2000 population. This approach is 

illustrated in Ernst (1986). This approach, unlike the 

other two approaches allows for an outcome to 

be considered a partial overlap, that is 

ijk c

Sk ij I

Sk

,, kji

< cijk < 10 , 

when an initial sample PSU partially intersects a new 

sample PSU. 

4. Perkins’ Method and its Generalization 

The method of Perkins (1970) has some of the 

same characteristics as that of Ernst (1986), at least for 

one PSU per stratum designs, which are the only type 

considered in this paper. (Perkins never generalized 

his procedure to other designs, while the Ernst (1986) 

procedure is not restricted to such designs.) In 

addition, in Perkins’ procedure, unlike the Ernst 

procedure, it is assumed that that the PSU definitions 

are the same in the initial and new designs. Since for 

CPI the PSUs in the 87 PSUs design and the 86 PSUs 

design are defined differently, the selection of PSUs 

for the 86 procedure was done using a generalization 

of Perkins’ procedure due to Johnson, Shoemaker, and 

Rhee (2002), which will be referred to as JSR. This 

procedure allows for different PSU definitions in the 

initial and new designs. However, although in the CPI 

application a new PSU is generally considered to be 

overlap if it is 30% covered by the initial sample, as 

explained in the Introduction, JSR actually attempts to 

select any new PSU that has any intersection at all with 

an initial sample PSU. 

Although the procedure of Ernst (1986) has 

several advantages as we have mentioned in Section 3, 

we proceed to describe JSR since this was the 

procedure used in selecting the 86 PSUs design and 

hence is needed in calculating the probability of a PSU 

in the 86 PSUs design being considered overlap. As 

mentioned in the Introduction, our description of the 

JSR procedure will be somewhat different than given 

Section on Survey Research Methods

2921
in that paper. 

For = ri ,,...,1 , let uj i = ,...,1 = Nk ,,,,,,1 π ij be 

the proportion of the 2000 population of S that is in 

and let 

ij I

ijk π be the proportion of the 2000 population 

of that is in . has the same meaning as 

in Section 3, which is the probability of initial stratum 

 being selected to provide information on the initial 

sample. However, for JSR is not obtained by 

solving a linear programming problem but instead is 

simply the proportion of the 2000 population of S that 

is covered by . 

SIij ∩ Sk i y

i I

i y

i I

Also, instead of calculating the variables 

using linear programming, we calculate for 

, where is 

the joint probability of the following four events: the 

piece of the new PSU in S is selected from ; is 

the initial sample PSU selected from ; the new PSU 

in S is selected from a piece of a PSU in S that resides 

in ; and the piece of the new PSU that is selected 

from resides in , that is is the new sample 

PSU selected from S. is obtained by letting 

ijk x

ijtk x

= ri ,,...,1 utj i = ,...,1, = Nk ,,,,,,1 ijtk x

i I ij I

i I

it I

it I Sk Sk

ijkt x

max{ if }0,

(max{ })0,

})1),/(min(1(

}1),/(min{

1

jtpy

py

py py

x

pyx py

itiit

u

l

iliil

iji ijiij itk

ijtk

ijiijijiijjk ijk

i

−× ≠

−

− =

=

∑

=

π

π

π π

π π

where itk π is the proportion of , based on the 

2000 population, that resides in . It then follows 

that 

it ∩ IS

Sk

iji

u

t

ijtk

ijik py

x

IIS

i

∑

= = 1 ),Pr(

and hence 

∑

∑

=

= =

r

i ij

u

t

ijtk

rjjk

i

i

r p

x

IIS

1

1

1 )...,,Pr( 1

where is the PSU in stratum that was 

in the initial sample. 

riI ,,...,1 i ij = i I

5. Calculation of PSU Weights After Sample Cut 

In order to obtain PSU weights after the sample 

cut that reflect the probability of selection, the key is 

that this weight, which is a random variable w, must 

satisfy wE = 1)( for each of the 75 retained PSUs and, 

in addition, that w = 0 for any PSU not among the 75 

PSUs. See Ernst (1989) for a discussion of this issue. 

Here the expectation is over all possible 87 PSUs 

design sample PSUs, 86 PSUs design sample PSUs 

selected using the overlap maximization procedure, 

and 75 PSUs design sample PSUs selected using the 

sample cut procedure. 

We assume that the weight before the sample cut 

satisfies the conditions that for each of the 

PSUs in sample for the 86 PSUs design and 

wE = 1)(

w = 0 if 

the PSU is not in sample for the 86 PSUs design. We 

only consider sample PSUs that are X PSUs in the 86 

PSUs design, since these are the only PSUs that are 

eligible to be cut in the 75 PSUs design. We also 

assume that the sample PSUs in the 87 PSU design 

were selected independently from stratum to stratum. 

This is not actually true since the sample PSUs in the 

87 PSUs design were selected using an overlap 

maximization procedure and, as shown in Ernst 

(1986), the use of most overlap maximization 

procedures, including Perkins’ procedure, destroys the 

stratum to stratum independence. See Case 2 below 

for further discussion of this issue. 

We proceed to describe the weighting procedure 

for the 31 PSUs that are X sample PSUs in both the 86 

PSUs design and the 75 PSUs design by considering 

the following cases. (In general more complex cases 

are possible, but in this work only the cases described 

below occurred.) 

Case 1. South region. 

Case 1.A. X PSUs in 86 PSUs design that overlap 

PSUs in 87 PSU design. 

There are 8 such PSUs. The PSU weight for each 

of these PSUs after the sample cut is the same as it is 

before the sample cut since each of these PSUs are 

retained in the 75 PSU design with certainty. 

Case 1.B. All other X PSUs. 

There are 10 sample X PSUs in the South region 

in the 86 PSUs design that are not considered as 

continuing from the 87 PSUs design. Of the 10, 7 

were selected with equal probability to remain in 

sample in the 75 PSUs design. For each of these 7 

PSUs, the PSU weight before the cut is multiplied by 

10/7 to obtain the base PSU weight after the cut. This 

weight is adjusted by a benchmark factor, which forces 

the sum of the weighted PSU population in the 7 

retained sample PSUs in the South region to agree with 

Section on Survey Research Methods

2922
the 2000 census population for the 10 strata from 

which these 7 PSUs were selected. Similar benchmark 

factor adjustment would be done in the other three 

regions. 

Case 2. Three other regions. 

For each of the three subcases of Case 2 described 

below, we present a formula for the probability that an 

X PSU was in sample for both the 87 PSUs design and 

the 86 PSUs design, since in these three regions an X 

PSU is in the 75 PSU sample if and only if these two 

events both occur. The reciprocal of this joint 

probability is the base PSU sample weight after the 

sample cut. Now if the selection of a PSU in the 87 

PSUs design and the selection of a PSU in the 86 PSUs 

design were independent events, then this joint 

probability would be the product of the probability of 

being in the 87 PSUs design and the probability of 

being in the 86 PSUs design. However, this 

independence does not hold because, as explained 

earlier, a procedure was used in selecting the 86 PSUs 

design sample that maximized overlap with the 87 

PSUs design sample and in general the use of such an 

overlap procedure destroys such independence. For 

the three sub-cases we present expressions for the joint 

probability that correctly take into account the use of 

the overlap maximization procedure in the selection of 

the PSUs in the 86 PSUs design. However, to avoid 

undue complexities, the calculation makes the 

incorrect assumption that the sample PSUs in the 87 

PSUs design were selected independently from stratum 

to stratum, when in fact these PSUs were also selected 

using an overlap procedure. In addition, we ignored 

the fact that a controlled selection procedure, as 

explained in JSR and summarized in Step 5 of Section 

1 of this paper, was used in selecting the PSUs in both 

the 87 and 86 PSUs designs and that controlled 

selection generally destroys the independence of 

sampling from stratum to stratum. As a practical 

matter we had no choice, since as far as we know the 

data to completely calculate the joint probabilities 

correctly no longer exists. We believe that taking into 

account the use of the overlap procedure in selecting 

the sample PSUs for the 86 PSUs design, as we have 

done, is the most important step that we could have 

taken in obtaining the correct joint probabilities. 

We now proceed to present these three sub-cases: 

Case 2.A. An entire sample PSU in the 86 PSUs 

design is contained in a single sample PSU in the 

87 PSUs design. There are 11 in these regions 

satisfying this condition. 

Sk

ij I

Sk

For each such , let be the probability of 

selection of the corresponding in the 87 PSUs 

design and let 

Sk pij

ij I

π ij be probability that the PSU selected 

from S in the 86 PSUs design is selected from a piece 

of . Let be as described in Section 4. Then the 

probability that is an overlap PSU, which is the 

probability that is in sample for the 87 PSUs 

design and is in sample for the 86 PSUs design is 

ij I i y

Sk

ij I

Sk

ijiijiji ijk py π py }1),/(min{ π (5.1) 

The reason that this is so is that is for stratum S

the joint probability that is the corresponding 

selected initial stratum and is the initial sample 

PSU in for the 87 PSUs design, since these two 

events are independent. 

py iji

i I

ij I

i I

}1,/min{π py ijiij is the 

probability that the new PSU in S will be selected from 

a piece of in the 86 PSUs design given the first two 

events and 

ij I

ijk π is the probability that the piece of 

selected is . 

ij I

Sk

Case 2.B. Two of the X PSUs, , in the 86 PSUs 

design have the following properties. Each of these 

PSUs was included in a single sample PSU in the 

87 PSUs design sample except that contains 

counties that were not included in an urban area in the 

87 PSUs design and hence were not in and not 

eligible for the CPI sample. 

Sk

ij I

Sk

ij I

Then (5.1) can be applied in this case as in Case 

2.A if we artificially consider the non-urban counties 

that are part of in the 86 PSUs design to be part of 

 in the 87 PSUs design but with an urban 

population of 0. That is, under these conditions , 

in the 86 PSUs design is contained in a single sample 

PSU in the 87 PSUs design and (5.1) applies. 

Sk

ij I

Sk

ij I

Case 2.C. There are three PSUs in the 86 PSUs design 

that constitute this case. Each of the PSUs in this 

design consists of two urban counties, county 1 and 

county 2, which in the 87 PSUs design were in 

different urban strata. 

Let be the 87 PSUs design strata containing 

county 1 and county 2, respectively; let be 

the PSUs in this design containing county 1 and county 

2, respectively; and let S be the 86 PSUs design 

stratum containing counties 1 and 2. Let be 

the probability of selection of in the 87 PSUs 

design and let 

21, II

2111, II

2111, pp

2111, II

2111 π ,π be the proportion of the 2000 

population of S in ,, respectively. Let 2111 II

Section on Survey Research Methods

2923
iy = 2,1,i , be the proportion of the 2000 population of 

S covered by . Then the probability that county 1 

and/or county 2 is in the 87 PSUs design sample and 

the PSU consisting of counties 1 and 2 is in the 86 

PSUs design sample is 

i I

Sk

212 21221 21

111 11111 11

}1),/(min{

}1),/(min{

k

k

py py

py py

π π

π π

+

References 

Causey, B. D., Cox, L. H., and Ernst, L. R. (1985). 

Applications of Transportation Theory to Statistical 

Problems. Journal of the American Statistical 

Association, 80, 903-909. 

Ernst, L. R. (1986). Maximizing the Overlap Between 

Surveys When Information Is Incomplete. 

European Journal of Operational Research, 27, 192-

200. 

_______ (1989). Weighting Issues for Longitudinal 

House and Family Estimates. Panel Surveys, 139-

159. New York, John Wiley. 

Johnson, W. H., Shoemaker, O. W., and Rhee, Y. W. 

(2002). Redesigning the Consumer Price Index 

Area Sample. Proceedings of the Section on 

Government Statistics, American Statistical 

Association, 1671-1676. 

Perkins, W. M. (1970). 1970 CPS Redesign: Proposed 

Method for Deriving Sample PSU Selection 

Probabilities Within 1970 NSR Strata. 

Memorandum to Joseph Waksberg, U.S. Bureau of 

the Census. 

Any opinions expressed in this paper are those of the 

authors and do not constitute policy of the Bureau of 

Labor Statistics. 

Section on Survey Research Methods

2924