OPTIMAL ALLOCATION FOR STRATIFIED TELEPHONE SURVEY DESIGNS

Robert J. Casady, U. S. Bureau of Labor Statistics

James M. Lepkowski, University of Michigan

Key Words: Random digit dialing, Telephone surveys

1. THE CURRENT STATUS OF TELEPHONE

SURVEY DESIGNS

The two stage random digit dialing design for

sampling telephone households, first proposed by

Mitofsky (1970) and more fully developed by

Waksberg (1978), has been widely employed in

telephone surveys. The Mitofsky-Waksberg

technique capitalizes on the fact that working

residential numbers (hereafter referred to as WRNs)

tend to be highly clustered within banks of

consecutive telephone numbers. Currently, only

about twenty percent of the possible telephone

numbers within the known area code, three digit

prefix combinations are WRNs. However, if a bank

of 100 consecutive telephone numbers can be

identified that has at least one known WRN then, on

average, over 50 percent of the numbers in the bank

will be WRNs. The Mitofsky-Waksberg technique,

which identifies 100-banks containing WRNs in the

first stage of sampling, greatly reduces the amount of

screening necessary to identify telephone numbers

assigned to households.

Alternatively, lists of published telephone

numbers have been employed as sampling frames.

These lists of published numbers are available for the

entire country from commercial firms such as

Donnelley Marketing Information Systems. A

straightforward selection of telephone numbers from

such lists provides a very high rate of WRNs

(typically at least 85%) but unfortunately does not

cover households with unpublished numbers.

Comparisons of telephone households with and

without published numbers (see, for example, Brunner

and Brunner, 1971) indicates that substantial bias may

result.

The purpose of this paper is to examine stratified

designs based on the BellCore Research (BCR) frame

as an alternative to list frames and MitofskyWaksberg design. As an example of frame

stratification, the BCR frame could be partitioned into

two strata: a "high density" stratum consisting of

residential numbers in 100-banks with one or more

listed numbers and a "low density" stratum consisting

of all the remaining numbers in the BCR frame.

stratum. The obvious cost difference of sampl

from the two strata can be exploited throu

differential sample allocation.

The next section examines the question of 

appropriate allocation of sample between the st

when simple random sampling is utilized within e

stratum. A key feature of the stratified teleph

sample approach is that it permits alterna

approaches to sample selection within the differ

strata. Several alternatives are presented 

discussed in Section 3. The paper concludes wit

general discussion contrasting the Mitofsky-Waksb

procedure and stratified designs.

2. THE ALLOCATION PROBLEM FOR

STRATIFIED TELEPHONE DESIGNS

2.1 Background

For the purposes of this paper we will assume 

the basic sampling frame is the collection of 

telephone numbers generated by appending four d

suffixes to the BCR list of area-prefix codes. We w

assume that each household in the target populatio

"linked" to one and only one telephone number in 

basic sampling frame.

We will also assume that we have access (poss

only indirect) to a directory based, machine reada

list of telephone numbers such as that availa

through Donnelley Marketing. It should be noted 

because many households choose not to list th

telephone numbers in a directory, any such direct

based frame will not contain all of the WRNs. A

directory based lists are, by nature, out of date so t

will omit some numbers that are currently WR

while including others that are no longer WRNs.

From a survey design point of view these 

frames tend to be radically different. The BCR fra

includes all WRNs so it provides complete "covera

of the households in the target population, but o

about 20 percent of the telephone numbers included

the BCR frame are actually WRNs. Thus, the 

rate" (and hence sampling efficiency) will be qu

low for a simple RDD sample design utilizing 

BCR frame. In contrast, a typical directory/list fra

covers only about 70 percent of the target househo

but the hit rate is 80 to 90 percent. In general 

sampling efficiency for a simple RDD design usin
stratum with a very low hit rate. The sample is

allocated to the strata so as to minimize cost

(variance) for a specified variance (cost). Hereafter

the low hit rate stratum will be referred to as the

residual stratum.

2.2 Basic Notation

Assume that the BCR frame of telephone numbers

has been partitioned into H strata based on a 100-bank

attribute which can be determined from the directory

based frame. The choice of 100-banks is arbitrary,

10-banks or any other sized banks would work just as

well. For the i

th stratum let

Pi = proportion of the frame included in the stratum,

hi = proportion of the telephone numbers in the 

stratum that are WRNs (i.e., the hit rate),

wi = average proportion of WRNs in the non-empty 

100-banks (i.e., the average hit rate for 

non-empty banks),

zi = proportion of the target population included in 

the stratum, and

ti = proportion of 100-banks in the stratum that 

contain no WRNs.

The average hit rate for the frame is h = hiPi

i=1

H

Â

and the proportion of empty 100-banks in the frame is

t = tiPi

i=1

H

Â . In general only the Pi 's will be known

with certainty. Data from a joint research project

involving the Bureau of Labor Statistics and the

University of Michigan were used to provide

approximate values for the parameters hi

 and wi for

the two strata in the example. Values for the

remaining parameters were calculated using the

algebraic relationships ti = 1- hi wi b g and zi = hiPi

h .

The approximations for all of the frame parameters

for the two stratum design are given in Table 1. The

values in the table imply that for the BCR frame

h @.211 and t @.605, which are in close agreement

with those given in Waksberg (1978).

2.3 The Basic Estimation Problem, Sample Designs

and Estimators

We assume the telephone numbers in the i

th

stratum are labeled 1 through Mi and we let

dij = 1 if the j

th nb. in the ith stratum is a WRN,

0 otherwise

R

S

T .

Ni denotes the number of WRNs in the i

th stratum 

N. denotes the number of WRNs in the population

We denote simple random sampling with

replacement (i.e., simple RDD) from the teleph

numbers in the BCR frame as design D0 and stratif

simple random sampling from the BCR frame (

independent simple RDD samples are selected fr

each stratum) as design D1. Under design D0

standard ratio estimator for m is given by Y0 = Y$

0 /

where Y$

0 and N$0 are the usual inflation estimators 

Y. and N. respectively. The estimator Y0

asymptotically unbiased for m and its variance

given by

var Y0 c h @

s2

mh

where m is the sample size and s2 is the populat

variance of the y's. For the design D1 the stand

ratio estimator of m is given by Y1 = Y$

1 / N$1 wh

Y$

1 and N$1 are the standard inflation estimators 

Y. and N. under stratified sampling. The estimato

is also asymptotically unbiased for m and

var Y1 c h @

zi

2

si

2 1+ 1- hi b gl i c h

i=1 mi

hi

H

Â (2

where l i = b g mi - m

2

/ si

2

 and mi , mi , and si

2

 are 

stratum sample sizes, means, and varian

respectively.

2.4 The Cost Model

There are costs associated both with determin

the value of the indicator variable d and the value

the characteristic of interest Y. The cost function 

determining the indicator variable is denoted by C1

with

C1a f d = c1 if d = 1

c0 if d = 0

R

S

T

This model allows for the possibility that the cos

determining that a telephone number is not a W

may be different than determining that a teleph

number is a WRN. In fact, the cost of determin

the status of telephone numbers that are WRNs

usually less. The cost of determining the value of

characteristic Y includes only the additional cost

determining the value of y after the value of d

been determined. Accordingly, let C2 a f ◊,◊ repres

this additional cost, then

a f R0 if d = 0
for both designs . Letting C Db0g and C(D1 ) represent

the total cost of conducting a survey under the two

respective designs it is straightforward to show that

E C Db0g= mc0c1+ g- a 1fhh (2.2)

and

E C Dbg1 = c0 mic1+ g- a f1 hih

i=1

H

 . (2.3)

2.5 Optimal Allocation for Y1

The stratum sample allocation that minimizes

varcY1h for a fixed expected total cost C* (or that

minimizes E C Db1g for a fixed variance V*) is

specified up to a proportionality constant by

mi

zisi

hi

1+b1-higli

1+ g- a f1 hi

F

HG IKJ

1/2

(2.4)

where the proportionality constant is determined by

substitution into the expected cost equation (or the

variance equation, as appropriate). Relative to RDD

sampling, the proportional reduction in variance (cost)

under optimal allocation for fixed cost (variance),

denoted by R Y1 c,Y0h, is approximately

-h

zi

si

hi

c h 1+b g 1-hi li c h 1+ g- a f1 hi

1/2

i=1

LH

N

M O

Q

P

s2c h 1+ g- a f1 h

2

 .(2.5)

2.6 Practical Problems Associated With Optimal

Allocation

The problem of specifying the values for the

parameters in the allocation equations is generic to

optimal allocation schemes. For our particular case

there are three basic types of parameters: frame

related (zi

 and hi), cost related (g and c0 ) and those

specific to the variable of interest (li and si

2

).

Currently, we have a fairly good working knowledge

of the frame related parameters for the two stratum

example and certain other specific stratification

schemes. In Section 4, we will discuss several active

research projects that should further expand our

knowledge in this area.

It is clear that g ‡1 but the actual value can vary

widely. Waksberg (1978) considers values of g

between 2 and 20. Potentially the variable specific

parameters pose the most serious problem. Usually

our knowledge regarding the values of these

parameters is limited and, in the case of multipurpose

Therefore, with caution, we assume that mi = m

si

2 = s2

 for i =1,2,K , H. Optimal allocation

achieved by

mi

zi

hi

c h 1+ g- a f1 hi

-1/2 (2

and the proportional reduction in variance is

R Y1 c h,Y0 @1-h

zi

1+ g- a f1 hi

hi

F

HG IKJ

1/2

i=1

LH

N

M

M

O

Q

P

P

2

c h 1+ g- a f1 h . (2

In the case of the two stratum example, the allocat

specified by (2.6) implies that the allocation relat

to the residual stratum (i.e., m1 / m2 ) is 2.54 w

g = 2 and 1.42 when g =10 . In the first case 

projected proportional reduction in variance 

R =.283 and in the second R =.077 . In fact

follows from (2.7) that as the relative cost 

determining the value of the variable of inte

increases, the relative benefit of optimal allocat

decreases.

The Mitofsky-Waksberg sample design, deno

by D3 , employs two stages of sample selection (i

non-empty 100-banks are selected in the first st

and WRNs are selected in the second stage). T

Mitofsky-Waksberg estimator, denoted by Y3,

unbiased for m . Under "optimal" within 100-b

sample allocation, the reduction in variance relative

simple RDD for the estimator Y3, denoted by R Y3 c,

, is approximately

-

c h 1+ g- a f1 h -t

1/2a f 1-r

1/2

+ raft 1/2 2

1+ g- a f1 h (2

where r is intra-bank correlation. At the natio

level Groves (1977) reports that r @.05 for econom

or social statistics. Using this value of r, toget

with the values of h and t from the two strat

example, the projected proportional reduction 

variance for the Mitofsky-Waksberg procedure

R=.281 when g = 2 and R=.060 when g =10.

The two methodologies appear to prod

essentially identical variance reduction for both val

of the cost ratio. However, too much should not

read into this simple comparison as the projec

reduction for each of the procedures is based 

simplifying assumptions that will not be strictly t

for any application. The only inference intended

that the two procedures appear to highly competit
efficiency of the estimators, not their expectations.

Unfortunately an extremely high price is paid for the

assurance of unbiasedness because sampling from the

residual stratum provides information on only a small

proportion of the population and at a relatively high

cost. If we are willing to settle for an estimate of the

population mean exclusive of those households linked

to telephone numbers in the residual stratum, we can

"truncate" the original frame by eliminating the

residual stratum and selecting a stratified RDD sample

from the remaining telephone numbers. For the two

stratum example the "truncated frame" would consist

only of those telephone numbers in the first stratum.

The hit rate for the sample from the truncated frame

would be .521, in contrast to a hit rate of .211 for the

entire frame. However, only about 94% of the target

population would remain in scope.

In what follows we assume that the truncated

frame is simply the original BCR frame less the

residual stratum. Accordingly, for the truncated

frame h* =ch -PKhK hb1-PK g is the hit rate, the

proportion of empty 100-banks is

t * = t -PKt b K gb1-PK g and m* = m-z b Km K g1-z b K g

is the population mean. Let design D4 be stratified

simple random sampling from the truncated frame,

and Y4 the standard ratio estimator of the population

mean. The estimator Y4 is asymptotically unbiased

for m*

, and, in general, it is biased for m . The

(asymptotic) bias is given by

BchY4 = m* -m =

zK bm-m K g

1-z b gK

 . (3.1)

In most practical circumstances the bias tends to zero

monotonically as the proportion of the target

population in the residual stratum becomes small,

although, as indicated by (3.1), this is not necessarily

the case. In any event, since the value of m-m K is

never known, an upper limit on the proportion of the

population in the residual stratum is usually the key

specification to be determined when considering the

use of a truncated frame. For the two stratum

example approximately 6% of the target population is

excluded from the sampling frame and, in almost all

cases, this would not be tolerable for Federal

agencies.

The equations for cost, variance, allocation, and

proportional reduction in variance (or cost) are

essentially the same as those presented in Section 2.

In fact, the only modifications required for equation

R Y4 c h,Y0 =1-

h 1+h* c ag -1fh

h*

c1+hag -1fh

 . (3

Thus for the two stratum design, the proportio

reduction in variance (cost) is approximately .49 w

g = 2 and .21 when g =10. In both cases 

reduction is substantially greater than achieved by 

two methods in the previous section. However, nea

6% of the population is not covered by the frame.

In an attempt to retain the relative efficiency

truncation while reducing the magnitude of 

coverage problem, BLS and the University 

Michigan are investigating several alternat

stratification plans in an effort to reduce 

proportion of the population in the residual stratu

One promising approach calls for the partitioning

the residual stratum in the two stratum example

form a new residual stratum which consists 

telephone numbers in 100-banks thought to 

primarily assigned to commercial establishments

not yet activated for either residential or commer

use. Estimated frame parameters for the result

three stratum design are given in Table 2.

These data were used to compute the projec

proportional reduction in variance for both the th

stratum design and the truncated three stratum desi

These results, together with a summary of the res

for the two stratum designs and the Mitofs

Waksberg design, are presented in Table 3. Tabl

also includes the projected reduction in variance fo

cost ratio of 20.

It appears that the proposed partitioning strat

was reasonably successful as the percent of 

population out of scope was reduced from nearly 

to approximately 2%. The projected proportio

reduction in variance for the truncated three strat

design is approximately .41 when g = 2 and .16 w

g =10. From an efficiency point of view, it occup

the middle ground between the highly effici

truncated two stratum design and unbiased desig

Of course the issue to be faced when considering s

a design is the coverage problem. For any particu

application the risk inherent in sampling from a fra

that does not include all of the target population m

be weighed against the potential gain in efficien

As expected, the standard three stratum design

slightly more efficient than the two stratum desi

However, the increase in efficiency is so small tha

is doubtful that the added cost of partitioning the B

frame into an additional stratum is justified except
sampling in others. The motivation for this type of

design is based on the following two considerations:

(a) Mitofsky-Waksberg sampling tends to be 

"administratively complex", and if the gain in 

efficiency is small, simple RDD is preferred.

(b) If the proportion of empty banks in a stratum

is "small" then Mitofsky-Waksberg sampling offers 

little, if any, increase in efficiency.

Thus, we propose to utilize simple RDD sampling in

strata with a "small" proportion of empty hundred

banks and Mitofsky-Waksberg sampling in the

remaining strata. A complete discussion of this topic

may be found in Casady and Lepkowski (1991).

4. CONCLUDING REMARKS

The strengths of the Mitofsky-Waksberg

technique for generating telephone samples are clear:

high hit rates in the second stage of selection, an

efficient method for screening out empty banks of

telephone numbers, and a conceptually ingenious

approach to sample generation. It is a remarkable

testimony to the strength of the technique that after

many years it is still considered to be the standard

method of random digit dialing with few serious

competitors . The weaknesses of the technique (first

stage screening and replacement of non-residential

numbers during the data collection) do not, on the

surface, seem to be important relative to the general

strength of the technique. However, these features

can cause substantial difficulty, especially in short

time period telephone survey operations.

Several alternatives are considered. In the two

stratum methods, telephone numbers are divided into

two groups, the high density stratum consisting of all

telephone numbers in 100-banks containing listed

numbers and the low density stratum consisting of all

remaining telephone numbers. In the three stratum

methods, the "all other numbers" or original low

density stratum is further subdivided, on the basis of

auxiliary data, into two strata. One of the new strata

is expected to contain nearly all residential numbers in

the original low density stratum and the other is now

the (new) low density stratum. For both the two and

three stratum design, two general alternatives are

considered: (1) selecting simple random samples from

all strata except the low density stratum frame where

the Mitofsky-Waksberg method is used and (2)

selecting simple random samples from all strata

except the low density stratum which is not sampled

at all.

residential numbers at the second stage, since the o

telephone numbers that must be dialed in the h

density stratum are those that are generated at 

beginning of the study.

Even for low cost ratios, the two and three strat

designs are as efficient as the Mitofsky-Waksb

approach. When numbers can be dropped from 

low density stratum, these alternatives are much m

efficient, at the price of unknown bias due 

excluding part of the target population. When c

ratios are high, the two and three stratum approac

are clearly superior.

A critical issue is the magnitude of the b

introduced by dropping the low density stratum. E

though the proportion of the population in the strat

is small, the magnitude of the bias may be relativ

large for some characteristics and for some subgro

of the population. Further empirical investigati

are necessary.

The cost of the auxiliary list frame is 

addressed in this investigation because the fra

information used in stratification was derived from

specialized research file. Further investigation

needed into this cost as it must be considered in 

practical application.

In order to improve the hit rates in the h

density stratum, smaller banks of numbers can

used. In another investigation we have found that 

banks will have hit rates in the neighborhood of 

compared to the .52 reported here for 100-ban

Working with 10-banks substantially increases 

size of files and processing operations used 

generate samples. And, the cost of a 10-bank fram

likely to be much higher than the 100-bank frame.

Clearly the results presented here are insuffici

to draw final conclusions about the overall value

these alternative designs. Further cost data 

empirical evidence on the size of the bias caused

eliminating the numbers from the low density strat

is needed before a final conclusion can be reached.

5. References

Brunner,J. A., and Brunner, G. A., "Are Voluntaril

Unlisted Telephone Subscribers Really Different?

Journal of Marketing Research, Vol. 8, February

1971, pp.121-124.

Casady, R. J. and Lepkowski, J. M., "Stratified 

Telephone Designs," unpublished report of the U

Bureau of Labor Statistics, Washington D.C.

Groves, R. M. ,"An Empirical Comparison of Two
Table 1. Approximate values of the frame parameters for a two stratum design based on

the BCR frame and the Donnelley list frame.

Proportion

of Frame

(Pi)

Proportion

of

Population

(zi)

Hit Rate

(hi)

Proportion

of Empty

100-Banks

(ti)

Hit Rate Within

Non-empty Banks

(wi)

1 .3804 .9402 .5210 .0300 .5371

2 .6196 .0598 .0204 .9584 .4900

Table 2. Estimated frame parameters for a proposed three stratum design based on the

BCR frame and the Donnelley list frame.

Stratum

Proportion

of Frame

(Pi)

Proportion

of

Population

(zi)

Hit Rate

(hi)

Proportion

of Empty

100-Banks

(ti)

Hit Rate Within

Non-empty Banks

(wi)

1 .3804 .9402 .5210 .0300 .5371

2 .2000 .0399 .0420 .9143 .4900

3 .4196 .0199 .0100 .9796 .4900

Table 3. Projected proportional reduction in variance (or cost) relative to simple RDD

sampling for five alternative telephone sample designs.

Sample Design Proportional Reduction in Variance or

Cost

Proportion of

Frame

g = 2 g =10 g = 20 Not in Scope

Two Stratum .2829 .0766 .0320 .0000

Two Stratum

(Truncated)

.4917 .2055 .1189 .0598

Mitofsky-Waksberg .2811 .0597 .0135 .0000

Three Stratum .3001 .0866 .0389 .0000

Three Stratum

(Truncated)

.4095 .1574 .0879 .0199