384 Journal of Official Statistics

Chandhok, P., Weinstein, R., and Gunlicks, C. (1990). Augmenting a Sample to Satisfy
Subpopulation Reliability Requirements. Proceedings of the Section on Survey
Research Methods, American Statistical Association, 696-701.

Cochran, W.G. (1977). Sampling Techniques. New York: John Wiley and Sons.

Ernst, L.R. (1990). Simultaneous Selection of Primary Sampling Units for Two Designs.
Proceedings of the Section on Survey Research Methods, American Statistical Associa-
tion, 688-693.

Goodman, R. and Kish, L. (1950). Controlled Selection — A Technique in Probability
Sampling. Journal of the American Statistical Association, 45, 350-372.

Sampford, MLR. (1967). On Sampling Without Replacement With Unequal Probabilities
of Selection. Biometrika, 54, 499-513.

Received March 1993
Revised December 1996


378 Journal of Official Statistics

between-PSUs-within-strata component, there is a relatively simple relationship between
the within-PSUs component of variance for these two estimators.

The function f behaves as follows: f(R)=1 whenever R =|R], that is, when R is an
integer. Furthermore, as can be shown by means of elementary calculus, within the
interval bounded by any two consecutive integers, f(R) has a single relative optimum, a
relative maximum, at [R]+ 1/2, with

1
AR\IR] + 1)
Thus f(1) = 1; f increases in the interval [1, 3/2], with (3/2) = 9/8 by (5.11); f decreases
in the interval [3/2, 2] with f(2) = 1; f increases in the interval [2, 5/2], with f(5/2) =
25/24; and so forth.

One of the comparisons in the next section will be of V(¥) and V(P’) for the CPS expansion.

FUR + 1/2) = 1+ (5.11)

6. Comparison of Methods for the CPS Expansion

In this section variances for the multiple WLs per stratum method estimator Y are first
compared to variances for three other methods for selecting the D2 sample for the formerly
planned CPS expansion, then vy) is compared to vir). These three other methods are
the independent sample, the independent supplement (both described in Chandhok, Wein-
stein, and Gunlicks 1990), and controlled selection (Ernst 1990). For each of these four
methods for the study to be described, the number of USUs selected from the D, noncer-
tainty PSUs for the D3 design is the same, m. As was the case for the multiple WLs
method, the expansion from D, to D» for the other three methods is presented for the
D, noncertainty PSUs only.

The independent sample method is used as a benchmark for evaluating the variances of
the other methods. For D2, PSUs are stratified for a one-PSU-per-stratum design of size m
USUs, and PSUs are selected to be in the D, sample independently of the D; sample PSUs.
This stratification procedure attempts to optimize an objective function that includes
estimates of both unemployment and civilian labor force, and in the rest of this section
we will refer to a design that uses it as optimal. This is the method that would be used
if the D, sample was selected for a straightforward redesign of a survey which did not
require an intervening D, sample.

The independent supplement method starts with the D, sample. It then restratifies all the
PSUs for a one-PSU-per-stratum design of size m-m* USUs, and selects PSUs from the
new strata independently of the D, sample PSUs. The resulting sample is added to that
of D, to complete D>.

In general, two-dimensional controlled selection (Goodman and Kish 1950; Causey,
Cox, and Ernst 1985) is a technique for selecting sampling units simultaneously satisfying
two criteria of stratification, while preserving the selection probabilities for both stratifica-
tions. In this particular application, the D, design consists of L strata and the D3 design
consists of (say) L’ strata, with both designs one-PSU-per-stratum. The D, stratification
for controlled selection is the same as for the independent sample method. To satisfy
the D> stratification criterion, the technique selects L’ PSUs, with one PSU from each
of the D2 strata. To satisfy the D, stratification criterion, the method ensures that L of these
L’ PSUs will be from separate D, strata and will constitute the D, sample PSUs, while the


Weidman, Ernst: Multiple Workloads per Stratum Designs 369

Example. Suppose a state has L = 4 strata and m* = 160 USUs in sample for D,. If
m = 368 USUs are needed for D>, then the smallest integral number of expected average
WLs of size 160/4 = 40 that will achieve this size is n = |(4)(368/160)] + 1 = 10. Then
R = 10/4 =2.5 is the average number of WLs per stratum, and 10-(4)|2.5] = 2 strata
will be chosen to receive [2.5] + 1 = 3 WLs. The remaining two will receive [2.5] = 2.

2.2. Allocation of workloads to PSUs within a stratum

How should the n;, WLs be allocated to the PSUs in stratum h? Let pj; be the probability of
selection of the ith PSU in the Ath stratum in D, and nj; be the number of WLs allocated to
this PSU in D>. Typically, when n;, PSUs are selected without replacement, each PSU is
selected either once with expected probability ,p,,; or not selected (Cochran 1977;
Sampford 1967). Variances are smaller for these procedures than if a PSU is allowed to
be selected more than once. However, the cited authors also assume that all pa; < 1/ny.
When this is not the case, we generalize the restriction on the number of times a PSU
can be selected to conditions (ii) of Section 1. The multiple WLs approach has the follow-
ing features, where

Tyg = Pn = ln Pail + 1)
We see that

ny; = 1 if PSU hi is in D, (2.2)
which is equivalent to design property (1) of the previous section;

yg = Mn Ppal OF Mae = Lp Pril + 1 (2.3)

which is condition (ii)

Thi = "Pri — AnPril (2.4)
Ni.
Sone = Me (2.5)
i=l

where N;, = number of PSUs in stratum h, which simply says that the total number of WLs
in stratum A is nj.
The combination of (2.3) and (2.4) imply condition (i), that is,

E(npi\th) = Pri (2.6)

If n, = 1, let the Dy sample PSU for stratum h be the D, sample PSU, since this allo-
cation satisfies (2.2)—(2.5). For n, = 2 a procedure for allocating WLs that satisfies (2.2)—
(2.5) is presented in Section 3.

2.3. Allocation of USUs within a PSU and total sample size

Let M, M,, and M,,; be the number of USUs in the total population, the Ath stratum, and the
hith PSU, respectively. We assume that p),;=M;;/M), and that the USUs within sample
PSUs are selected in a manner such that each USU in the population has the same
probability of selection, that is m/M. Consequently, the WL size for Dz in the Ath stratum,


382 Journal of Official Statistics

Table 6. 1970 ratios of total variances for other options to the independent sample

State Unemployed Civilian labor force

cs IS MW AL cs Is MW AL

Alabama 1.001 1.019 1.016 1.051 1.000 1.028 1.064 = 1.025
Arizona 1.000 1.003 1.014 1.043 0.998 1.069 1.016 1.018
Arkansas 1.001 1.032 0.991 1.020 0.999 1.036 0.957 0.976
Colorado 1.000 1.031 1.024 1.054 1.000 1.212 1.065 1.059
Georgia 1.000 1.010 1.002 1.002 1.000 1.024 0.990 0.990
Idaho 1.002 1.259 1.041 1.048 1.003 1.155 1.045 1.002
Indiana 0.999 1.028 1.033 1.127 1.018 1.112 1.003 1.090
Towa 0.998 1.030 1.001 1.011 0.997 1.055 1.001 1.005
Kansas 1.000 1.026 1.009 1.034 0.996 1.035 0.993 1.009
Kentucky 1.001 1.020 1.028 1.062 0,997 1.050 1.040 1.044
Louisiana 1.000 1.007 1.012 1.041 0.999 1.054 1.037 1.045
Maryland 1.000 1.020 1.000 1.001 1.000 1.196 0.952 0.952
Minnesota 1.002 1.011 1.014 1.029 0.996 1.057 1.013 1.011
Mississippi 0.999 1.047 1.003 1.012 1.001 1.132 0.980 0.981
Missouri 1.000 1.026 1.010 1.059 1.005 1.120 0.979 1.018
Montana 0.997 1.032 1.043 1.036 1.009 1.261 0.979 0.990
Nebraska 1.000 1.020 1.006 1.020 1.000 1.078 0,983 0.998
Nevada 1.000 1.035 0.998 1.040 0.992 1.855 0.960 0.994

New Mexico 0.999 1.013 1.040 1.056 1.011 1.084 1.053 1.049
North Dakota 0.999 1.071 1.011 1.011 0.986 1.106 0.978 0.978
Oklahoma 1.000 1.012 1.009 1.009 0.993 1.117 0.995 0.995
Oregon 0.999 1.015 1.021 1.054 0.999 1.216 0.981 1.005
South Carolina 1.003 1.054 0.993 1.016 1.002 1.135 1.009 1.023
South Dakota 0.996 1.054 1.020 1.020 0.997 1.092 0.977 0.977

Tennessee 1.001 1.047 1.005 1.026 1.000 1.060 0.993 1.005
Utah 1.000 1.019 1.002 1.039 0.995 1.128 0.980 0.987
Virginia 0.999 1.031 0.998 0.998 1.018 1.112 0.970 0.970

Washington 0.999 1.034 1.012 1.051 1.008 1.148 1.015 1.030
West Virginia 1.000 1.002 1.012 1.041 1.001 0.994 1.002 1.032

Wisconsin 0.999 1.028 1.015 1.025 0.999 1.086 0.989 0,988
Wyoming 0.999 1.014 1.012 1.053 1.000 1.212 1.067 1.050
Mean 1,000 1.034 1.013 1.035 1.001 1.130 1.002 1.010

CS =Controlled Selection

IS = Independent Supplement

MW =Multiple Workloads (7)

AL = Alternative MW Estimator (7)

Finally, we compare the variances of the two multiple WL estimators ¥ and Y’. In 1980
Y’ has smaller between-PSUs variances in 23 states for unemployed and all states for CLF,
and the means are also less for 2’. The 1970 results only have two unemployed cases for
which ?’ has slightly larger between-PSU variances. (As the proportion of between-PSUs
variance due to between-strata variance decreases for a state (not shown), the ratio of f to
¥’ between-PSUs variances increases, as suggested by the absence of a between-strata
component for ¥’) On the other hand, total variances are smaller in most of these cases
for ¥. This is because the between-PSUs component makes up a small fraction of the total


374 Journal of Official Statistics

These three terms are, respectively, the between-strata, between-PSUs-within-strata, and
within-PSUs components of variance.
4.1, Between-strata variance

Let Y, = rN 1Ypi, the characteristic total for stratum h. Let S denote the set of strata h for
which nj, = IRL 1 and let € =n — URI, the number of such strata. If € = 0, then the
between-strata component of variance is 0. Otherwise, by (2.6), (4.4), and the fact that
LRIY, is a constant, we have

Wiss ys exo] =v, (oS ym Th vs) = Zz Y (> mts)

1 i=1 1 i=1

=3 ¥(S5e - yn) = (5) Y% (=%) 7)

hES

Since the set S is a simple random sample of £ out of L strata, (4.7) is (€/R)’ times the
variance of the mean from a simple random sample. Using Theorem 2.2 of Cochran
(1977), we obtain that (4.7) reduces to

“L-oe [e yy
rar (7) “

4.2. Between-PSUs-within-strata variance

To evaluate the second bracketed term in (4.6), first let p*(k) = P(nj, = k) and observe that
for all h

PUR) =1+1R|-R, p*UR]+1) =R-[RI (4.9)
and p*(k) = 0 for all other k. Consequently
L Ny WRH1 Nn
Ey Is Vp (= outs) |= S Sov (= An:Y pal = ‘)e (k) (4.10)
hal i=l h=1 kar) i

To evaluate the variance of the term in parentheses in (4.10), let mj; = mai — Ltn Pri] and

expand
ny = =1)= 3

Ny
Vv (s OY pilin, = ‘)=4 sv (Som ny —
ia
1[% Yh
=| SS Vveniln, =

Ne Me Yu Yi
+ Ly Cov( nis Miglin, = ky . (41D
i=l j=l Phi Phy

ij


Weidman, Ernst: Multiple Workloads per Stratum Designs 371

Observe that by (3.2), nj, of the N;, elements of s), are in T,,; for each s;,. That is, each sp,
corresponds to a set of nj, PSUs out of the N;, PSUs that receive the larger of the two pos-
sible numbers of WLs for the PSUs in D>. Consequently, the problem of obtaining a set of
P(sp)'s which satisfy (3.3) is equivalent to the problem of finding joint probabilities of
selecting without replacement ni, of the stratum h PSUs, for which the probability of selec-
tion of the PSU iis m,;, i= 1,....Np. If, = | this can be accomplished by simply select-
ing a single PSU from the stratum with probability TM;, i= 1,....Ny. If my = 2 then the
joint selection probabilities for the sets of n, PSUs can be obtained by using any of a num-
ber of procedures for sampling without replacement with probability proportional to size.
The best known of these are the procedures of Brewer-Durbin (Cochran 1977) for nj, =2
and its generalization by Sampford (1967) for nj, = 3.

We have thus shown how to obtain unconditional probabilities P(s;,) for selecting the
number of D2 WLs from each PSU in stratum h. We then let the conditional probability that
sp is selected for D> given that the hith PSU was in the D, sample, denoted PCs, li ) be simply

ni P(Sh)

Posi) ~ MhPhi

(3.4)
where np; is the ith element of s,. Note that (3.4) satisfies (2.2), since if mj,j = 0 then
P(s;|i) = 0.

We show that the conditional probabilities (3.4) yield P(s;) as the unconditional
probability for selecting each s,. This follows since

Ni

Ny
So pnPli) = ro = Pls)
oi

i=l

Consequently, by (3.3), these conditional selection probabilities result in unconditional
probabilities satisfying (2.3)—(2.5).

Finally, we must demonstrate that for each i= 1, ...,N,, the conditional probabilities
sum to 1, that is L,,P(s,|i) = 1, or equivalently, by (3.4)

So meP On) = Pris = Aye Ni (3.5)
Sh

To obtain (3.5) let

Thi = (5n 2 ai = bn Pri lds
and observe that by (3.3)
Pn) = Un pail + D> Plow) +b pail S> Plow)
5h

itu ety
= (Le Pail + DC Pri — Ltn Pail) + berm Pail — Mn Pri + Ltn Pail) = Mn Pri

Example. This example illustrates the procedure just described. Consider a stratum h
with N,, = 4 and pps given in Table 1. Let n, = 5. Then the values of nj, pj; and mp; are
as given in Table 1.

Furthermore, nj, = 2 and we can, therefore, use the Brewer-Durbin procedure to obtain”
pairwise joint probabilities 1,;;, i,j = 1,...,4, i<j, where mij denotes the probability


370 Journal of Official Statistics

denoted m,, is
=m,

~ MR

(Unless m,, is already an integer, we round it up to the next integer when defining WL size.)
Within each PSU Ai in the Ath stratum for which n,; > 0, j,;m, USUs are selected with
equal probability.

For the three-stage sampling procedure just outlined, the expected number of D2 sample
USUs is m, but the actual number selected is a random variable that depends on the first
stage of sampling. This is because, by (2.7), the WL size is not the same for each stratum.

Example. Here we attach sizes to the strata of the previous example and calculate the
range in total sample size that is possible due to the random selection of strata to receive
additional WLs in D>. Recall that m= 368 and R=2.5. If the M,, in the four strata are
88,000, 80,000, 78,000 and 74,000, then M = 320,000 and the m,s are 41, 37, 36, and
35. The number of USUs in Dy can range from (3)(35 + 36) + (2)(37 + 41) = 369 to
(3)(37 + 41) + (2)(35 + 36) = 376. (Note that the average WL size of 37.25 here is
smaller than the expected average WL size of 40 in the previous example. This is because
rounding up the number of WLs to the nearest integer allows a slight decrease in WL size
to attain a specified number of USUs in D3.)

An alternative three-stage procedure for which the number of sample USUs would
always be m would begin by selecting the number of WLs, n,, assigned to the hth stratum
so that E(n,) = nM,/M, that is proportional to size. For every sample, n;, would also be
within 1 of E(n,). The second and third stages would be selected as described above,
except now the WL size in each stratum would be m/n. This alternative approach will
not be discussed further in this article.

m, (2.7)

3. Selecting PSUs to Receive Additional Workloads

In this section our principal goal is to obtain probabilities for the sets of WLs selected from
stratum h for the D2 design, conditioned on the D, sample PSU from this stratum. These
conditional probabilities will satisfy (2.2) and yield unconditional selection probabilities
for the D, WLs which satisfy (2.3)—(2.5).

In actuality, we work backwards. We first demonstrate how to obtain unconditional
selection probabilities for the sets of D. WLs satisfying (2.3)—(2.5) and then specify
conditional selection probabilities which yield these unconditional selection probabilities
and satisfy (2.2). To this end, let s;, denote a random vector of length N;, which specifies the
number of D2 WLs to be selected from each PSU in stratum h. That is, s, is of the form
Sh = (My1, Mp2, ---sMpy,)» Where the elements in s;, satisfy (2.3) and (2.5). Let

Thi = {54 Me = Wn Pra + 1}, i=1,...,N, G.1)
My,

nh, = My — Sod, Pri (3.2)
fai

We seek a set of probabilities P(s;,) for the s,s which satisfy (2.4), that is for which

Te = D> PO) = MPa —VaPrel T= Dy Nn (3.3)
SnET i


Weidman, Ernst: Multiple Workloads per Stratum Designs 383

Table 7. 1970 ratios of between PSU variances for other options to the independent sample

State Unemployed Civilian labor force

cs Is MW AL cs IS MW AL

Alabama 1.12 3.36 3.00 2.62 1.03 2.87 5.33 1.26
Arizona 1.07 4.21 18.63 16.44 0.89 4.68 1.84 0.34
Arkansas 1.06 2.67 0.53 0.32 0.99 1.57 0.34 0.14
Colorado 1.01 432 3.59 2.57 1.00 5.31 2.33, 1.42
Georgia 1.04 2.42 1.25 1.25 1.02 1,91 0.64 0.64
Idaho 1.10 13.47 2.95 1.67 1.06 464 2.05 0.26
Indiana 0.89 4.00 4.52 2.30 1.47 3.90 1.08 0.44
Towa 0.78 3.63 1.10 0.97 0.83 4.29 1.09 0.66
Kansas 0.95 3.49 1.90 1.48 0.92 171 0.85 0.64
Kentucky 1.08 3.35. 4.22 4.00 0.84 3.67 3.13 1.43
Louisiana 0.95 2.65 3.89 3.30 0.97 2.63 2.14 1.46
Maryland 1.00 4.62 1.09 1.09 1.00 4.92 0.04 0.04
Minnesota 1.12 1.51 1.69 1.44 0.87 2.75 1.39 0.73
Mississippi 0.93 4.84 1.28 1.08 1.03 3.70 0.59 0.39
Missouri 0.95 3.99 2.17 2.16 1.09 3.25 0.61 0.45
Montana 0.88 2.16 2.54 1.70 1.19 6.24 0.58 0.49
Nebraska 0.99 2.83 1.52 1.36 1.00 2.78 0.62 0.59
Nevada 1.02 5.81 0.78 0.82 0.86 15.02 035 0.25

New Mexico 0.91 3.06 7.26 3.27 1.41 4.13 2.97 1.32
North Dakota 0.91 6.34 1.85 1.85 0.72 3.09 0.57 0.57
Oklahoma 1.02 2.72 2.27 2.27 0.83 3.86 0.88 0.88
Oregon 0.89 2.25 2.72 2.15 0.98 5.40 0.61 0.32
South Carolina 1.17 3.64 0.68 0.41 1.10 9.67 1.56 0.67
South Dakota 0.90 2.53 1.56 1.56 0.95 2.50 0.63 0.63

Tennessee 1.10 8.16 1.77 1.81 1.00 3.01 0.76 0.50
Utah 0.97 2.66 1.18 1.18 0.94 2.67 0.73 0.39
Virginia 0.95 2.87 0.87 0.87 1.33 3.07 044 0.44

Washington 0.94 3.29 1.78 1.68 1.25 5.86 1.48 0.66
West Virginia 0.93 160 4.23 2.05 1.12 0.37 1.22 0.52

Wisconsin 0.96 2.17 1.62 1.38 0.98 2.94 0.74 0.37
Wyoming 0.67 5.69 5.04 5.03 1.03 35.91 12.08 2.60
Mean 0.98 3.88 2.89 2.33, 1.02 5.11 1.60 0.69

CS = Controlled Selection

IS = Independent Supplement

MW =Multiple Workloads (¥)

AL= Alternative MW Estimator (?')

variance (< .07) for every one of these estimates and f(R) > 1 except when R is an integer.
In cases when the between-PSUs component is a larger proportion of the total variance, we
would expect to see the comparisons of ¥ and ?’ be more mixed, moving toward favoring
¥’ more often as the proportion increases.

7. 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.


Weidman, Ernst: Multiple Workloads per Stratum Designs 373

Recall from Section 2 that the final stage of sampling is carried out so that all USUs
have equal overall probability of selection. Irrespective of how this is done, the unbiased
Horvitz-Thompson estimator ¥ of Y is given by

(4.1)

where ypj is the total value for characteristic Y for a sample USU from PSU hi, and the
summation is over all sample USUs. Some y,,;s may possibly appear more than once in
this summation if with replacement sampling is used at the final stage. Such an estimator
is called self-weighting because of the constant coefficient or ‘“‘weight’’ M/m for all USUs
in (4.1). We proceed to develop a formula for vy), the variance of the estimator /. There
will be three terms in V(Y), reflecting the three stages of sampling.

We first obtain an alternative expression for Y. Let

Fae = Mi SYnaglrya tm) if ry > 0
J

=0if my =0 (4.2)

Given n,; > 0, Pui is an unbiased estimator of the total, denoted Y;,;, for PSU hi. Then
combining (2.7), Os) an and a we obtain

(4.3)

where
MM Mri
RM RP hi

ny (44)

is a random variable whose value for each sample is determined by the first two sampling
stages.
The variance of ¥ can then be written in the form

. LOM . LON, -
V(¥) = ViEDEs (= Sots) + E\V2E3 (Sdn)

hl i=l hal i=1

LM,
+ E\E,V3 (Sd) (4.5)
heal i=t

The subscripts on the expectations denote the three stages of sampling. Since whenever
nyj > 0, E3(Yui) = Yair then
LM,

vy y y Ex(ay)¥yi| + Ei bP Vy (= ay %s)|
LM, .
+E, EX aceincwn, (or

hel i=l

V(¥) =



Journal of Official Statistics, Vol. 12, No. 4, 1996, pp. 365-384

Multiple Workloads per Stratum Designs

Lynn Weidman! and Lawrence R. Ernst’

This article introduces an approach to expanding a stratified sample design, D,, with one
primary sampling unit (PSU) selected per stratum to a larger design, D2. Define a workload
(WL) to be the sample size in a given stratum in D,. The three-stage approach selects the
number of WLs for each stratum, the PSUs to receive the additional WLs in each stratum,
and the ultimate sampling units. Procedures are given for selecting PSUs in the key second
stage, satisfying the following conditions when a stratum in D> is to have s=2 WLs: (i) the
expected number of WLs in a PSU is s times the probability that it was selected to get the
single WL in Dj; and (ii) the actual number of WLs assigned is within one of the expected
number. These conditions are a generalization of probability proportional to size, without
replacement sampling. The properties and variances of this approach are compared to those
from three alternative expansion procedures via application to a proposed, but since cancelled,
expansion of the Current Population Survey.

Key words: Stratified sample design; PSU selection; workload; variance decomposition.

1. Introduction

The sample expansion procedure presented here was motivated by a planned (but since
cancelled) expansion of the U.S. Census Bureau’s Current Population Survey (CPS) that
was to be selected in two phases. Phase 1 would be the traditional sample redesign of
the CPS based on recently available decennial census data. This design must meet monthly
variance requirements on estimates of the number of persons unemployed for the nation,
the eleven largest states, New York City and Los Angeles. The remaining states have
annual variance requirements. (Denote this first phase design as D,.) At a later date, addi-
tional sample would be selected from these remaining states in phase 2 to meet monthly
state variance requirements. (Denote this second phase design as D>.) Because of the
expenses incurred in recruiting and training new interviewers, and the time required to
develop their skills, it is desirable to maintain as many D, sample primary sampling units
(PSUs) as possible for D2. (For this reason, when redesigning the U.S. Census Bureau’s
major household surveys each decade, procedures to maximize overlap of PSUs between
the old and new designs are used when feasible.) Although the CPS application was the

' Lynn Weidman is Principal Researcher, Continuous Measurement Staff, U.S. Bureau of the Census, Washing-
ton, D.C. 20233, U.S.A.

2 Lawrence R. Ernst is Chief, Research Group, Office of Compensation and Working Conditions, U.S. Bureau of
Labor Statistics, Washington, D.C. 20212, U.S.A. and was formerly Assistant Chief, Statistical Research
Division, U.S. Bureau of the Census, Washington, DC 20233, U.S.A.

‘The views expressed in this article are attributable to the authors and do not necessarily reflect those of the U.S:
Bureau of the Census or the U.S. Bureau of Labor Statistics. The authors would like to thank the referee and the
associate editor for their constructive comments.

© Statistics Sweden


376 Journal of Official Statistics

which we combine with (2.1) to conclude

LM > . M LOM,
> DAG vi =F Mn Si (4.19)

hal i= hal i=1

E

Finally, if the Sai are the same for all Ai, with common value denoted S3, then by summing
(4.19) over all hi we obtain that the within-PSUs variance is approximately M ? S3/m. This
is approximately the sampling variance for the standard estimator of population total
from a simple random sample with replacement, for a sample of size m selected from
a population of size M, for M large, with variance S}. Similar assumptions lead to the
same approximate within-PSUs variance for the other options investigated for the two-
phase sampling application, a result which will be used in some of the comparisons in
Section 6.

5. An Alternative Estimator of Y

The estimator Y defined by (4.1) was selected because it is an unbiased estimator of Y and
because it assigns each USU the same coefficient M/m. However, for the Dz sample
design, Y has the disadvantage that it results in a non-zero between-strata component of
variance, which makes variance estimation much more complex. In this section we present
an alternative unbiased estimator {” of Y for which the between-strata variance is 0. We
will also obtain expressions for the between-PSUs-within-strata component and the
within-PSUs component of V(¥’), and compare the within-PSUs components of V(¥’)
and V(?’).
Let

L

v= Mi y* Syn (5.1)
7

fat MAM GI

Since, by (2.7)

M, _ MR
NyMy, mn),

(5.2)

Y’ assigns weights to the USUs which are a function of the number of D> WLs, n;, in each
stratum. Also let

Raj
ahi = <a (5.3)
th
Then by (4.2), (5.1), (5.2), and (5.3)
a LM “
Hav alihs (5.4)
h=1 i=l

and V(Y’) can be written in the form (4.6) with a,; replaced by a}.
Furthermore, by (2.6), (4.4), and (5.3)

R
Ex(ajy) = 7 F2tani) = 1
th


Weidman, Ernst: Multiple Workloads per Stratum Designs 375

where the substitution of nj,; for n,; is justified by the fact that conditional on n, these two
variables differ by a constant.

To evaluate V(n);lnq =k) and Cov(nj;, njjlng =k), first observe that nj; is a 0-1
variable and that n},; = 1 if and only if mj = [np pyi|-+ 1. We then let

Tpi(k) = Ply = Uy = K) = kPa — lk Pail (4.12)

myij(k) = Ply = 1, mhyj = Un, = © (4.13)

where the last equation in (4.12) follows from (2.4). This is equivalent to the notation used
in Section 3, except now we have converted 71,;, 7,;; to functions to indicate their depend-
ence on the value of n,. Furthermore, to compute 7),;;(k), observe that by (3.2), if m, =k
then nj, =k — ENA Lk ppil. Inj, = 1, then pik) = 0 for all i,j, while if nj, = 2 then myij(k)
depends on the sampling method employed to select the number of WLs in stratum h. For
example, if nj, = 2 and the Brewer-Durbin method is used, then the 7,;(k) are the joint
Probabilities for selecting pairs of units given in Cochran (1977, p. 262), while if
nj, = 3 and Sampford’s (1967) method is used then the ,,;(k) are the joint probabilities
for pairs given in that reference.
From (4.12) and (4.13) we have, since nj, is a 0-1 variable, that

Vnjalay = 1) = Elopal ta: =) — EQ4a| Mag = OP = tue) — THAW (4.14)
and
Cov(nyis Myjlty = K) = Elnpintyjlty = K) = myilk) — mrilk) my(k). (4.15)

Finally, we combine (4.10), (4.11), (4.14), and (4.15) to conclude

L Nn LURHI Ny
E [is Vp (Soon) => 1 [Sou - mi) Mi
A= i=l

h=l k=|R}

Nin Nn

+> » (api (K) — tril my (K)) ny p(k) 4.16)
At i

4.3. Within-PSUs variance

Finally, we evaluate the third bracketed term in (4.6) under the assumption that all sample
WLs within a PSU are selected by simple random sampling, either with replacement for
large M;,; or without replacement with a negligible finite population correction factor.
Appropriate modifications are necessary for other within-PSU selection procedures.

From (4.2) it follows that if n,,; > 0, then
MiiS5ui

V3(Pni) = — tty
!

(4.17)
where S},; is the population variance of the y,;; in PSU hi.
Then by (4.4), (2.6), (2.7), and the relation pj = Mpi/M),
End Mi SSur_ MMi Sri 4M Mi Sh

EqlazV3(Pu)] = = = 4.18)
OVS) Rr Rom, Rm (4.18)



372 Journal of Official Statistics

Table 1. Phir My Pris And Tj

i

1 2 3 4

Phi A8 24 -16 12
Mn Phi 24 1.2 8 6
Thi 4 2 8 6
Table 2. P(s)

ij

1,2 1,3 1,4 2,3 24 3,4
Sh (3,2,0,0) (3,1,1,0) — (3,1,0,1) (2,2,1,0) — (2,2,0,1)— (2,1,1,1)
Tig = P(Sp) _.0277 -2535 1188 -1188 0535 A277

Table 3. P(s,|i)

Sh

i (3,2,0,0) (3,1,1,0) (3,1,0,1) (2,2,1,0) (2,2,0,1) (2,1,1,1)

1 .0346 3169 1485 .0990 0446 3564
2 0462 2113 0990 1980 0892 3564
3 0000 3169 0000 1485 -0000 5346
4 -0000 -0000 1980 0000 0892 -7128

that PSUs hi and hj are the two PSUs that receive the maximum number of D, WLs. This
results in the probabilities P(s;,) given in Table 2.

Then, from Table 2 and (3.4), the conditional probabilities P(s;, ja given in Table 3 are
obtained.

Remark, We have demonstrated how we can select any additional number of WLs for
Dy, in a stratum satisfying (2.2)—(2.5) when initially one PSU had been selected in D,. A
natural question is whether an expansion to a D2 design satisfying (2.2)—(2.5) can also
always be obtained if the D, design was other than a one-PSU-per-stratum design. The follow-
ing simple example illustrates that this is not always possible. Consider a two-PSUs-per-
stratum without replacement design, D,, in which N;, = 4 for stratum h, with pj; = .45,
Pir = 40, Pr3=-10, Pra =.05. If nj, =3 for D2, then nj, pay = 1.35, nj, py = 1.20 and
hence PSUs hl and h2 must receive at least 1 WL in D2. However, if PSUs h3 and h4
had been the D; sample PSUs, then since only 1 additional WL is selected for the D2 design,
it would not be possible for both PSUs h1 and h2 to meet the minimum WL requirement.

4. Variance Decomposition for a Self-Weighting Estimator

In Sections 4 and 5 we develop variance formulae for two unbiased estimators of a
population characteristic total Y, based on the three-stage sampling procedure of this
article. The estimator of this section (Y) is self-weighting and has a between-strata com-
ponent of variance, while in the following section we eliminate this variance Component
by using an alternative estimator (’) that is not self-weighting.


368 Journal of Official Statistics

at the same time as the D, sample PSUs, but can be selected at a later date; (b) all D; PSUs
are retained in D2; and (c) only large PSUs can receive more than one WL.

2. Expanding an Existing Design

The presentation in this and the next three sections considers the expansion from D, to Dy
for stratified PSUs only. For each non-stratified PSU the expansion is obtained by selecting
an appropriate number of USUs to supplement the D, sample.

Expansion from D, to D2 by the multiple WLs approach usually involves a three-stage
sampling process. First, the total number of WLs are allocated among the strata. Then the
WLs in each stratum are allocated among its PSUs. Finally, USUs which make up the WLs
are selected within the designated PSUs. We now proceed to describe each of these stages.
The details of the multiple WLs approach to PSU selection will be given in Section 3.

2.1. Allocation of workloads to strata

Recall from Section 1 that in most states the Current Population Survey is a multi-stage
design with each USU having the same overall probability of selection. The PSUs are
stratified so that the strata are of approximately the same size and, as a result, the number
of USUs in sample in each stratum will be close to a specified target WL. This target is the
number of USUs that can be most efficiently handled by a single interviewer during an
interviewing period. In order to minimize the number of interviewers needed for Dy
and assure that each is able to efficiently handle his/her WL, a large increase in total
sample size should be made by adding an integer number of WLs to each stratum. This
can only be accomplished, while simultaneously adding the minimal number of WLs
required overall for D2, by allowing the number of WLs to vary among strata.

How should WLs be allocated to each stratum? One fairly straightforward method is to
let the number of WLs vary by at most one among the strata, and select a simple random
sample of strata to receive the higher number. (This sampling of strata in combination with
a self-weighting estimator has a between-strata component of variance. However, by
limiting the variation in the number of WLs each stratum can be allocated, this
between-strata component will be reduced.) We describe this method notationally and
then present a simple example.

Let m” denote the number of. D, sample USUs and L the number of strata, so that m“/L is
the expected average WL size in D>. If m is the desired number of D2 sample USUs, then
the minimal number of expected average WLs needed to attain this sample size is mL/m”.
If this is an integer, then this is the number of D, WLs, denoted by n; otherwise, we round
up the number of D, WLs to the next integer |mL/m*] + 1 =n. (For any number x, let [x]
denote the greatest integer not exceeding x.) Now R = n/L is the average number of WLs
per stratum in D5, so each stratum is to be allocated either |R| or [R] + 1 WLs. This is
accomplished by selecting a simple random sample of n — L|R] strata to receive [R] + 1
WLs and the rest to receive |R] WLs. (If R is an integer, n — LR] = 0 and all strata receive
RWLs.) Letting ,, denote the number of WLs assigned to the Ath stratum by the first stage
of sampling, note that

E(n;,) = R, h (2.1)



Weidman, Ernst: Multiple Workloads per Stratum Designs 379

remaining L'-L PSUs will fall in a leftover ‘‘stratum’’ of PSUs that will be in the D,
sample but not in the D; sample. See Ernst (1990) for further details.

6.1. Properties of the methods

The independent sample method uses an optimal D, stratification, but is the only one of the
four methods that does not assure that D, sample PSUs are retained in D2. Controlled
selection is the only method that selects PSUs from optimal stratifications for both designs
while retaining all D; sample PSUs in D3. However, unlike the other three methods,
controlled selection requires that the D; and D2 sample PSUs be selected simultaneously,
and consequently cannot be used for an expansion planned after D, is in place. (This
technique also has the disadvantage of introducing a between-strata component of
variance in both the D, and D, designs.) If we want to have D,;CD, but are unable to
select D> at the same time as Dj, then multiple WLs and the independent supplement
are the only two methods of the four compared that are operationally feasible. However,
they will probably have larger variances than the other two methods because they do not
use an optimal Dz stratification. The decision as to which of the four methods should be
used in any situation depends on the importance attached to the different properties and
the variances of the methods relative to each other.

Tables 4 through 7 present the ratios of variances for controlled selection, independent
supplement, and multiple WLs methods (both estimators, ¥ = MW and ’ = AL), to the
independent sample method. Tables 5 and 7 show between-PSUs variances and Tables 4
and 6 show total variances. These total variances include the within-PSUs component
from both certainty and noncertainty PSUs. For all four methods, 1980 census data
were used to obtain the stratifications, since 1990 census data were unavailable at the
time these computations were done. The variables used were number of unemployed
persons and number of persons in the civilian labor force (CLF). The ratios were computed.
for 31 states. Averages of these ratios over these 31 states were also computed. The
remaining states were omitted for various reasons, as described in Ernst (1990).

For each of the four methods (in this and the next three paragraphs we refer to the multi-
ple WL estimator ¥ only), the within-PSUs variances were obtained by computing the
simple random sampling with replacement variance for sample size m and multiplying
by a design factor to account for the fact that clustered, systematic sampling was actually
used within each PSU. For the multiple WLs method, this approach to computing the
within-PSUs variances is at least partially justified by the results at the end of Section
4. The within-PSUs component of each variance is thus computed to be the same for
all four methods and the differences among the methods are due solely to differences in
the between-PSUs component for all methods, and also the between-strata component
for the controlled selection and multiple WLs methods, which are the only methocs
among the four methods to have such a component.

Tables 4 and 5 compare the methods at the time of stratification using 1980 data.
Tables 6 and 7 use 1970 data to simulate a ten-year lag between stratificatson and
the collection of the survey data, which would be roughly the average lay, time for
the two-phase CPS. Because the between-PSUs components are such a sma‘ proportion
of the total variances in these designs, the ratios in Tables 4 and 6 are all extremely close


Weidman, Ernst: Multiple Workloads per Stratum Designs 381

Table 5. 1980 ratios of between PSU variances for other options to the independent sample

State Unemployed Civilian labor force

cs IS MW AL cs IS MW AL

Alabama 1.01 2.40 0.99 0.92 0.91 3.57 4.00 2.96
Arizona 0.89 4.53 0.12 0.08 0.76 2.54 451 3.26
Arkansas 0.70 3.14 1.80 1.33 1.12 135 2.61 1.69
Colorado 0.85 5.44 0.64 0.57 0.96 6.68 2.25 1.83
Georgia 1.02 1.96 0.83 0.83 0.87 1.77 2.65 2.65
Idaho 0.99 12.38 0.58 0.33 1.05 14.87 12.15 5.42
Indiana 1.05 9.48 0.94 0.89 0.65 6.14 5.39 3.36
Towa 1.01 3.06 0.69 0.63 0.98 454 3.10 1.97
Kansas 0.89 155 1.10 1.48 0.92 171 0.85 0.64
Kentucky 0.91 448 0.81 0.87 0.90 3.50 4.19 1.54
Louisiana 1.12 3.37 0.63 0.57 1.02 2.53 5.86 5.72
Maryland 1,00 5.13 0.59 0.59 1.00 43.15 20.81 20.81
Minnesota 1.14 1.30 0.39 0.38 1.02 4.21 2.33 0.84
Mississippi 1.26 459 2.76 2.52 1.08 1.46 3.44 3.15
Missouri 1.06 3.98 1.06 1.12 1.05 4.74 2.85 2.32
Montana 1.18 5.37 0.60 0.57 0.86 166 1.61 1.47
Nebraska 0.79 3.38 0.88 0.84 0.97 3.05 6.13 5.60
Nevada 0.98 5.91 0.42 0.48 1.07 487 = 3.58 2.84

New Mexico 1.10 2.12 0.50 0.45 1.19 4.73 4.77 4.05
North Dakota 0.95 3.86 0.79 0.79 0.99 8.55 3.60 3.60
Oklahoma 0.85 2.27 0.70 0.70 0.95 5.55 3.66 3.66
Oregon 1.05 4.00 0.71 0.76 1.17 5.31 7.78 6.68
South Carolina 1.43 19.29 15.38 14.53 1.00 4.89 12.74 3.56
South Dakota 0.87 2.72 0.63 0.63 0.92 3.53 2.84 2.84

Tennessee 0.89 3.81 0.83 0.86 1.01 3.06 3.37 2.91
Utah 0.93 3.56 1.04 1.07 1.00 452 5.96 4.55
Virginia 1.06 4.57 2.02 2.02 1.25 8.91 7.85 7.85

Washington 0.88 1.84 0.47 0.42 1.04 6.85 5.86 4.38
West Virginia 0.59 2.14 4.11 3.91 0.87 0.41 5.02 2.46

Wisconsin 1.07 4.68 2.59 2.61 1.00 3.77 4.77 3.55
Wyoming 1.04 26.75 1.04 0.80 0.82 31.75 21.40 5.20
Mean 0.99 5.26 1.50 1.43 0.98 6.64 5.81 4.04

CS = Controlled Selection

IS = Independent Supplement

MW =Multiple Workloads (7)

AL= Alternative MW Estimator (?’)

both unemployed and CLF.) For the ten-year lag of 1970 the situation is reversed, with all
but four unemployed ratios greater than 1.0 and the majority of CLF ratios less than 1.0. In
all cases the mean between-PSU variances are smaller for the benchmark than for f.

At the time of PSU size measurement in 1980, the mean between-PSU variances are less
for multiple WLs than for the independent supplement for both unemployed and CLF, but
in the CLF case each method has smaller variance for about half the states. The between-
PSU variances for the 1970 data are smaller for ¥ than for IS in 22 states for unemployed
and 29 states for CLF. The mean is less for ? in each case, so the multiple WL method is”
clearly preferred to the independent supplement for this CPS example.


Weidman, Ernst: Multiple Workloads per Stratum Designs 367

Condition (i) is the general probability proportional to size criterion. Condition (ii) is a
generalization of without replacement sampling to include the case when the expected
number of times a PSU is selected in D2 exceeds 1; that is, when the probability of its
selection in the D, sample is greater than 1/n;, Condition (ii) is motivated by the fact
that minimizing the variability in the number of times that a PSU is selected tends to lower
between-PSU variances, the same motivation for preferring without replacement sampling
to with replacement sampling.

The desire to satisfy properties 1 and 2 motivated the approach of expanding the sample
by selecting additional WLs from the D, strata, and property 3 motivated the use of
conditions (i) and (ii) in the selection of the additional WLs. A different approach to satis-
fying properties 1 and 2, independent supplement (Chandhok, Weinstein, and Gunlicks
1990), has also been investigated at the U.S. Census Bureau and is described in Section
6. However, in contrast with the multiple WLs approach, under independent supplement
small PSUs can receive more than one WL and, as a result, it is not as successful as the
multiple WLs approach at satisfying property 3.

In most applications, including the CPS, the selection of PSUs to receive additional
WLs under the multiple WLs approach constitutes the second stage of a three-stage
sampling process. The final stage, that of selecting the USUs, satisfies the condition
that the probability of selection of each USU in the state is the same, allowing for self-
weighting estimates. This condition was motivated by property 4, since in certain
situations a self-weighting estimator will minimize within-PSUs variance and total
variance (Cochran 1977, Sec. 10.10). Variance formulae for this combination of USU
sampling and self-weighting estimator are derived in Section 4.

The first stage of the multiple WLs procedure is the random determination of the
number of additional WLs to be selected from each D stratum for the D2 design. Note
that, as a result of this process, the number of WLs in each stratum is a random variable,
and there will be a between stratum component of variance when our sampling process is
combined with self-weighting estimates. Thus even our recommended sampling and
estimation combination does not satisfy property 5. Consequently, in Section 5 we
introduce an unbiased estimation procedure as an alternative to self-weighting, which in
combination with multiple WLs does not have a between-strata component of variance,
and allows us to satisfy property 5. We will show analytically in Section 5 and empirically
in Section 6 that there is a trade-off, however, since this alternative estimator generally
yields larger within-PSUs variances than the self-weighting estimator.

In the remainder of this article we look at the details of the multiple WLs procedure.
Section 2 describes all three stages of the procedure for expanding from D, to Dp. Section
3 details the procedures for selecting PSUs to receive the additional WLs, which is the
only sampling stage for which the selection methodology is not routine. Variance formulae
are derived in Section 4 for the self-weighting estimator. In Section 5 variance formulae
are obtained for an alternative estimator having no between-strata component of variance.
Finally, Section 6 presents, as an example, the state variances of the self-weighting multi-
ple WLs estimator for the CPS application and compares them with those from the three
other options investigated at the U.S. Census Bureau and the alternative multiple WLs esti;
mator. The multiple WLs per stratum method is the only one of the four methods that have
all three of the following desirable properties: (a) the D2 sample PSUs need not be selected


366 Journal of Official Statistics

motivation for the procedure to be discussed, there are potential applications to other
sample expansion problems as well.

For the CPS most of the states are divided into PSUs. PSUs smaller than a specified size
(population), the noncertainty PSUs, are combined into strata of approximately equal
size within states. One PSU per stratum is then selected with probability proportional
to size. Ultimate sampling units (USUs), which are small clusters of adjacent housing
units, are then sampled from each of the stratified PSUs in sample and the non-stratified
(certainty) PSUs. This is done in such a manner that each USU within a state has the
same overall selection probability. (A few states are treated as one large non-stratified
PSU.) Our interest is in how to increase sample in the stratified PSUs when the total
sample for the state must be increased to meet the D2 variance requirements. (The sample
in the non-stratified PSUs is increased in a straightforward manner by selecting additional
USUs.)

In formulating what we call the multiple workloads (WLs) procedure, we were guided
by the desirability of obtaining a sampling scheme with the following properties:

1. All D, sample PSUs are D2 sample PSUs.

2. The D2 sample PSUs may be selected subsequent to the selection of the D, sample
PSUs, which would allow the procedure to be used if it is necessary to expand the
sample after the D, PSUs are selected.

. The between-PSUs variance should be low.

4. The within-PSUs variance should be low.

5. Variance estimation should not be unduly complex.

w

The most natural approach to satisfying property 3 is to select the D, PSUs from an opti-
mal D, stratification. Two procedures for doing this which have been investigated at the
U.S. Census Bureau, independent sample (Chandhok, Weinstein, and Gunlicks 1990) and
controlled selection (Ernst 1990), are discussed in Section 6. Unfortunately, independent
sample violates property 1 and controlled selection violates property 2.

Consequently, we developed the multiple WLs approach. Let the number of USUs
selected from each sample PSU in a D, stratum, called a workload (WL), be the number
that can be efficiently handled by a single interviewer during an interview period. The
multiple WLs approach selects PSUs to receive additional full WLs in D2. For each
stratum h, let the number of PSUs (not necessarily distinct) selected for D2 be denoted
by mp. The D2 sample PSUs from stratum h consist of the single D; sample PSU and
nj-1 additional PSUs selected to satisfy:

i. the expected number of times a PSU is selected for D2 is n;, times the probability that
it was selected for D,; and

ii. the number of times that a PSU is selected for D2 is within one of the expected
number in (i).

Example. Consider a stratum h with four PSUs and n,, = 5. If the D selection probabilities
for these PSUs are .48, .24, .16 and .12, then by condition (i) their expected numbers of
selections in D, are 2.4, 1.2, .8 and .6. By condition (ii) the number of times each PSU
can be selected for D2 is 2 or 3 for the largest one, 1 or 2 for the second largest, and 0
or 1 for the smallest two.


Weidman, Ernst: Multiple Workloads per Stratum Designs 377

and hence
- LM
EXES@ = S030 Ena) ¥nn = ¥ (5.5)
fat

From (5.5) it follows that ¥’ is an unbiased estimator of Y, and the between-strata
component of V(¥’) is 0.

The between-PSUs-within-strata component of V(?’) can be obtained by proceeding as
in Section 4.2 with a}; substituted for a,; and, by (5.3), the final expression in (4.11)
modified by multiplying it by R7/K-. Hence, with these changes, we obtain, analogously
to (4.16), that

L Np. L URHI [ Nn YZ
E,|SOVe Seta) |= yy [Siem - Th)
h=1 i=l hi

h=1 k=[R}
Na, Yat ke

+E Les — mony) a rw 66)
ot

Aj

There is no clear relation between the values of (4.16) and (5.6). The expression within the
brackets is multiplied by 1/R? in (4.16) and 1/K in (5.6), and I/k? = 1/R? for k = |R] while
1/k? < UR? for k =[R]+ 1. However, the expression within the brackets is a complex
function of k. It is possible, for example, for this term to be larger when k =[R] than
when k =[R] + 1, and also for the opposite to be true.

To obtain the within-PSUs component of VY’), first observe that by (5.3) and (4.18)

RM My Soni

Exlai2v,o1 = eC (5.7)
Furthermore, by (4.9)
R (GR pM) (1 +IRI-R | R-IRI
E,(~)= =R(— 4) arr 5.8
(2) Ry i r( inl +i) FR) (6.8)
where
(Ry = RUS R+ AR) 69)

LIAR] + 1)
We then combine (5.7) and (5.8) to obtain

LOOM
Ei [SS esetvdio =1(4 ie ¥ Somusi =r)

hal i=1 hel i=l

LM
Y Sms
feat it

(5.10)

and that, using the same reasoning as in Section 4.3, if $3,; = S} for all hi, then the within-
PSUs variance of Y’ is approximately f(R)M?S3/m.

Finally, it follows from (5.10) and (4.19) that the ratio of the within-PSUs component of
V(2’) to the within-PSUs component of V(Y) is approximately f(R). Thus, unlike the


380 Journal of Official Statistics

Table 4. 1980 ratios of total variances for other options to the independent sample

State Unemployed Civilian labor force

cs IS MW AL cs IS MW AL

Alabama 1.000 1.009 1.000 1.021 0.999 1.028 1.032 1.043
Arizona 0.999 1.017 0.996 1.003 0.998 1.015 1.035 1.030
Arkansas 0.998 1.014 1.005 1.023 1.003 1.008 1.037 1.037
Colorado 0.998 1.051 0.996 1.008 0.998 1.260 1.057 1.051
Georgia 1.000 1.004 0.999 0.999 0.998 1.012 1.025 1.025
Idaho 1.000 1.165 0.994 1.003 1.000 1.096 1.077 1.044
Indiana 1.000 1.058 1.000 1.055 0.995 1.068 1.058 1.086
Towa 1.000 1.018 0.997 1.004 1.000 1.021 1.012 1.013
Kansas 0.999 1.008 1.001 1.018 1.001 1.055 1.046 1.047
Kentucky 0.999 1.022 0.999 1.021 0.998 1.039 1.050 1.030
Louisiana 1.001 1.016 0.997 1.012 1.000 1.016 1.051 1.065
Maryland 1.000 1.013. 0.999 0.999 1.000 1.110 1.052 1.052
Minnesota 1.004 1.008 0.984 0.991 1.000 1.061 1.025 1.005
Mississippi 1.001 1.014 1.007 1.014 1.001 1.008 1.044 1.047
Missouri 1.000 1.017 1.000 1.021 1.001 1.092 1.046 1.052
Montana 1.005 1.114 0.990 0.995 0.993 1.033 1.031 1.030
Nebraska 0.998 1.019 0.999 1.008 1.000 1.023 1.057 1.060
Nevada 1.000 1.021 0.998 1.002 1.002 1.139 1.093 1.070

New Mexico 1.001 1.015 0.993 1.011 1.005 1.106 1.107 1.105
North Dakota 0.999 1.060 0.996 0.996 1.000 1.135 1.046 1.046
Oklahoma 0.998 1.013 0.997 0.997 0.999 1.077 1.045 1.045
Oregon 1.000 1.025 0.998 1.012 1.001 1.034 1.053 1.058
South Carolina 1.000 1.010 1.008 1.009 1.000 1.014 1.043 1.011
South Dakota 0.995 1.061 0.987 0.987 0.998 1.065 1.048 1.048

Tennessee 0.999 1.013 0.999 1.007 1.000 1.031 1.036 1.036
Utah 1.000 1.012 1.000 1.009 1.000 1.048 1.068 1.057
Virginia 1.000 1.014 1.004 1.004 1.003 1.080 1.069 1.069

Washington 0.997 1.018 0.989 1.002 1.000 1.059 1.049 1.049
West Virginia 1.000 1.001 1.003 1.021 0.999 0.997 1.020 1.025

Wisconsin 1.000 1.017 1.007 1.016 1.000 1.042 1.058 1.048
Wyoming 1.000 1.093 1.000 1.018 0.999 1.135 1.089 — 1.037
Mean 1.000 1.030 0.998 1.009 1.000 1.062 1.050 1.046

CS = Controlled Selection

IS = Independent Supplement

MW =Multiple Workloads (?)

AL = Alternative MW Estimator (?’)

to 1. Thus, in the CPS case, there is not much difference between the methods. In order to
compare the methods more generally, we concentrate on the between-PSUs variances.
For 1980 the between-PSUs variances of unemployed for ¥ are less than the benchmark
for 21 of 31 states. The influence of the large ratio for South Carolina is the primary reason
for the average ratio of the 31 states being greater than 1. The CLF ratios are = 2.25 for all
but one state. (The result for the unemployed is surprising. Since the 1980 D, stratification
used for the benchmark is optimal while ¥ is based upon a stratification optimal for the smaller
Dy, it might be expected that the variances of the benchmark would be smaller. The fact that
they are not might be due to the attempt to choose the D; stratification to optimize a function of