Stratification by Size Revisited

Alan H. Dorfman and Richard Valliant, U.S. Bureau of Labor Statistics

Richard Valliant, Room 4915, 2 Massachusetts Ave. NE, Washington DC 20212

Keywords: balanced sample, best linear unbiased

predictor, robustness, superpopulation model.

Stratification is one of the most widely

used techniques in finite population sampling.

Strata are disjoint subdivisions of a population,

the union of which exhaust the universe, each of

which contains a portion of the sample. Two of

its essential statistical purposes are to:

(1) allow for efficient estimation, especially in

the case of stratification by size, and

(2) deal statistically with subpopulations or

domains by controlling their sample

allocations.

Stratification by size is typically considered as

serving purpose (1) by creating strata in an

efficient way and optimally allocating the sample

to the strata. Using model-based analysis, we

show that, in the situation where stratification by

size is generally used, optimal allocation of a

weighted balanced sample achieves exactly the

same variance as unstratified, best linear unbiased

(BLU) prediction coupled with weighted balanced

sampling. In other words, stratification by size

has no advantage over the optimal, unstratified

procedure. This and other theoretical findings are

illustrated with simulations using real populations.

1. A Stratified Linear Model and Weighted

Balanced Samples

Let h denote a stratum and i a unit within

the stratum. The target variable for unit hi is Yhi .

The population contains H strata with the number

of units in each stratum being Nh (h = 1, , H)

and the population size being N Nh h

H

= ∑ =1. A

sample of nh units is selected from stratum h with

the total sample size being n nh h= ∑ . Denote the

set of sample units in stratum h as sh and the set

of nonsample units as rh. Assume that a separate

linear regression model holds within each stratum:

E M (Yh )= Xhβh, varM (Yh )= Vhsh

2

(1)

where Yh is Nh × 1, Xh is N p h × h,

Vh = diag(vhi) is Nh × Nh, and βh is a ph ×1

parameter vector. The model in stratum h is

denoted by M(Xh:Vh ) and the BLU predictor is

then the sum of the BLU predictors in each

stratum:

T T( h:

h )

h

H

=

=

∑ X V

1

.

In stratum h define a root(v)-balanced sample to

be one that satisfies

1 1 2

1 2

nh

sh sh sh

Nh h

Nh h Nh

′ =

′

′

−

1 V X

1 X

1 V 1

/

(2)

where 1sh is a vector of nh 1’s, 1Nh is a vector of

Nh 1’s, Vsh is the n n h × h diagonal covariance

matrix for the sample units, and Xsh is the n p h × h

matrix of auxiliaries for the sample units. Any

sample satisfying (2) will be denoted by

B(Xh:Vh ), and, when (2) is satisfied in each

stratum, the entire sample is a stratified weighted

balanced sample.

If the model has a certain structure given

in Theorem 1 below, then a weighted balanced

sample is the best that can be selected in the sense

of making the error variance of the BLU predictor

small. Let M(Xh ) be the vector space generated

by the columns of Xh. A straightforward

application of Theorem 2 in Royall (1992) yields

the following stratified result.

Theorem 1. Suppose that model (1) holds in

stratum h for h = 1, , H . If both V 1h Nh and

Vh 1Nh (Xh )

1 2 ∈ M , then the BLU predictor

achieves its minimum variance when each stratum

sample is B(Xh:Vh ). In that case, the BLU

predictor reduces to

( / )

T N /v

n

Y

v

h h

h

hi

h i s hi

H

h

=

= ∈

∑ ∑

1 2

1 2

1

1

, (3)

and the error variance is

( )

( )

var ( )

/

M

h

h h h h h

T T h

n

− = N v − N v











∑ 

1 1 2

2

2

s (4)

where ( )

vh

1/2

= vhi N

i

N

h

h 1 2

1

/ ∑ =

 and vh =

vhi N

i

N

h

h ∑ =1

.

In a stratified weighted balanced sample,

the optimal estimator, thus, reduces to a sum of

mean-of-ratios estimators, which, for later

reference, we will write as T (v ) MRS

1 2

.

2. Optimal Allocation for Stratified Balanced

Sampling
06/16/97 2

The optimum allocation to the strata of a

weighted balanced sample can be easily calculated.

Assume that the cost of sampling is

C( ) C chnh hn = 0 +∑ where C0 is a fixed cost and

ch is the cost per unit sampled in stratum h.

Theorem 2. Assume that model (1) holds, that

V 1h Nh and Vh 1Nh (Xh )

1 2 ∈ M , and that a weighted

balanced sample B(Xh:Vh ) is selected in each

stratum. The allocation of the sample to the strata

that minimizes the error variance of the BLU

predictor, subject to the cost constraint

C( ) C chnh hn = 0 +∑ , is

( )

( )

n

n

N v c

N v c

h h h h h

h h h h h

=

′ ′ ′ ′ ∑ ′

1 2

1 2

/

/

s

s

 for h = 1, , H.

When optimal allocation is used and all

costs are equal, the BLU predictor (3) becomes

( )

( )

/

T /

n

N v

Y

v

h h h h

hi

hi h

h sh

= ∑ ∑ ∑

1 1 2

s 1 2s(5)

and its error variance (4) can be rewritten as

varM (T − T)=

( )

( )

1 1 2

2

2

n

N v N v

h h h h h h h h

/ ∑ s −∑ s (6)

3. The Case of a Single Model for the

Population

An important special case is having a

single model that fits the whole population.

Assume the model in each stratum is

E M (Yh )= Xhβ , varM (Yh )= Vhs

2

, (7)

with Xh Vh and defined as in (1). Expression (7)

is just another way of writing the unstratified

model M(X:V). Thus, strata can be ignored in

calculating the BLU predictor and its error

variance. If V1N and V 1 (X)

1 2

N ∈ M , then, by

Theorem 1 with H=1, a weighted balanced sample

s ∈B(X:V) is optimal for the BLU predictor. In

that case, the BLU reduces to

( )

T

n

Nv Y v i i s

= ∑

1 1 2 1 2

(8)

with error variance

( )

( )

varM T − T = n (Nv ) − Nv









− s2 1 1 2

2

.

On the other hand, suppose we select a

stratified weighted balanced sample and use the

optimal allocation given in Theorem 2 above for

the equal cost case. Using (5) with sh = s , the

BLU predictor with the optimal allocation is

( )

( )

/

T

n

N v Y v

h h h hi hi h sh

= ∑ ∑ ∑

1 1 2 1 2

which is exactly equal to (8). In other words,

stratification with optimal allocation of a stratified

weighted balanced sample gains nothing at all

compared to the strategy of selecting an

unstratified sample with overall weighted balance.

A situation where a common model may

hold for the whole population is one where a

single auxiliary variable x is available. The

auxiliary can be used for stratification by size as

well as for estimation. Strata are formed by

ordering the units from low to high based on x so

that the first stratum contains the N1 units with the

smallest x values, the second stratum contains the

next N2 smallest units, and so on. Take the

special case of model (7) given by

E M (Yh )= Xhβ , varM (Yhi) hi = s x

2 g

(9)

where, in many populations, 0 ≤ g ≤ 2 . When (9)

is true, we can use the idea of the minimal model

introduced by Dorfman and Valliant (1997). The

minimal model is the one with least variables

satisfying V1N and V 1 (X)

1 2

N ∈ M . When

varM (Yhi) hi = s x

2 g

, this is

E M (Y ) x x hi = bg hi + b hi

g

g

g

2

2

. We denote this model

by M(x x x )

g 2 g g

, : . With the variance

specification in (9), the optimum allocation in

Theorem 2 becomes

( )

( )

n

n

N x c

N x c

h h h h

h h h h

=

′ ′ ′ ∑ ′

g

g

/

/

2

2

(10)

When the optimal allocation (10) is used and costs

are all equal, the error variance (4) of the BLU

predictor in a stratified weighted balanced sample

reduces to the variance for an unstratified,

weighted balanced sample:

( )

( )

( )

( )

var

/

M T T

n

− = Nx − Nx













s

2 g 2 g

1 2

. (11)

4. Comparisons with Other Strategies

In this section we denote the polynomial

model having E M (Y ) x x i i J i

J

= d0 +d1 + +d and

varM (Yi) i = s v

2

 by M(d0, ,dJ:v).

When strata are formed on the basis of a

size measure x, an oft-studied procedure is the

separate ratio estimator, defined as
06/16/97 3

T N Y

x

x

RS h hs

h

h hs

H

=

=

∑

1

where xh

xhi Nh i

Nh

= ∑=1, y Y n hs hi h i sh= ∑∈, and

x x n hs hi h i sh

= ∑∈. Under the working model

M(0,1: x), TRS is unbiased with variance equal to

varM ( RS ) ( )

h

h

h

hr h

h hs

T T

N

n

f

x x

x

− = s ∑ −

2

2

1 ,

where f

h = nh Nh

 and xhr xhi (Nhnh )

i sh

= − ∑∉.

If one is completely confident that M(0,1: x) is

correct, then the optimal sample for TRS would be

to pick the nh units with the largest x’s in each

stratum. Even more extreme is the globally

optimal strategy of the simple ratio estimator and

the n largest units in the population.

Confidence in any single model is seldom

this high and having protection against model

failure is usually prudent. If the true model is

M(d 0, ,d J: x), then the estimator has a bias:

( )

( ) ( )

E T T N x

x

x

x

x

M RS h h j j

hs

j

hs

h

j

j h

J

h

− = −













=

∑ ∑d b

0

where ( )xhx N

j

hi

j

h i

Nh

= ∑=1

 and ( )

x x n hs

j

hi

j

h i sh

= ∑∈.

If a stratified (unweighted) balanced sample, i.e.

one that is balanced in each stratum ( ( ) ( )x x hs

j

h

j

=

for j = 1, , J ), is selected, then TRS reduces to

the stratified expansion estimator

T S NhYhs

h

H

0

1

=

=

∑ .

Denote a stratified (unweighted) balanced sample

by s (J)

∗

 and a simple (unstratified) balanced

sample of order J by s(J). When the working

model M(0,1: x) holds, the optimal allocation of

the sample to strata for TRS is n N x h ∝ h h.

Protection against bias under the

polynomial model M(d 0, ,d J: x) is afforded

either by simple balanced sampling with the ratio

estimator or stratified balanced sampling with the

separate ratio estimator. Royall and Herson

(1973) showed that if n N x h ∝ h h

, then under

M(d 0, ,d J: x) the strategy [s (J) TRS ]

∗

, is more

efficient than [s(J),T(0,1: x)] in the sense that

EM [T(0,1: x) T] EM (TRS T)

2 2

− ≥ − .

But, because the separate ratio estimator

does not flow from a model satisfying the

conditions of Theorem 1, the strategy [s (J) TRS ]

∗

,

is not the best that we can do. When

varM (Yi) i = s x

2

, as in M(0,1: x), the minimal

model is M(x x x)

1 2

, : . Now, suppose that the

correct model contains some higher order

polynomial terms. Specifically, let

M(d0,d1 2, ,dJ: x) denote the model with

E M (Y ) x x x i i i J i

J

= d0 + d1 2 + d + +d

1 2

1

 and

varM (Yi) i = s x

2

. If the sample has weighted

balance—equation (2) above—so that

( )

( )

1

1 2 1 2 n

x

x

x

x

i

j

i

j

s

∑ = for j = 0 1 J

1

, 2 , , , , (12)

then the BLU predictor T(x , x: x)

1 2

 under

M(x x x)

1 2

, : is protected against bias if the model

is really M(d0,d1 2, ,dJ: x). By Theorem 1,

when (12) is satisfied, T(x , x: x)

1 2

 reduces to the

mean-of-ratios estimator T (x ) MR

1 2

=

( ) Nx n Y x i i s

1 2 −1 1 2 ∑ and has error variance

[ ( ) ]

( )

varM TMR x T ( )

n

Nx Nx

1 2 2 1 2 1 2

− = −













s . (13)

By Theorem 1, this error variance will be less than

or equal to any that can be achieved under

M(d0,d1 2, ,dJ: x) using TRS .

5. Formation of Strata

A question traditionally posed when

stratifying by size is how to form the strata. When

a common model holds for the entire population

as in section 2 and V1N, V 1 (X)

1 2

N ∈ M , we

know that the BLU predictor with a weighted

balanced sample is the best strategy. That is,

stratification in this common circumstance is

unnecessary. However, various methods of strata

formation are used in practice, and it is interesting

to investigate their properties.

One set of methods are known as equal

aggregate size rules. Units are sorted from low to

high based on x. Strata are then formed in such a

way that each contains about the same total of the

size variable or a monotone transformation of it.

Equalizing ( ) N xh h

g 2

 leads to several stratification
06/16/97 4

rules. When g = 0, equal values of ( ) N xh h

g 2

correspond to equal numbers of units Nh in each

stratum. When g = 1, we have equal aggregate

square root of size, and g = 2 gives equal

aggregate x.

The equal aggregate size rules can be

derived using only model-based arguments. Due

to limited space, we will only summarize the

result. When the model is E M (Yh )= Xhβ ,

varM (Yhi) hi = s x

2 g

 with V1N, V 1 (X)

1 2

N ∈ M , the

sample is sh ∈B(Xh:Vh ), and an equal number of

units is allocated to each stratum, then the error

variance of the BLU predictor is minimized if

strata are constructed to have equal ( ) N xh h

g 2

 in

each. Moreover, if strata are constructed in this

manner, an equal allocation is the optimal

allocation in the equal cost case. However, an

optimally allocated stratified, weighted balanced

sample yields exactly the same variance as an

unstratified sample with weighted balance.

Another method of stratification is known

as the cum f rule due to Dalenius and Hodges

(1959) that we will include in the simulation

reported in section 6 but will not describe in detail

here.

6. Some Empirical Results on Strata

Formation

In this section we will illustrate the

different methods of strata formation and their

effects on estimation in a simulation study. The

three populations used are known in the literature

as Hospitals, Cancer (Royall and Cumberland

1981), and Beef (Chambers and Dunstan 1986).

Figure 1 shows scatterplots of the three. Four

methods of stratification were used with H=5:

(1) equal numbers of units Nh in each stratum,

(2) equal cum f in each stratum.

(3) equal aggregate total of x in each stratum

(the cum x rule),

(4) equal aggregate total of x in each stratum (the

cum x rule).

We used the four methods of stratification listed

above and also did unstratified sampling. Five

200 400 600 800

1000 2000

Hospitals

20000 40000 60000 80000

0 100 200 300

Cancer

5000 10000 15000 20000

0 500 1000 1500

Beef

Figure 1. Scatterplots of three populations.

combinations of estimators and sample selection

methods were used:

(a) T(x , x : x )

g 2 g g

, which is minimal when

varM (Yi) i = s x

2 g

, and pp(x )

g 2

 sampling with

g = 1, 2 ,

(b) T (x ) MRS

g 2

, the stratified mean-of-ratios

estimator and pp(x )

g 2

 sampling with

g = 1, 2 ,

(c) T0S, the stratified expansion estimator, and

stratified simple random sampling (stsrs)

without replacement,

(d) TRS , the separate ratio estimator, and stsrs

without replacement, and

(e) TLS , the separate regression estimator, defined

below, and stsrs without replacement.

The separate regression estimator is defined as

TLS T S Nhbhs(xhxhs) h

= 0 +∑ − with bhs =

(xhi xhs )yhi (x x ) s

hi hs s h h ∑ − ∑ −

2

. Note that

T (x ) MRS

g 2

 is the Horvitz-Thompson estimator in

pp(x )

g 2

 sampling and that T0S is both the

Horvitz-Thompson and the Hájek estimator in

stsrs.

For each method of stratification a sample

of n=30 was divided equally among the five strata

giving nh = 6 in each stratum. As noted in

section 5, when strata are formed to equalize
06/16/97 5

( ) N xh h

g 2

, costs are all equal, and

varM (Yhi) hi = s x

2 g

, then an equal allocation is

optimal. In addition, equal allocation is one

method traditionally used with the cum f

method.

Both unrestricted and restricted sampling

techniques were used in the simulation.

Unrestricted pp(x )

g 2

 was implemented using the

random order, systematic method described by

Hartley and Rao (1962). Restricted pp(x )

g 2

sampling was done by selecting a sample with the

random-order method and then checking its

closeness to weighted balance on four moments

within each stratum. The balance measures

( )

( ) ( ) ( )

( )

e s

n x x x

s

j h

sh

j

h

j

h

jxh

=

−

−g 2 g 2

, j = 0 1 2

1

, 2 , ,

were calculated in each stratum where

( ) ( )

sjxh hi(xhi x x )

j

h

j

i h

Nh

= −









− ∑ =

p

g 2 g 2

2

1

1 2

 and

( )

p ( )

g g

hi hi h h = x N x

2 2

. For the pairs

(j = = )

1

2 ,g 1 and (j = 1,g = 2), ej(sh )= 0 , and

balance on those moments is trivially satisfied.

For the non-trivial cases, if ej(sh )≤ 0.1256613 for

all measures in every stratum, then the sample was

retained; otherwise, it was discarded and another

drawn. This technique retains only about 10% of

the best-balanced samples.

Balancing on the other moments above, in

addition to j = g , protects the minimal estimator

against different polynomial terms not in a minimal

working model without losing any precision under

the working model. With the weighted balance

conditions above, the mean-of-ratios estimator

T (x ) MRS

g 2

 is equal to the minimal estimator

T(x , x : x )

g 2 g g

, but in unbalanced samples there

may be important differencesa point that the

simulation results will illustrate.

Unrestricted and restricted stsrs samples

were used for estimators (c)-(e) above. In the

unrestricted samples, a simple random sample was

selected without replacement in each stratum and

retained regardless of its configuration. For

restricted samples, a without-replacement srs was

selected in each stratum and checked for simple

balance on the moments ( )

xsh

j

, j = 0 1 2

1

, 2 , , . As

above, only about 10% of the best-balanced

samples were retained.

For each combination of stratification,

sampling method, and estimator, 1,000 samples

were selected. For restricted samples this means

that samples were selected until 1,000 were

retained. The root mean square errors for each

estimator were computed as rmse(T)=

[∑s=

(T − T) ]

2

1

1000 1 2

1000 . Figure 2 presents

results, using a rowplot of the type devised by

Carr (1994). In each column, the ratio of each

rmse to the minimum rmse among the estimators

for the population is plotted. Black dots represent

restricted samples while open circles are for

unrestricted samples. The narrow triangles are

cases where the ratio was truncated at 2 to avoid

scaling problems. Some observations are:

• In Hospitals and Cancer, the minimal

estimator with unstratified, restricted pp(x )

1 2

sampling has the smallest rmse or very near it.

In Beef the stratified, minimal estimator with

pp(x )

1 2

 sampling is best.

• Unrestricted sampling is generally inferior to

restricted, balanced sampling.

• The minimal and mean-of-ratios estimators

have about the same rmse’s in weighted

balanced samples as expected. In contrast,

T (x ) MRS

g 2

 can have much higher rmse’s than

T(x , x : x )

g 2 g g

 in unrestricted pp(x )

g 2

sampling.

• The estimators used when sampling is

stsrsexpansion, ratio, and regressionare

improved by balanced sampling, but are

generally inferior to the minimal estimator

with weighted balance as anticipated in section

4.

• For a given selection method ( pp(x )

1 2

 or

pp(x)), stratification with weighted balance

within strata yields rmse’s very near those of

unstratified sampling and weighted balance for

the minimal or mean-of-ratios estimator in

Hospitals and Cancer. This is expected since

the minimal and mean-of-ratios estimators are

equal in weighted balanced samples, and an

optimally allocated, stratified, weighted

balance sample also has overall balance. Beef

is the exception because restricted, stratified
06/16/97 6

sampling achieved better overall, weighted

balance than unstratified sampling.

• In contrast, stratification with balanced

sampling can substantially improve the

expansion, ratio, and regression estimators.

7. Discussion

Rules for stratification by size have been in

the literature for many years, e.g., Mahalanobis

(1952). More recently (Wright 1983) the method

has been justified as a means of approximating the

optimum selection probabilities derived by

Godambe and Joshi (1965). Thus, there has been

some recognition that stratification by size may

entail a loss of efficiency, but the method remains

a common tool of practitioners. Exact modelbased optimality can be obtained through

stratified, weighted balanced sampling and

optimum allocation, but the stratification by size is

superfluous, unless the strata are needed for other

reasons, such as estimating domain characteristics

or controlling for differential costs.

Acknowledgements

Any opinions expressed are those of the authors

and do not constitute policy of the Bureau of Labor

Statistics.

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Dorfman, A.H. and Valliant, R. (1997), “The Hájek

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Godambe, V.P., and Joshi, V.M. (1965), "Admissibility

and Bayes Estimation in Sampling Finite Populations -

I," Annals of Mathematical Statistics, 36, 1707-1723.

Hartley, H.O., and Rao, J.N.K. (1962), “Sampling with

Unequal Probabilities and without Replacement,”

Annals of Mathematical Statistics, 33, 350-374.

Mahalanobis, P.C. (1952), "Some Aspects of the Design of

Sample Surveys," Sankhya, 12, 1-7.

Royall, R.M. (1992), "Robustness and Optimal Design

Under Prediction Models for Finite Populations,"

Survey Methodology, 18, 179-185.

Royall, R.M. and Cumberland, W.G. (1981), "An

Empirical Study of the Ratio Estimator and Estimators

of its Variance," Journal of the American Statistical

Association, 76, 66-77.

Royall, R.M. and Herson, J. (1973), “Robust Estimation in

Finite Populations II: Stratification on a Size

Variable,” Journal of the American Statistical

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Wright, R. (1983), “Finite Population Sampling with

Multivariate Auxiliary Information,” Journal of the

American Statistical Association, 78, 879-884.

Equal Nh

Expansion, srs

Ratio, srs

Regression, srs

Equal cum sqrt(f)

Expansion, srs

Ratio, srs

Regression, srs

Equal cum sqrt(x)

Expansion, srs

Ratio, srs

Regression, srs

Minimal, pp sqrt(x)

Mean-of-ratios, pp sqrt(x)

Equal cum x

Expansion, srs

Ratio, srs

Regression, srs

Minimal, pp(x)

Mean-of-ratios, pp(x)

Unstratified

Expansion, srs

Ratio, srs

Regression, srs

Minimal, pp sqrt(x)

Mean-of-ratios, pp sqrt(x)

Minimal, pp(x)

Mean-of-ratios, pp(x)

Hospitals

1 1.2 1.5 1.8 2

Cancer

1 1.2 1.5 1.8 2

Beef

1 1.2 1.5 1.8 2

Figure 2. Ratios of rmses for estimators in stratified and unstratified sampling to the minimum

 for the Hospitals, Cancer, and Beef populations; S=1000, n=30, nh=6 in all strata for stratified rmse

designs.

Stratification/Estimator

Ratio to minimum rmse

restricted sampling unrestricted sampling truncated value