1

International Conference on Establishment Surveys, II

June 2000, Buffalo, New York

THE THEORY AND PRACTICE OF MAXIMAL BREWER SELECTION 

WITH POISSON PRN SAMPLING

Phillip S. Kott And Jeffrey T. Bailey, National Agricultural Statistics Service

Phillip S. Kott, NASS, Room 305, 3251 Old Lee Hwy, Fairfax, VA 22030, USA

pkott@nass.usda.gov

K.R.W. Brewer suggests that when estimating the total of a single item for which there is control

(auxiliary) data, one employ a ratio or regression estimator and draw the sample using probabilities

proportional to the control values raised to a power between 1/2 and 1. Brewer's sample selection

scheme can be expanded to multiple targets by drawing overlapping Poisson samples for a number of

items simultaneously using permanent random numbers (PRN's). We can call the result an example

of "Maximal Brewer Selection" (MBS). This paper develops the theory behind MBS and the

calibration estimator rendering it practical. It goes on to describe how this estimation strategy is being

used at the National Agricultural Statistics Service. 

Key Words: Model; Model-assisted; Randomization; Calibration; Delete-a-group jackknife

1. INTRODUCTION

K.R.W. Brewer’s (1963) article in the Australian Journal of Statistics is one of the truly remarkable works in the survey

sampling literature. It discusses a model-based approach to survey sampling theory, contrasts that approach with the

conventional randomization paradigm, and shows how the two can be used in tandem. All this seven years before Royall

(1970) set the survey world buzzing with prediction theory (another name for the model-based approach) and almost

three decades before the publication of Särndal et al. (1992) made model-assisted survey sampling (which uses models

and randomization in tandem) the new conventional wisdom.

This paper builds on one small result in Brewer’s impressive opus and some of his work since then. Suppose we are

interested in estimating a population (P) total, T = 3P yi

, with a random sample (S) of size n. We suspect that the yi

follow the model 

 yi = $xi

 + k,i

, (1)

where E(,i *xi

) = E(,i,j *xi

, xj

) = 0 (ij), and Var(,i *xi

) = Fi

2

 is known for all i (but k need not be known). 

Equation (1) is a useful model for many establishment surveys. Whether or not it is correct, the following estimator

is nearly randomization unbiased for large n (and randomization consistent under a number of sampling designs), 

 t = (3P xi)3S (yi /Bi)/3S (xi /Bi),

where Bi is the selection probability of unit i. Of course, in order for t to be practical, the population sum 3P xi must be

known, and the individual xi

 must also be known for all units in the sample. In what follows, we further require xi

 to be

known for all units in the population. Such an x is called a "control" variable for the target variable y. 

Brewer showed that when Bi % Fi the randomization-expected model variance of t was (asymptotically) minimized for

fixed sample size n. In this sense, Bi = nFi / 3P Fk ) if less than or equal to 1 for all i ) is the optimal selection scheme

given sample size n and estimator t . Godambe (1955) has a similar result for randomization unbiased estimators.

It is sometimes assumed that the Fi have the form xi

g

, where 0 # g # 1. If that is the case, then when g = 1, the optimal

selection scheme (i.e., randomization-expected model-variance minimizing) is probability proportional to size,

Bi = nxi / 3P xk , and t collapses into the Horvitz-Thompson mean-of-ratios estimator (n-13S(yi /Bi); this is Godambe's

1955 result). When g = 0, the optimal sampling scheme is self-weighting, Bi

 = n/N. For establishment surveys, however,
2

g is usually between 1⁄2 and 1. Brewer has said (out loud, if not in print) that a sensible value for g in many surveys is

3/4. 

Sadly, Brewer's suggestion that the unit selection probabilities be in proportion to some known control value, xi

, raised

to the 3/4 power has not been implemented much in practice. One problem is that many real establishment surveys have

multiple targets of interest with varying relevant control values. Recently, however, several survey organizations have

come to use calibration estimators in place of traditional expansion and ratio estimators. This has allowed the National

Agricultural Statistics Service (NASS) to begin implementing a multivariate version of Brewer's suggestion in its

Crops/Stocks Survey (CS). Interna lly, NASS calls this procedure "multivariate probability proportional to size"

sampling. A better name would be "Maximal Brewer Selection" (MBS). This method of sample selection has proven

more flexible than the stratification approaches NASS has traditionally used (see Bosecker 1989). 

Section 2 fills out the theory of Brewer selection when there is a single target and control. Section 3 describes a simple

extension for multiple targets each with its own control variable. Briefly, a Brewer selection probability is assigned to

each population unit for every target variable of interest. The largest of these for each unit is then used for the actual

sample selection. Section 4 addresses a several questions that NASS needed to resolve before making MBS practical

to use. Poisson PRN sampling allows the agency to focus on different combinations of target variables in different survey

periods. Section 5 further describes NASS's experience with this new sampling scheme. Section 6 offers some

comments including one that describes a method for co-ordinating samples to minimize overlap. 

2. BREWER SELECTION 

2.1. Some Theory

Suppose we have target values, yi

, which we believe (roughly) obey the model in equal (1). We will call tC

 = 3S aiyi

,

based on a sample S with n members, a calibration estimator for T if the calibration equation

 3S aixi = 3P xi (2)

is satisfied, and each ai = Bi 

-1[1 + OP

(1/on)], where Bi is (again) the selection probability of unit i, and OP refers to an

asymptotic probability order with respect to the randomization rather than the model (see Isaki and Fuller 1982 for a

development of asymptotics in a finite population context). This is a bit of a generalization of the definition of a

calibration estimator in Deville and Särndal (1992). 

One obvious choice for the ai is Bi 

-1(3P xk /3S

[xk /Bk]). This renders tC equal to t in Section 1. The choice satisfies the

calibration equation, and the ai are sufficiently close to the Bi

-1 as long as the design and population are such that 

(3S

[xk /Bk

] ) 3P xk

)/3P xk is OP

(1/on). 

The model variance of tC as an estimator of T is

 E,

[(tC ) T)2

] = E,

[(3S aiyi ) 3P yi

)

2

]

 = E,

[(3S ai,i ) 3P ,i

)

2

]

 = 3S ai

2

Fi

2 ) 23S aiFi

2

 + 3P Fi

2

. (3)

Since each ai . 1/Bi

, E,

[(tC ) T)2

] . 3S Fi

2 /Bi

2 ) 23S Fi

2 /Bi

 + 3P Fi

2

Technically, the relative difference between the left and right hand sides of the above equation is OP

(1/on). For our

purposes, this defines when the two sides of an equation are approximately equal. 

The randomization expectation (denoted using the subscript "P") of the model variance of tC is

 EP

{ E,

[(tC ) T)2

]} . 3P Fi

2 /Bi ) 3P Fi

2

. (4)
3

Under mild conditions, this is the same as the model expectation of the randomization means squared error of tC

. Isaki

and Fuller called that last quantity the "anticipated variance" of tC

, presumably meaning “the anticipation under the

model of the randomization mean squared error or variance” (randomization mse and variance are virtually identical

under the designs Isaki and Fuller had in mind). We will use their term here, but keep in mind an alternative meaning

for “anticipated variance:” the model variance anticipated before sampling. 

If we restrict ourselves to a randomization estimator like tC, a sensible policy is to choose selection probabilities so that

the right hand side of equation (4) is minimized for a given sample size n. Since n = 3P Bi, it is a simple matter to set

up a Langrangian equation, the solution to which is Bi = nFi/3P Fk. For this solution to be valid each Bi must be no greater

than 1. We assume that to be the case for the time being.

The anticipated variance of tC

 is (asymptotically) minimized by setting the unit probabilities of selection equal to

nFi/3P Fk no matter which method it used to draw the sample. In fact, the same minimum variance is obtained if the

sample size itself is allowed to be random with an expected value equal to n. Poisson sampling is a simple example of

a sampling scheme with a random sample size. 

2.2. The Selection Scheme 

Suppose we have a working assumption about the Fi

 in equation (1). In particular, suppose Fi

 is believed to be

proportional to xi

g

 for some g between 1/2 and 1. Let us reparameterize the model as 

 yi = $(xi + [3P xk /3P xk

g

]xi

g

,i), (5)

where (again) E(,i *xi

) = E(,i,j *xi

, xj

) = 0 (ij), and (now) Var(,i *xi

) = F

2

 . We have chosen this parameterization so

that F is invariant to changes in scale (units of measurement) of the yi

 and xi

. Notice that when g =1, F

2

 is the relative

variance of yi

 under the model. Thus, F

2

 for any g is in some sense a generalized relative variance for yi

. 

Observe that Fi

2

 in equation (4) now equals $

2

 [3P xk /3P xk

g

]

2

xi

2gF2

. Since under the model T . $ 3P xk, the relative

anticipated variance of tC

 is 

 EP

{ E,

[(tC ) T)2

]} 3P xi

2g(Bi

-1) 1)

 ))))))))))))))) . ))))))))))))) F

2

.

 E,

(T2

) (3P xi

g

)

2

Similarly, the asymptotic anticipated coefficient of variance for tC under the model in equation (5) can be defined as

 [3P xi

2g(Bi

-1 ) 1)]1/2

 ACV(tC) = ))))))))))))))) F. (6)

 3P xi

g

Observe that ACV(tC) decreases, all other things held constant, as any of the Bi increases.

The right hand side of equation (6) attains its minimum for a fixed expected sample size, nE = 3P Bi, when Bi =

nExi

g

/3P xk

g

 if all these selection probabilities are bounded by 1. Furthermore, at that minimum, ACV(tC) # F/3⁄4nE. Near

equality holds when all nExi

g

/3P xk

g

 << 1.

Equation (6) further tells us that if we knew F, we could be assured of meeting meeting an ACV target, say, C. We do

this by setting Bi = min{1, nTxi

g

/3P xk

g

} and nT $ (F/C)2. 

We can call nT the “targeted sample size.” The expected sample size, nE = 3P Bi, is less than or equal to nT. Equality

holds only when all the nTxi

g

/3P xk

g are bounded by 1, which we are not requiring. Nevertheless, setting the selection

probabilities at Bi

 = min{1, n T

xi

g

/3P xk

g

} assures ACV (tC

) # F/3⁄4nT

 under the model in equation (5) .
4

In practice F

2

 must be guessed at or estimated from previous data, say by 

 3f wixi

g

ei

2 

 s2 = )))))))))) ,

 3f wixi

g

where f denotes the previous sample, wi

 is the weight for unit i in that sample, ei

 = [3F xk

g /3F xk

](yi

 - bxi

)/(bxi 

g

),

b = 3f wiyi

 /3f wixi

, and F is the previous population. Alternatively, 

 

 (3f wkxk

g

) 3f wi

(yi

 - bxi

)

2

/xi

g

 s2 = ))))))))))))))))))))) .

 

 (3f wiyi

)

2

When the model holds, ei . ,i. That is one justification of our choices for the ei and s2. Another follows. If the

selection probabilities were Bi = n*xi

g

/3P xk

g

 << 1 for all i, then the relative randomization variance of tC as an estimator

for 3F yi under Poisson sampling (which is what NASS uses) would be roughly [3F (yi ) Bxi)

2

/Bi]/(3F yi)

2

, where

B = 3F

 yi

 /3F

 xi

. This can be reasonably estimated with the sample actually drawn by [3f wi

(yi ) bxi

)

2

/Bi

]/(3f wiyi

)

2

 = s2

/n*.

Thus, our choice for defining s2 is in some way robust to model failure. 

We will call the a sample selection procedure where each Bi

 = min{1, nTxi

g

/3P xk

g

} and 1⁄2 #g # 1 “Brewer selection.”

This name applies whether or not the choice of nT

 depends on F in equation (5). 

3. MULTIPLE TARGETS

Suppose we have M target variables, and yim denotes the unit i y)value for the m’th target. Each target has its own

(maybe unique, maybe not) control variable, and xim denotes the unit i x)value for the m’th control. Furthermore,

suppose each target/control pair is believed to obey the following model:

 yim = $m(xim + [3P xkm /3P xkm

g

]xim

g

,im), (7)

where E(,im *xim) = E(,im,jm *xim, xjm) = 0 (ij), and Var(,im *xim) = Fm

2

 for all m.

A set of weights, {ai}, can often be constructed for a sample S that satisfies the M calibration equations

 3S aixim = 3P xim, m = 1, ...., M, 

such that every ai = Bi

-1[1 + OP

(1/on)], where Bi is (again) the selection probability of unit i. Each calibration estimator

tC(m) = 3S aiyim provides a model unbiased estimator for Tm = 3P yim under the model in equation (7).

One potential way to construct these weights is with the formula inspired by linear regression:

 ai

 = Bi

-1 + (3P xk ) 3S Bk

-1xk

)(3S ckBk

-1xk

’xk

)

-1ciBi

-1xi

’, (8)

where xi

 = (xi1, xi2, ..., xiM) is a row vector, and the choice for the ci

 is arbitrary as long as 3S ckBk

-1xk

’xk

is invertible.

Popular choices are ci

 = 1/xi1 when M = 1 (so tC

 becomes t from Section 1), and ci

 = 1 (when one xim is constant across

i). Brewer (1994) suggests ci = (1 ) Bi)/zi, where zi is some composite measure of size across the M controls. We will

return to this question of setting the ci

 in Section 4.

Given target ACV’s (denoted Cm) for all M target variables under the model in equation (7) and known F-values (Fm)

for each variable, we can be assured of meeting these target ACV’s when every

 Bi

 = min{1, max{nT1hi1 

(g), ..., nTmhiM 

(g)}}, (9)
5

where nTm = Cm /Fm, and him 

(g) = xim

g

/3k xkm

g

.

Observe that Bi in equation (9) can also be expressed as

 Bi = max{Bi1, ..., BiM}, (10)

where Bim = nTmhim

(g) is Brewer selection for variable m. Consequently, the selection scheme in equation (10) can be

called “Maximal Brewer Selection (MBS).” This name applies whether or not each target sample size nTm is set equal

to Cm /Fm.

4. APPLYING MBS

4.1. Poisson PRN Sampling

Brewer selection can be shown to minimize ACV(tC) for a fixed nE under the model in equation (5) and conversely to

minimize the expected sample size given a target ACV. Maximal Brewer selection when M > 1 does not necessarily

minimize the expected overall sample size given M target ACV’s. Sigman and Monsour (1995) sketch a method for

determining selection probabilities that are optimal (i.e., expected-sample-size minimizing) in this sense.

Although not optimal, MBS is relatively simple and conveniently flexible when combined with Poisson PermanentRandom-Number (PRN) sampling (Ohlsson 1995 uses the term “PRN;” the concept can be found in Brewer et al. 1972).

In such a design, every population unit i is independently assigned a random number pi ) a PRN ) from the uniform

distribution on the interval [0, 1). Unit i is selected for the sample if and only if pi < Bi.

Poisson sampling, whether employing PRN’s or not, has the well-known property that the joint selection probability of

two distinct units i and k is equal to the product of their individual selection probabilities; that is, Bik = BiBk. This greatly

eases randomization variance estimation. This method of sampling also assures that 3S zi /Bi . 3P zi, since the relative

variance of 3Szi /Bi is less than (3P zi

2

/Bi)/(3P zi)

2

, which is O(1/n) under very mild restrictions on the zi and Bi (see Isaki

and Fuller 1982). 

Poisson PRN sampling furthermore allows us to think of a sample drawn with MBS inclusion probabilities as the union

of M Poisson PRN samples each drawn using the same PRN’s and individual Brewer selection probabilities. This is

convenient when we are interested in estimates of different combinations of target variables in different surveys. 

For example, NASS makes estimates for potatoes in Minnesota in June and December, row crops (e.g., soybeans and

corn) in March, June, and December, and small grains (e.g., wheat and barley) in March, June, September, and

December. It wants to contact the same farms throughout the year, but has little interest in sampling a farm for the

September survey if it has not historically had small grains. Thus, Poisson PRN samples of farms using the same PRN’s

can be drawn for potatoes, row crops, and small grains, each with its own Brewer selection probabilities. The union of

all three is the overall sample in June. Similarly, the union of the row-crops and small-grains samples is the overall

sample in March. Bailey and Kott (1997) discuss NASS’s use of MBS and Poisson PRN sampling in Minnesota in

greater detail. 

Two additional points should be made at this time. One is that NASS actually draws the row-crops sample itself using

MBS with individual row crops (soybeans, corn, etc.) serving as the target variables. The other is that MBS as practiced

by the agency is the result of individual Brewer selections and Poisson PRN sampling. MBS is the cart and the

individual Brewer selections the horse. 

The overall MBS sample may not be the most efficient (expected-sample-size minimizing) way to meet multiple ACV

targets. It is, however, the most efficient way of combining individual Brewer-selected samples. 
6

4.2. A Count Control Variable

One potential target variable in an establishment survey is the number of units in P that still exist during the survey

period. An obvious control variable for this target is unity, which can be assigned to each unit in P. Such a control is

called a “count variable.”

Whether or not the number of units still in existence is really of direct interest to survey managers, setting one component

of xi, say xi0 equal to 1 for all i is a sensible policy. For one thing, it assures us that tC(m) will be randomization unbiased

when yim > 0, but xim = 0; that is to say, when survey managers are surprised that unit i has a positive quantity of target

variable m. 

4.3. Calibration and Variance Estimation

NASS determines its calibration weights by first employing equation (8) with ci = 1 ) Bi. Brewer (1994) calls such a

weighting scheme “cosmetic calibration,” because the estimator can be put in prediction form ( tC(m) = 3S yim + 

(3P xi ) 3S xi)bm, where bm is defined below equation (11)) when xi contains a count-variable component. He argues that

with cosmetic calibration individual weights rarely fall below unity. Weights below unity are deemed undesirable by

many. 

Under the weighting that results from employing equation (8) with ci

 = 1 ) Bi

, when Bi

 is 1, ai

 is also 1. Cosmetic

calibration weights lower than unity, although rare, can still occur. NASS uses an iterative process described below that

has, so far, successfully eliminated all weights less than unity. When plugging ci

 = 1 ) Bi

 into equation (8) produces

an aj < 1, Bj in the equation is set equal to unity, and the equation run again for all i. This process is continued until all

ai $ 1. 

The estimator tC(m) is model unbiased not only under the model in equation (7), but also under the more general model:

 yim = xi(m + uim, 

where (m is an unspecified M-vector, and E(uim*xi) = 0. 

In order to be able to estimate the model variance of tC(m), we need to add the assumptions E(uimujm*xi, xj) = 0, and

E(uim

2

*xi

) = Fim

2

 < 4. In sharp contrast to the design stage, we are allowing the unit variances to be unspecified as long

as they are finite. 

Following the same reasoning that produced equation (3) leads to 

 E,[(tC(m) ) Tm)

2

] = 3S ai

2

Fim

2 ) 23S

 aiFim

2

 + 3P Fim

2

. (3')

When n is large, we can make use of the near equalities 3 S aiFim

2 . 3S Fim

2

/Bi . 3P Fim

2

, and conclude 

 E,[(tC(m) ) Tm)

2

] = 3S (ai

2 ) ai

)Fim

2

 < 3S ai

2

Fim

2

.

For a Poisson sample, the randomization mean squared error of tC(m) is 

 EP[(tC(m) ) Tm)

2

] . 3P ëim

2

(Bi

-1 ) 1), 

where ëim = yim ) xiBm, and Bm = (3P ckxk

’xk

)

-1 3P ckxk

’ykm (since 3S aiyim ) 3P yim = 3S ai

ëim ) 3P ëim = 3S ëim /Bi

 + 

(3P xk ) 3S Bk

-1xk

)(3S ckBk

-1xk

’xk)

-13S ciBi

-1xi

’ëim ) 3P ëim . 3S ëim /Bi + (3P xk ) 3S Bk

-1xk

)(3S ckBk

-1xk

’xk)

-13P cixi

’ëim )

3P ëim = 3S ëim /Bi ) 3P ëim). When the ck are all equal, the vector Bm is often called the “finite-population” or “census”

regression coefficient.
7

An estimator for both the model variance and randomization mean squared of tC(m) is

 v(t

(m)) = 3S (ai

2 ) ai

)eim

2

. (11)

where eim = yim ) xibm, and bm = (3S ckBk

-1xk

’xk)

-1 3S ckBk-1xk

’ykm are the sample analogues of ëim and Bm, respectively.

4.4. The Delete-a-Group Jackknife

The problem with v in equation (11) is that is requires eim to be calculated separately for each target variable. That is

one reason why NASS uses a delete-a-group (DAG) jackknife variance estimator (Kott 1998). The DAG jackknife is

also convenient when estimating the variances of domain totals and of ratios. 

 The Poisson sample is randomly divided into 15 replicate groups, denoted S1, S2, ..., S15 (some groups can have one more

member than others). The complement of each Sr is called the jackknife replicate group S(r) = S ) Sr. NASS then creates

15 sets of replicate weights. For the rth set: ai(r) = 0 when i 0Sr; and 

 ai(r) = ai + (3P xk ) 3S(r) akxk)(3S(r) ckakxk’xk)

-1ci

aixi’

otherwise. This choice assures ai(r) . ai for i 0 S(r) when 15 is deemed large. Moreover, these two equalities will prove

helpful. Under the model, because the ,im are uncorrelated across units, 

 3S ai(r),im ) 3S ai,im = ) 3Sr ai,im + (3P xk ) 3S(r) akxk)(3S(r) ckakxk’xk)

-13S(r) ci

aixi’,im . ) 3Sr ai,im. 

Even without the model, 

 3S ai(r)ëim ) 3S aiëim = ) 3Sr aiëim + (3P xk ) 3S(r) akxk)(3S(r) ckakxk’xk)

-13S(r) ci

aixi’ëim 

 . ) 3Sr ai

ëim + (3P xk ) 3S(r) akxk

)(3S(r) ck

akxk

’xk

)

-13S(r) ciBi

-1xi

’ëim 

 . ) 3Sr ai

ëim + (3P xk ) 3S(r) akxk

)(3S(r) ck

akxk

’xk

)

-13S ciBi

-1xi

’ëim 

 . ) 3Sr ai

ëim + (3P xk ) 3S(r) akxk

)(3S(r) ck

akxk

’xk

)

-13P cixi

’ëim

 = ) 3Sr aiëim

when 15 is deemed large.

The DAG variance estimator for tC(m) is :

 vJ

(tC(m)) = (14/15) 3

15 (3S ai(r)yim ) tC(m))

2

, (12)

which WESVAR (Westat 1997) calls JK1.

It is easy to see that under the model in equation (7) and the error structure assumed above, the model expectation of

vJ(tC(m)) when 15 (as well as n) is assumed to be large is approximately 3S ai

2

Fim

2

 (since 3S ai(r)yim ) tC(m) = 3Sai(r)yim )

3S aiyim = 3S ai(r),im ) 3S ai,im . )3Sr ai,im). 

We sketch below a proof that the randomization expectation of vJ

(tC(m)) is approximately 3P ëim

2Bi

-1 when 3P ëim . 0.

This last near equality obtains exactly when ci = 1/((xi’) for some row vector ( (since then 3P ëim = 3P (xi’ciëim =

(3P cixi

’ëim = 0). In practice, NASS does not deliberately choose a ci

 with this property, however. This can cause the

DAG jackknife to be randomization-biased. NASS sets ci

 = 1 ) Bi

 and includes within xi

 (for calibration purposes) a

component xi0 = 1. Thus, when all the Bi are small, ci . 1 = 1/((xi’) for ( = (1, 0, ..., 0). When some Bi are large, the

randomization mean squared error is smaller than 3Pëim

2Bi

-1

, so whatever small bias in vJ(tC(m)) is caused by 3P ëim not

being near zero is likely to be overwhelmed by 3P ëim

2Bi

-1 being larger than 3P ëim

2

(Bi

-1 ) 1).

Let nr be the size of Sr. When 15 is large, n/nr . 15 and 14/15 . 1. The replicate group Sr can be viewed as a random
8

subsample of S, and qr

 = (n/nr

)3Sr ëim /Bi

 is a nearly randomization-unbiased estimator of 3S ëim /Bi

, which is approximately

0 when 3P ëim = 0. The randomization variance of qr

 with respect to the subsampling is approximately (n/nr

)3S ëim 

2

/Bi

for each r. Now 3S ai(r)yim ) tC(m) . 3S ai(r)ëim ) 3S aiëim . )3Sr ëim /Bi . )qr /15. We can conclude that the randomization

expectation of vJ

(tC(m)) in equation (12) with respect to the subsampling when 3P ëim = 0 is approximately 3S ëim 

2

/Bi

. 

5. MORE ON THE NASS EXPERIENCE

NASS prepares different samples in the various US states. NASS integrated its crops, stocks, and livestock surveys

in the mid 1980s. Stratified simple random samples were drawn using a priority stratification scheme. For example,

Stratum 1 might be large hog farms, Stratum 2 large crop farms that are not large hog farms, and so on, depending on

the priorities of the target variables. Simple expansion estimates were generated from the sample data. Livestock

variables were removed from the integrated Crops/Stocks (CS) Survey in the mid 1990s. 

In the 1997/98 growing year, NASS drew a MBS Poisson PRN sample for the CS in one state, Minnesota. This proved

very successful (Bailey and Kott 1997) . In 1998/99, this selection method was used in four states. By 1999/2000, 14

states had MBS Poisson PRN samples. Plans are to use MBS exclusively in the following year. 

Rather than explicitly adding xi0 to the other xim in selection equation (9), NASS has set a minimum value for Bi

 at roughly

0.01. For the most part, the same control variables have been used in the selection equation and the calibration (equation

(8)), although a count (intercept) variable has been added to every calibration. Figure 1 provides a chart of how many

control variables were used in each of the 14 1999/2000 CS states.

NASS has set g in equation (9) equal to 0.75. Brewer (1999) seems to show a slight preference for g = .6. Table 1 reports

estimated s-values in one state (PA) based on June 1999 survey data and various values for g. Crop and stock target

variables for a single commodity (e.g., corn) use the same control value. One thing to notice is the s seems to increase

as the fraction of the sample with positive x-values (called “the commodity population”) and positive y-values decreases.

The second is that NASS’s choice of g = .75 everywhere needs to be explored more thoroughly. In principle, the best

choice for g minimizes s asymptotically. 

Nonresponse has been handled using the pre-existing imputation scheme, which relies on the old priority stratification.

DAG jackknife variances are estimated treating non-response as a second phase of sampling and pretending that

respondents were reweighted using the priority strata as the reweighting groups. If the models supporting the imputation

scheme are correct, this will (if anything) bias mean squared error estimates upward. 

6. COMMENTS 

The change in the Crops/Stocks Survey from an estimation strategy featuring stratified simple random sampling and a

simple expansion estimator to Poisson PRN sampling with maximal Brewer selection probabilities and a (cosmetic)

calibration estimator has proven very successful at NASS. The Agency is currently exploring the use of the new strategy

in other surveys as well. In the interest of honest disclosure, NASS actually uses collocated sampling (Brewer et al.,

1972) sampling rather than Poisson sampling. This modestly reduces the sample-size variability. Mean squared errors

are estimated as if Poisson samples were drawn. 

Kott and Fetter (1999) show how Poisson PRN sampling can easily to adapted to limit the number of times a single unit

is selected across co-ordinated surveys. Let Bi

 (q) be the unit i selection probability for survey q ( = 1, 2, ... Q). Unit i

is in the sample for survey q when its PRN, pi, is in the interval [Ji,q -1, Ji,q), where Ji,0 = 0, and Ji,f = Bi

(1) + ... + Bi

(f)

. 

For this sequential interval Poisson (SIP) sampling methodology described above to work, Ji,q cannot exceed unity.

Fortunately, it is a simple matter to generalize SIP sampling a bit. We can redefine Ji,f as Bi

(1) + ... + Bi

(f) ) I(i), where I(i)

is the largest integer less than Bi

(1) + ... + Bi

(f). When Ji,q -1 > Ji,q, the interval [Ji,q -1, Ji,q) is similarly redefined as the union

of [Ji,q -1, 1) and [0, Ji,q). 

The larger I(i) the greater the number of survey samples in which unit i can find itself (that number will either be I(i) or
9

I(i) + 1) . This is another reason NASS needs to explore the value at which g in equation (9) is set. The smaller the

value, the less likely a particular unit with large control values with be selected for a sample. 

It may also be that the best choice for g varies by target variable. Worse, Var(,im) % xim

 g may not even be the appropriate

specification. Oddly, this widely used specification began as an approximation of axim + bxim 

2

 (see Cochran, 1963, p.

256), which has prompted the belief that 1⁄2 must be the lower bound of g in practice. In the NASS application, the

quality of control information is better for larger values. Consequently, it is possible that the best g for some target

variables is, in fact, less than 1⁄2.
10

REFERENCES

Bailey, J. T. and P.S. Kott (1997), “An Application of Multiple List Frame Sampling for Multi-purpose Surveys,

Proceedings of the Section on Survey Research Methods, American Statistical Association, pp. 496-500.

Bosecker, R. (1989), Integrated Agricultural Surveys, NASS Research Report Number SSB-89-05, Washington, DC:

National Agricultural Statistics Service.

Brewer, K.R.W. (1963), “Ratio Estimation and Finite Populations: Some Results Deductible from the Assumption of

an Underlying Stochastic Process,” Australian Journal of Statistics, 5, pp. 93-105. 

Brewer, K.R.W. (1994), “Survey Sampling Inference: Some Past Perspectives and Present Prospects,” Pakistan

Journal of Statistics, 10(1)A, pp. 213-233.

Brewer, K.R.W.(1999), “Cosmetic Calibration with Unequal Probability Sampling,” Survey Methodology, forthcoming.

Brewer, K.R.W., Early, L.J., and Joyce, S.F. (1972), “Selecting Several Samples from a Single Population,” Australian

Journal of Statistics, 14, pp. 231-239. 

Cochran, W.G. (1963), Sampling Techniques, 2nd Edition, New York: Wiley.

Deville, J-C. and Särndal, C-E. (1992), “Calibration Estimators in Survey Sampling, Journal of the American Statistical

Association, 87, pp. 376-382. 

Godambe, V.P. (1955), “A Unified Theory of Sampling from Finite Populations,’ Journal of the Royal Statistical

Society, B17, pp. 269-278. 

Isaki, and Fuller, W.A. (1982), “Survey Design Under the Regression Superpopulation Model,” Journal of the

American Statistical Association, 77, 89-96.

Kott, P. S. (1998). Using the Delete-A-Group Jackknife Variance Estimator in NASS Surveys, RD Research Report No.

RD-98-01, Washington, DC: National Agricultural Statistics Service.

Kott, P.S. and Fetter, M.J. (1999), “Using Multi-phase Sampling to Limit Respondent Burden Across Agriculture

Surveys,” Proceedings of the Survey Methods Section, Statistical Society of Canada, forthcoming. 

Ohlsson, E. (1995), “Coordination of Samples Using Permanent Random Numbers,” in Cox, B.G., Binder, D.A.,

Chinnappa, N., Christianson A, Colledge, M.J., and Kott, P.S. (eds) Business Survey Methods, New York:

Wiley, pp. 153-169. 

Royall, R. M. (1970), “ On Finite Population Sampling Under Certain Linear Regression Models,” Biometrika 57, pp.

377-387. 

Särndal, C-E, Swensson, B., and Wretman, J. (1992), Model Assisted Survey Sampling, New York: Springer.

Sigman, R.S. and Monsour, N. J. (1995), “ Selecting Samples from List Frames of Businesses,” in Cox, B.G., Binder,

D.A., Chinnappa, N., Christianson A, Colledge, M.J., and Kott, P.S. (eds) Business Survey Methods, New

York: Wiley, pp. 133-152. 

Westat, Inc. (1997), A User's Guide to WesVarPC

®

, Version 2.1, Rockville, MD: Westat.
11

Figure 1: Number of Control Variables in Each State
12

Table 1: Target Variable Calculations in PA With Different Values for g

Commodity 

Commodity

Population 

Survey:

Response

Rate

Survey:

Positive

Reports

Survey:

% with

Item

s with

 g = 0.5

s with 

g = 0.6

s with 

g =0.75

s with

g = 0.9

Alfalfa Acres 18006 84.3% 372 60.7% 1.26 1.26 1.27 1.31 

Wheat Stocks 8079 84.4% 29 6.2% 16.88 14.93 12.59 10.82 

Barley Acres 5206 84.3% 122 46.0% 1.39 1.40 1.45 1.55 

Corn Stocks 21268 82.4% 314 36.1% 2.64 2.51 2.43 2.47 

Corn Acres 21268 84.3% 559 78.5% 0.75 0.74 0.76 0.81 

Oat Stocks 11824 84.4% 114 22.2% 2.81 2.85 2.95 3.13 

Oat Acres 11824 84.3% 250 54.8% 1.39 1.41 1.47 1.55 

Other Hay 19478 84.3% 446 65.5% 1.30 1.28 1.27 1.31 

Potato Acres 829 84.3% 67 59.4% 0.82 0.82 0.87 0.99 

Rye Acres 4210 84.3% 103 40.5% 1.95 1.98 2.08 2.24 

Soybean Stocks 7030 83.9% 79 18.4% 3.90 3.73 3.56 3.48 

Soybean Acres 7030 84.3% 234 67.1% 0.95 0.95 0.96 1.00 

Tobacco Acres 979 84.3% 9 33.3% 1.33 1.35 1.41 1.48 

Wheat Acres 3836 84.3% 230 67.9% 0.98 1.03 1.14 1.29