“No” Is the Easiest Answer: 

Using Calibration to Assess Nonignorable Nonresponse

in the 2002 Census of Agriculture

Phillip S. Kott

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The 2002 Census of Agriculture adjusted for whole-farm nonresponse by

dividing the potential farms on its list into size classes and then weighting the

respondents within each class to account for the nonrespondents.

Unfortunately, to assign the size-class memberships, a consistent measure

of size was needed for all potential farms not just census respondents.

Subsequently-collected census information sometimes contradicts these

class assignments. By defining indicator variables for census respondents

based on “corrected” class assignments, instrumental-variable calibration

can be used to construct an alternative set of nonresponse weights.

Assuming the response model underpinning these new weights is correct,

the bias from using the original set of nonresponse weights can be assessed.

The potential bias in the estimated farm count is caused by nonfarms on the

list being more likely to respond to the census than farms. 

KEY WORDS: Weighting class; Calibration; Response model; Instrumental

variable; Measurement-error model.

))))))))))))))))))))))))))))))))))))))))))))))))))))))))))

Phillip S. Kott is Chief Research Statistician, Research and Development Division,

National Agricultural Statistics Service, 3251 Old Lee Highway, Room 305, Fairfax,

Virginia, 22030. This paper was prepared for the Joint Statistical Meetings, August 2005,

Minneapolis, Minnesota.
1

1. Introduction

Despite its name, it is helpful to think of the 2002 US Census of Agriculture as a

survey. It’s core was indeed a census of all places capable of producing $1,000 or

more of annual agricultural sales ! what the National Agricultural Statistics Service

(NASS) defined as farms. This core, however, had, to be supplemented to two

directions. First, not all entities on the Census Mailing List (CML) maintained by NASS

responded to the Census of Agriculture. Second, not all places NASS defined as farms

were on the CML. 

To compensate for nonresponse and undercoverage, Census records were

reweighted. Unlike most surveys, the completed records in the 2002 Census of

Agriculture had original weights of 1. These weights were adjusted first to account for

the nonresponse and then the undercoverage. How both of these were done is

explained in some detail on the NASS web site (Kott, 2004a).

Our attention here will be focused on the adjustment for nonresponse in the

2002 Census of Agriculture. NASS divided the entities receiving Census forms into

mutually exclusive “response groups” based on what NASS believed to be each entity’s

county of operation, its size class as measured by expected sales, and whether or not

the entity responded to an agency survey since 1997. Potential farms above a certain

expected size or without expected sales information were removed from this

categorization and handled separately, as were entities whose forms were returned as

undeliverable by the Post Office. 

Reweighting within mutually exclusive groups is a well-known and much-used

procedure. See, for example, Lohr (1999, pp. 266-267). There are two ways to defend

this practice theoretically. One is with a response model, in which every unit in a

groups is assumed to be equally likely to respond, irrespective of its Census item

values. This approach, known as quasi-randomization, treats the response

(nonresponse) process as a phase of random sampling. 

The alternative defense is to assume a prediction model for each of the item

values on the census. The response/nonresponse mechanism is assumed to be
2

ignorable; that is to say, the same prediction model applies to respondents and

nonrespondents. The unit values for an item are independently distributed random

values with a common mean within each response group. The response groups under

theses assumptions are more correctly thought of as prediction-model groups.

Both the response and prediction models have one obvious defect when used in

the Census of Agriculture. Groups are defined using expected 2002 sales before

enumeration rather than the actual sales reported on the US Census of Agriculture. It

is more reasonable to assume similar behavior from farms in the same actual-sales

group than from entities in the sample expected-sales group. An obvious example: two

farms in the sample expected-sales group but in different actual-sales groups can have

vastly different actual sales, a Census item. 

The single most important Census item is whether an entity meets the definition

of a farm (the item value is 1 if it does, 0 otherwise). Here, again, actual sales is a

much better predictor for the item value than expected sales. 

Why then did NASS use expected sales in creating response groups? The 

answer applies to many survey using reweighting not just the Census of Agriculture. In

order to apply reweighting, the entire sample, or in this case the entire categorizable

population, must be assigned to response groups. Since NASS only knows the actual

sales for respondents, it could not use actual sales in creating groups.

This paper explores an alternative approach to reweighting. Section 2 first puts

reweighting for nonresponse into a general quasi-randomization theory of calibration. A

mild extension allows the set of variables in the weight equation, the predictors of

response/nonresponse, to contain some variables not in set used in the calibration

equation. As long as there are the same number of variables in each set, instrumentalvariable calibration (Estevao and Särndal, 2000) can be employed. Furthermore, the

response-predicting instrumental variables need only be defined for the responding

units. 

Section 3 explores using the instrumental-variable calibration with 2002 Censusof-Agriculture data. Instrumental-variable calibration turns out to have some

undesirable properties in this application. Calibrated weights can be less than 1, and
3

estimated quasi-randomization variances are much higher than with simple reweighting

within groups. Still, under the assumption that the model using the actual sales-size

groups is correct, estimates based on simple reweighting can be biased downward

because potential farms that are not farms are more likely to respond than entities

meeting the definition of a farm. “No” is the easiest answer to give on a survey. 

Section 4 offers some concluding remarks. It looks at a prediction-model where

the distribution of an item value given the calibration variables need not be the same for

respondents and nonrespondents; that is to say, the response/nonresponse

mechanism may be nonignorable. This situation can be handled by assuming a

prediction model where the item values are conditioned on the instrumental variables

rather than the calibration variables. A direction for future research is also discussed. 

2. A Quasi-Randomization Approach to Calibration 

Poststratification

In the quasi-randomization approach to reweighting for nonresponse, each unit k

in the original sample (or population) has a positive Poisson probability of responding to

k

the survey, p . This probability may be a function of covariates associated with the unit,

but whether unit k responds to the survey is independent of whether other units

respond. 

Suppose we have a sample. k Unit k in the sample has original sample weight a . 

k

In the case of the Census of Agriculture, all a = 1. Folsom and Singh (2000) assume

that the unit-k Poisson probability of response has the form:

k k p = p(x *), (1)

where xk is a known row vector of Q covariates, * an unknown column vector of the

same size, and p(z) is a known continuous, monotonic function. The unit Poisson

probabilities in equation (1) are assumed independent of each other (hence the name)

and of all the Census item values.
4

When the popu U j lation total, 3 x, is known, Folsom and Singh argue that * in

equation (1) can be estimated implicitly by finding a set of calibration weights for each j

j j j in the respondent subsample, S, satisfying w = a /p(xg) for all k 0 S, and the calibration

equation:

S j j U j

 3 wx = 3 x. (2) 

Observe tha k k t under this framework using the “true” sampling weight, a /p(x *),

for each k in S would come asymptotically close to solving the calibration equation. 

Consequently, the g that satisfies the calibration equation exactly should be a

consistent estimator for *.

It is convenient to think of the calibration adjustment for nonresponse as the

k k k

factor, f(x g) = 1/p(x g). This factor when multiplied by the original weight, a , results in

k

the calibration weight, w When p(t) has the awkward form p(t) = 1/(1 + t) so that

f(t) = 1/p(t) is linear, Fuller, Loughin, and Baker (1994) point out that the usual linear

calibration weights result. That is because with f(t) = 1 + t, and

g = (3S j axj j U j S j j'x) (3 x !3 ax)', 

-1

k k k U j S j j S j j j k

 f(x g) = f(g'x ') = 1 + g'x ' = 1 + (3 x !3 ax)(3 ax'x) x ', (3) -1

 

and 3S wjxj S j j j U j = 3 a f(xg)x = 3 x. 

Suppose xk takes the form of a vector of response group indicators. It’s q-th

member is 1 when k is in response group q, zero otherwise. Since the groups are

mu k k tually exclusive, all functions of x * are equivalent to a linear function of x (because

k kq q kq q kq q

f(x *) = f(3 x * ) = 3 xf(* ) = 3 x * *). Thus, the seemingly awkward model, 

Q Q Q

pk k k k kq q = p(x *) = (1 + x *) is the same as p = 3 x * *. -1 Q

k If all of the a = 1, as in the 2002 Census of Agriculture, then the calibration

equation is solved when each *q q q * = n /N , the respondent size within group q divided by

the population size within the group. Equivalently, for responding unit k in group q, 

wk q q = N /n . Calibration weighting preserves estimates of the population sizes of
5

potential farms within each group. This type of reweighting is usually given the name

“poststratification.”

Instrumental-variable calibration

Estevao and Sarndal (2000) point out that, in general, the variables in the weight

equa k k ktions, w = a f(x g), need not coincide with the variables in the calibration equation

k

(2). One can replace x with another row vector with the same number of components,

say zk. Finding a vector g that satisfies the slightly-revised calibration equation, 

S j k j U j 3 af(z g)x = 3 x. (4)

is “simply” a matter of solving Q equations with Q unknowns. Such a solution can

often, but not always, be found. 

The k k components of z that are not in x are called instrumental variables. 

Because o kf this, the vector z is called an instrumental-variable vector when it does not

exact kly coincide with x , and weights that satisfy the calibration equation in (4)

instrumental-variable calibration weights. 

Observe that the underlying response model assumption is that the Poisson

probability of unit k responding has the form:

k k k

 p = p(z *) = 1/f(z *);

k

that is to say, response is a function of the components of z . Unlike the components of

xk k , the components of z need only be known for the members of the respondent

sample. Furthermore, calibration weights can often be found without the population 

z U j -total, 3 z , being known. 

In our empirical analysis of the Census of Agriculture in the following section, we

w kill replace components of x with analogously-defined components that substitute

ac k k tual 2002 sales for expected sales. Our new z , like x , will be a vector with a single 

1-valued component and Q!1 0-valued components. Consequently, all calibration
6

factors are essentially linear. They can be written as the following generalization of

equation (3):

k k U j S j j S j j j k

 f(z g) = f(g'z ')= 1 + (3 x !3 ax)(3 az 'x) z '. (5) -1 

Equation (3) k k is the special case where z / x . 

Given a S k k U k calibration estimator 3 w y for 3 y based on Poisson response

probabi k k lities of the form p = p(z '*), Kott (2004b) argues that under certain conditions

a S k k good estimator for the quasi-randomization mean squared error of 3 wy is

S k k k

 v = 3 (w ! w )e , (6) 2 2

 

where 

k k k S j j j j S j j j j

 e = y ! x ( 3 af'(zg)z 'x) 3 af'(zg)z 'y, (7) -1

f(t) = 1/p(t), and f'(t) = df(t)/dt. In the Census-of-Agriculture context under investigation

j

in the next section, all a = 1. Moreover, f(t) is linear, so f'(t) = 1.

 

3. An Evaluation of the Nonresponse-Adjustment Methodology in the 2002

Census of Agriculture

 

As noted in the introduction, NASS removed certain potential farms from the

population before creating reweighting groups for the 2002 Census of Agriculture. In

the analysis to be presented here, these and other problem entities (like those that

turned out to be associated with more than one farms) are removed from the data set

beforehand. 

NASS actually created response groups within counties. When too few potential

farms responded in a putative group, collapsing rules were followed. In order to focus

on the repercussions of using expected-sales in response-group formation and to

avoid sticky small-group problems, the analysis here forms groups at the state level in
7

47 states (NASS used whole-farm imputation rather than reweighting in Alaska, Hawaii,

and Rhode Island.) 

Paralleling the within-county routines actually used by NASS, reweighting is first

done within states by creating five mutually exclusive response groups:

X-Group 1: Expected 2002 sales less than $2,500; 

X-Group 2: Expected 2002 sales between $2,500 and $9,999;

X-Group 3: Expected 2002 sales between $10,000 and $49,999 and previously

reported survey data from 1997 or later;

X-Group 4: Expected 2002 sales greater than or equal to $50,000 and reported

survey data from 1997 or later;

X-Group 5: Expected 2002 sales greater than or equal to $10,000 and no reported

survey data from 1997 or later.

All the units in the data set belonged to one of these five groups. Mathematically, the

covar k k1 k2 k5 kg kg iate x = (x , x , ..., x ), where x =1 when k is in X-Group g and x =0 

o ktherwise. Reweighting within states using x in equation (3) can be compared to

k

instrumental-variable calibration weighting using z in equation (5), with 

zk1 =1 if unit k had reported 2002 sales less than $2,500, 0 otherwise; 

zk2 =1 if unit k had reported 2002 sales between $2,500 and $9,999, 0 otherwise; .

zk3 =1 if unit k had reported 2002 sales between $10,000 and $49,999 and previously 

reported survey data from 1997 or later, 0 otherwise;.

zk4 =1 if unit k had reported 2002 sales greater than or equal to $50,000 and reported 

survey data from 1997 or later, 0 otherwise;

zk5 =1 if unit k had reported 2002 sales greater than or equal to $10,000 and no 

reported survey data from 1997 or later, 0 otherwise. 

The sets o k k f components making up x and z are completely analogously, with

expected sales in the x-definitions replaced by actual sales in the z-definitions. We
8

ca klled the original five response groups that defined the components of x the 

“x-groups.” In a parallel fashion, there are also five mutually exclusive groups related

k

to the components of z . We call them the “z-groups.” 

Reweighting using the x-groups is justified if one assumes that all potential farms

in the same x-group are equally likely to responds. Instrumental-variable calibration

w kith the z is justified if one assumes that potential farms in the sample z-group are

equally likely to respond. The latter assumption is the more reasonable.

One way to summarize the two approaches is to say that the “x-group method”

treats the x-groups as both the response groups and the calibration groups. The latter

are the groups for which the calibration-weighted totals equal the population totals (the

calibration equation (2) holds). The “z-group method” treats the z-groups as the

response groups while the x-groups remain the calibration groups.

The empirical analysis here focuses on one item: the estimated number of farms

(entities capable of having at least $1,000 in agricultural sales in 2002). This is called

the “farm count.” Summary statistics for the two reweighting methods are contained in

Table 1. The biases and relative biases are for the x-groups method assuming the 

z-group method is (asymptotically) unbiased. Estimated standard errors are computed

using the square root of the right hand of equation (6). In these calculations, when the 

x-group (z-group) method is used, it is assumed to be unbiased.

If the probability of response is really constant within z-groups, but we reweight

as if they were constant within x-groups, as NASS did at the county-level for the 2002

Census of Agriculture, then the number of farms (in the categorized population) is

undercounted by as estimated 2.3%. There is a mild overcount of farms with annual

sales of less that $10,000, but that is overwhelmed by the undercount of farms with

more than $10,000 in annual sales, especially among those farms that did not respond

to a NASS survey from 1997 to 2002, what we have called Z-Group 5). 

Why this happens is pretty obvious. Using the x-group methodology, the

average response rate for farms in the Z-Group 5 is estimated to be 80%. Using the 

z-group methodology it is only 55%. Since all potential farms in this group are farms

(they have annual sales above $1,000), the x-group methodology does not adjust their
9

weights enough. Using either methodology, it is clear that the response rate goes

down as sales goes up. With the x-group methodology, however, the proper weighting

is confounded by potential farms that have higher expected sales than actual sales. 

Unlike it analogous z-group, not all entities in X-Group 5 (or 4, 3, or 2) are farms. 

Observe that the estimated standard errors using the z-group method are always

double those of using the x-group method. That may be the price one pays for

creating response groups based on information available only from respondents. 

Table 2 displays the variability of the average weight within z-groups across

states for the two methods as well as the variability of the estimated relative bias from

using the x-group method relative to the z-group method. Observe that the minimum

value of the weight is less than 1 in Z-Group 2. In all, instrumental-variable calibration

results in weights less than unity three out of 5 × 47 = 235 times. This happens in two

states for Z-Group 2 and in one state for Z-Group 1, although in the latter the weight

rounded to 1. As a result of this, the estimated number of farms is less than the

responding number of farms in these groups, an absurdity. By contrast, simple

poststratification never allows the calibration weight to be less than 1.

 Only one out of 47 states has a smaller estimated farm count using the 

z-group method. From this, we can infer that the downward bias in the x-group method

relative to the z-group method is significant. For the majority of states, the estimated

relative bias of the x-group method is in the -1 to -3% range. 

Table 1. Summary Statistics on the Two Methods of Reweighting for Nonresponse on the 2002 

 Census of Agriculture 

 Using x-groups Using z-groups

Z-group Unadjusted Adjusted response estimated Adjusted response estimated Bias Relative

 counts counts rate se counts rate se bias 

 

 1 408243 465059 88 219 446322 91 531 18737 0.040

 2 288040 330097 87 197 329755 87 1401 342 0.001

 3 256280 297589 86 187 309230 83 7321 -11641 -0.039

 4 216022 257410 84 150 268045 81 327 -10635 -0.041

 

 5 53107 66327 80 110 96127 55 344 -29800 -0.449

 

 Total 1221692 1416482 86 256 1449479 84 721 -32997 -0.023

Bias and relative bias are estimates for using the x-group methodology when the z-group methodology is

(asymptotically) unbiased.
10

Table 2. The Variability of the Weights and the Relative Biases Across States

Less than $2,500 in sales: Z-Group 1

 Lower Upper

 Variable Mean Min Quartile Median Quartile Max 

 x-weight 1.14 1.04 1.12 1.14 1.17 1.21

 z-weight 1.10 1.00 1.07 1.10 1.12 1.16

 relative bias 0.04 -0.01 0.02 0.03 0.06 0.14

Between $2,500 and $9,999 in sales: Z-Group 2

 Lower Upper

 Variable Mean Min Quartile Median Quartile Max 

 x-weight 1.15 1.05 1.13 1.15 1.18 1.24

 z-weight 1.17 0.94 1.12 1.15 1.22 1.37

 relative bias -0.01 -0.13 -0.05 -0.02 0.02 0.20

Between $10,000 and $49,999 in sales and appearance on a NASS survey since 1997: Z-Group 3

 Lower Upper

 Variable Mean Min Quartile Median Quartile Max 

 x-weight 1.17 1.06 1.13 1.17 1.20 1.26

 z-weight 1.22 1.06 1.16 1.19 1.27 1.53

 relative bias -0.04 -0.19 -0.06 -0.02 -0.01 0.04

Over $50,000 in sales and appearance on a NASS survey since 1997: Z-Group 4

 Lower Upper

 Variable Mean Min Quartile Median Quartile Max 

 x-weight 1.20 1.06 1.17 1.20 1.24 1.32 

 z-weight 1.25 1.06 1.19 1.23 1.30 1.52

 relative bias -0.04 -0.15 -0.06 -0.04 -0.02 0.09

Over $10,000 in sales and no appearance on NASS survey since 1997: Z-Group 5

 Lower Upper

 Variable Mean Min Quartile Median Quartile Max 

 x-weight 1.22 1.05 1.18 1.22 1.26 1.40 

 z-weight 1.70 1.22 1.43 1.65 1.85 2.72

 relative bias -0.26 -0.52 -0.32 -0.25 -0.17 -0.06

All farms

 Lower Upper

 Variable Mean Min Quartile Median Quartile Max 

 x-weight 1.16 1.05 1.14 1.16 1.19 1.24

 z-weight 1.19 1.05 1.15 1.19 1.22 1.26

 relative bias -0.02 -0.06 -0.03 -0.02 -0.01 0.01

x-weight is the average weight – adjusted count/unadjusted count -- in a state using the x-groups method; 

z-weight uses the z-group (instrumental-variable calibration) methodology; 

relative bias is the estimated relative bias in a state of the x-group methodology assuming the z-group

methodology is (asymptotically) unbiased.
11

4. Concluding Remarks

The introduction discussed two theoretical justifications for handling

nonresponse by reweighting within mutually exclusive groups: response modeling and

prediction modeling. In the former, every unit in the same response group was

assumed equally likely to respond to the survey (or, in the case analyzed in depth here,

the census). In the latter, the unit values for an item of interest were assumed to have

a common mean within response groups (which more accurately could be called

“prediction-model groups” in this context). 

Under either justification the adjusted method was the same: each unit in the

original sample (or, in the case here, the categorized population) was assigned to one

of the response groups, and the respondents were reweighted so that the sum of the

new weights among respondents within groups equaled the sum of the old weights

among both respondents and nonrespondents. 

In Section 2, a instrumental-variable calibration methodology was developed and

justified under a response model in which, again, every unit in a response group was

equally likely to respond. Now, however, only the responding units needed to be

assigned to a response group. This was made possible because there was an equal

number of calibration groups to which every unit in the original sample had been

assigned whether or not it responded. 

A prediction-model approach 

Instrumental-variable calibration can be given a general prediction-model

justification. In brief, suppose the prediction model for a item of interest y has the form:

y =k k z $ k + , , (8)

where E(,k k |z ) = 0 for both respondents and nonrespondents. Moreover, suppose the

cal kibration-variable row vector, x , has an analogous form:
12

xk k k = z ' + > , (9)

where E(>k k |z ) = 0 for both respondents and nonrespondents. 

Equation (9) is often referred to as a measurement-error model. Note that

although E(,k k k k |z ) = 0 and E(> |z ) = 0 are assumed to hold for both respondents and

nonrespondents k k , E(, |x ) = 0 need not be zero for both respondents and

nonrespondents. It is for this reason that the response/nonresponse mechanism is

called “nonignorable” in the title. 

Under the prediction and measurement-error models in equations (8) and (9)

S k k

respectively, the difference between a calibration estimator, t = 3 w y and its target, 

T U k = 3 y , is 

S k k U k t !T = 3 wy ! 3 y 

S k k k k U kk k

 = 3 w[(x ! > )' $ + , ] ! 3 [ ' (x ! > )' $ + , ] -1 -1 -1

S k k k U k k

 = 3 w(, ! > ' $) ! 3 (, ! > ' $), (10) -1 -1

when the weights S k k U k are such that the calibration equation, 3 w x = 3 x , holds, and ' is

invertible. 

Equation (10) appears to support the notion for any calibration estimator would 

be unbiased unde S k kr the prediction and measurement-error models so long as 3 w x = 

3U k x holds. The fault with this reasoning is in treating the calibration weights as

cons ktants given the z . Conventional calibration weights, those not involving

k k k k

instrumental variables, have the form w = a f(x g). They are functions of the x , which

are random variables under the measurement-error model. 

The weights in instrumental-variable calibration are designed to be functions of

k k k

the z rather than the x . Nevertheless, they too are functions of the x through the

ca k k klibration-equation-solving g in w = a f(z g). Consequently, the instrumental-variable

cal kibration estimator is not strictly prediction-model unbiased given the z . 

Nevertheless, it is a consistent estimator under mild conditions so long as g converges
13

to a vector with finite components as the respondent sample grows arbitrarily large. 

Note that although f(.) is used to define g, its inverse, the Poisson response probability

p(.), need not be correctly specified under this prediction-model justification for

instrumental-variable calibration. 

k k k

If both the , and > are uncorrelated across units given z , then under mild

cond kitions, e in equation (7) is approximately:

k k k S j j j j S j j j j

 e = y ! x ( 3 afN(zg)z 'x) 3 afN(zg)z 'y)

-1

k k k k S j j j k k S j j j k k

 = z $ + , ! (z ' + > )( 3 afN(zg)z '[z ' + > ]) 3 afN(zg)z '(z $ + , ) -1

k k k k

 . z $ + , ! (z ' + > )' $ -1

k k = , ! > ' $,

-1

which allows a prediction-model interpretation to the mean-squared-error estimator in

equation (6). 

Caveats and a possible extension

There are two obvious drawbacks to the instrumental-variable calibration

methodology laid out in Section 2. The first is that the theory underpinning it is

asymptotic (since g k k k k k in w = a f(z g) is a consistent estimator for * in p = p(z *)). That

is not much of a limitation when reweighting for nonresponse in the Census of

Agriculture at the state level because both calibration and response group sizes are in

the hundreds. It is a limitation had we tried to apply the methodology at the county

level where some group sizes are small to nonexistent. Since one of the purposes of

the Census of Agriculture is to measure farm activity at the county level, this limitation is

very problematic. In principle, the limitation appears to be shared by simple

poststratification as well, but one can define the fraction of responses in a group as the

exact conditional probability of response in the group and avoid asymptotics entirely. A

group is only too small when its respondent size sinks below one, making estimation

impossible. 

A more general drawback of instrumental-variable calibration is that the
14

me kthodology is limited to a response (or prediction-model) vector (z ) with the same

number of k components as the calibration vector (x ). Chang and Kott (2005) develop

an approach to calibration where the calibration vector is larger than the response

vector. This is made possible because their approach loosens the restriction that the

calibration equation holds exactly. 

Even when the number of components in the response and calibration vectors

coincide, there may be no set of weights in the proper form that satisfy the calibration

equation exactly. For example, suppose one assumes the probability of response in

k k the 2002 Census of Agriculture was logistic (p(z *) = [1 + exp(!z *)] ). Consequently, -1

k k the calibration weight for each farm has inverse-logistic form (w = 1 + exp(!z *)) and

must exceed 1. In Section 3, we saw that the solution of the 5-equation-5-unknown

problems results in some weights less than 1 in three states. It is not hard to show that

these solutions are unique. Thus, no set of weights in inverse-logistic form satisfy the

calibration equation exactly in these states. Chang and Kott’s loosening of the

calibration-equation constraint is appealing in this situation.
15

References

Chang, T. and Kott, P.S. (2005). “Using Calibration to Fit Nonresponse and 

Noncoverage Models.” ASA Proceedings of the Survey Research Methods Section, to

appear.

Estevao, V.M., and Särndal, C-E. (2000). “A Functional Form Approach to Calibration.”

Journal of Official Statistics, 16, 379-399. 

Folsom., R.E. and Singh, A.C. (2000). “The Generalized Exponential Model for

Sampling Weight Calibration for Extreme Values, Nonresponse, and Poststratification.” 

ASA Proceedings of the Section on Survey Research Methods, 598-603.

Fuller, W.A., Loughin, M.M., and Baker, H.D. (1994). “Regression Weighting for

the 1987-88 National Food Consumption Survey.” Survey Methodology, 20, 75-85.

Kott, P.S. (2004a). “Collected Technical Notes on Weighting and its Impact on the

2002 Census of Agriculture.” http://www.nass.usda.gov/research/reports/

COMBINED.pdf

Kott, P.S. (2004b). “Comment on Demnati and Rao.” Survey Methodology, 30, 27-28.

Lohr, S. L. (1999). Sampling Design and Analysis. Duxbury Press, Pacific Grove, CA.