AN APPLICATION OF REGRESSION AND CALIBRATION ESTIMATION

TO POST-STRATIFICATION IN A HOUSEHOLD SURVEY

Bodhini Jayasuriya, Richard Valliant, U.S. Bureau of Labor Statistics

Richard Valliant, 2 Massachusetts Av., N.E., Rm 4915, Washington, D.C. 20212

Key Words: General regression estimator, Principal

person method, Replication variance

ABSTRACT

This paper empirically compares three estimation

methods—regression, calibration, and principal person—

used in a household survey for post-stratification. Posttratification is important in many household surveys to

adjust for nonresponse and the population undercount that

esults from frame deficiencies. The correction for

population undercoverage is usually achieved by adjusting

estimated people counts in each post-stratum to equal the

corresponding population control counts typically available

rom an external source such as a census. We will compare

estimated means from the three methods and their

estimated standard errors for a number of expenditures

rom the Consumer Expenditure Survey sponsored by the

Bureau of Labor Statistics in an attempt at understanding

how each estimation method accomplishes this step in

post-stratification.

1. INTRODUCTION

In large household surveys, post-stratification is a

means of reducing mean square errors by adjusting for

differential response rates among population subgroups and

rame deficiencies that often result in undercoverage of the

arget population. In general, the population is subdivided

nto groups (post-strata) at the estimation stage based on

nformation that affect the response variables. The

estimator is constructed in such a way that the estimated

otal number of individuals falling into each post-stratum is

equal to the true population count. Post-stratum

population counts are typically available from an external

census for numbers of persons but not for numbers of

households. If household estimates are needed, a single

weight must be assigned to each household while using the

person counts for post-stratification. Regression

estimators of totals or means accomplish this by using

person counts in each household’s auxiliary data.

Calibration estimation, with a least-squares distance

unction, is closely related to regression estimation but

possibility that each person in a household ma

different weight. The weight associated with the

person is then assigned to the household. Thi

method is difficult to analyze theoretically. The r

estimator discussed in this paper, while easily adju

the population under count, automatically pr

household weight that is not based on any particul

its members. Lemaître and Dufour (1987)

Statistics Canada’s use of the regression estimat

regard.

There are a growing number of precedents fo

of regression estimators in surveys both in the th

literature and in actual survey practice. Statistic

has incorporated the general regression estimato

generalized estimation system (GES) software th

used in many of its surveys. Fuller, Loughin an

(1993) discuss an application to the USDA Na

Food Consumption Survey. One of the attra

regression estimation is that many of the 

techniques in surveys including the post-stra

estimator mentioned above are special cases of r

estimators. It also more flexibly incorporates auxi

than other more common methods. Other works r

regression estimation and post-stratification 

Bethlehem and Keller (1987), Casady and Vallian

Deville and Särndal (1992), Deville, Särndal, and

(1993), and Zieschang (1990).

In this study we compare the regression estim

the PP estimator currently in use at the Bureau 

Statistics (BLS). The ordinary least-squares r

estimator has the disadvantage that it can 

nonpositive weights. A number of ways are sug

the literature on how to overcome this problem. 

the most flexible is the calibration method intro

Deville and Särndal (1992) which can remove any

weights as well as control extreme weight

calibration estimators produced by these new we

also compared to the original regression estimato

PP estimator.

In Section 2, the three different estima

presented. Section 3 is an application of these pr

to the Consumer Expenditure (CE) Survey at B

same setting as in Zieschang (1990). We com
2. REGRESSION, CALIBRATION, AND

PRINCIPAL PERSON ESTIMATION

First, we give a brief introduction to the regression

estimator. A sample s of size n is selected from a finite

population U of size N . Let the probability of selection of

he i

th unit be p i . The sample could be two-stage and the

unit could be either the primary sampling unit or the

econdary sampling unit. There is no need here to

complicate the notation with explicit subscripts for the

different stages of sampling. Let the variable of interest be

denoted by y and suppose that its value at the i

th unit, yi

 ,

s observed for each i s Î . Assume the existence of K

auxiliary variables xx x 1 2 K ,,, K whose values at each i s Î

are available. Define xi = (, ,, ) xx x ¢ i i iK 1 2 K , for each i UÎ ,

where xik denotes the value of the variable xk at unit i .

Let X = ¢ ( ,, ) X X 1 K K denote the K-dimensional vector of

known population totals of the variables xx x 1 2 K ,,, K . The

egression estimator is then motivated by the working

model x :

yx x x i i i K iK i = + ++ + bb b e 11 2 2 K (2.1)

or i N = 1, , K . Here, b b 1 , , K K are unknown model

parameters. The ei are random errors with

Eξ ξ () () i ii ε εσ = = 0 2 and var for i N = 1, , K . The term

“working model” is used to emphasize the fact that the

model is likely to be wrong to some degree. In the CE, yi

might be the total food expenditures by the consumer unit

CU) and the xik 's might be various CU characteristics like

numbers of people of different ages, or CU income, that

have an effect on the CU’s expenditure on food. The

variance of expenditures might be dependent on CU size so

hat having si

2

 proportional to the number of persons in

he CU might be reasonable. Then, a linear regression

estimator of the population total of y is defined to be

$$ $ $ yR = +- y ¢ p p b g X x b (2.2)

where y$p denotes the p -estimator (or Horvitz-Thompson

estimator) of the population total of y , i.e., y ay $ i i i s p = åÎ ,

where ai i = 1/ p . Also, x$ $ , , $ pp p = ¢ x x 1 K K b g is the vector

of p -estimators of the population totals of the variables

x x x 1 2 K ,,, K and 

$ $ , , $ b = ¢ = L ¢

N

M O

Q

P Î

-

Î

b b å å s s

1 2

1

K K 2

iii

i s i

iii

i s i

a ay e j xx x . (2.3)

Even if model (2.1) fails to some degree, y$R will still have

easonable design-based properties because, even though

h dl i i h i $ id i

The regression estimator y$R can also be expres

weighted sum of the sample yi's with i th weight,

w a

a

i i

iii

i s i

i

i

= +- ¢F ¢

H

G I

K

J

L

N

M

M

O

Q

P Î P

-

1 å 2

1

X x 2

xx x $ p s s

b g .

From (2.4) it is easily seen that the known populat

are exactly reproduced for the auxiliary variables.

The estimator of b in (2.3) does not accoun

correlation among the errors in model (2.1). In 

populations, units that are geographically near ea

e.g., CU's in the same neighborhood, may be co

Using a full covariance matrix V may be mo

optimal (e.g., see Casady and Valliant 1993 

1992). Though use of a full covariance matrix

lower the variance of $b, the elements of V will d

the particular y being studied, and estimation 

generally a nuisance. Consequently, it is intere

practical to consider the simple case of V = diag(

leads to (2.2). Note that when the design

var ($ ) p R y is estimated, it will be necessary to use 

that properly reflects clustering and other

complexities.

The regression estimator has the disadvantage

weights can be unreasonably large, small or even 

The calibration estimators of Deville and Särnda

introduced next, add constraints to restrict the si

weights. Calibration estimators are formed by min

given distance, F, between some initial weight

final weight, subject to constraints. The constr

involve the available auxiliary variables thus inco

them into the estimator. The regression 

presented above is a special case of the c

estimator in which F is defined to be the general

squares (GLS) distance 

F w a ac w a i i ii i i ( , ) ( / )/ = -1 2 2 for i n = 1, , K , w

known, positive weight (e.g., c c ii i = = s

2 or ) 1 a

with unit i , and wi, the final weight. The tota

distance Fw a ( ) i i i s , ∑∈ is minimized subject

constraints, wi i i sx X åÎ = . In this form, the weig

regression estimator of the population total of y

(2.4) can be written as,

w agc i ii i = ¢ - ( ) 1l x

for i n = 1, , K where

gu u () , = +1

for u ÎÂ and l is a Lagrange multiplier evaluat

minimization process. The calibration weights c

the unreasonably extreme values resulting in
chosen in such a way as to reflect the desired restrictions

on the weights. Choosing L>0 ensures that the weights

are positive, and U is picked to be appropriately small to

prohibit large weights. The calibration weights must be

olved for iteratively; one easily programmed algorithm is

given in Stukel and Boyer (1992).

In most household surveys, post-stratification serves

primarily as an adjustment for undercoverage of the target

population by the frame and the sample. In the U.S., there

are no reliable population counts of households to use in

post-stratification. Consequently, population counts of

persons are used for the post-strata control totals. This

disagreement in the unit of analysis (the household) and the

unit of post-stratification (the person) when a household

characteristic is of interest led to the development of the

PP method that is used in the CE and Current Population

Surveys.

In the PP method described in Alexander (1987), a

household begins the weighting process with a single base

weight, ai , that is then adjusted for nonresponse. The

adjusted weight is assigned to each person in the household

and the person weights are then further adjusted to force

hem to sum to known population controls of persons by

age, race, and sex. This last adjustment can result in

persons having different weights within the same

household. The household is then assigned the weight of

he person designated as the "principal person" in the

household. This method has an element of arbitrariness

and is difficult to analyze mathematically. The regression

and calibration estimators can be formulated in such a way

hat population person controls are satisfied, all persons in

a household retain the same weight, and no arbitrary choice

among person weights is needed to assign a household

weight.

3. AN APPLICATION

We compare the three estimators (i.e., regression,

estricted calibration (with L=.5, U =4), and principal

person) by an application to the estimated means and their

estimated standard errors for a number of expenditures

rom the CE Survey sponsored by the Bureau of Labor

Statistics.

The CE Survey gathers information on the spending

patterns and living costs of the American consumers.

There are two parts to the survey, a quarterly interview and

a weekly diary survey. The Interview Survey collects

detailed data on the types of expenditures which

espondents can be expected to recall for a period of three

n = 5156 CU’s were used. The CE Survey’s prim

of analysis is the consumer unit, an economic fam

a household. A consumer unit (CU) consists of in

in the household who share expenditures. Thus, t

be more than one CU in a household.

Five different sets of auxiliary variables were

They were chosen by testing the adequacy of mo

for the selected expenditures with different combin

the available auxiliary variables. The 56 post-str

on age/race/sex currently in use in the CE were 

The combinations of auxiliaries used to form the

weights are given in Table 1. The number of 

variables in each model is given within parenthese

on this information, weights (2.5) were computed

given in (2.6)—regwts—and (2.7)—calwts. For

regression and restricted calibration weights, w

equal to the adjusted base weight, i.e., 1/ p i

nonresponse adjustment.

Table 1. Weights and their corresponding auxiliary

variables. Number of cells are in parantheses.

Weight Auxiliary Variables

regwts0 age/race/sex (56)

regwts1 inter., age/race/sex, region, urban×region (18)

regwts2 intercept, age/race/sex, region, urban×region,

age of reference person, housing tenure, family

income before taxes (24)

calwts0 age/race/sex (45)

calwts1 inter., age/race/sex, region, urban×region (18)

calwts2 intercept, age/race/sex, region, urban×region,

age of reference person, housing tenure, family

income before taxes (24)

calwts3 intercept, age/race/sex, region, urban×region,

family income before taxes (truncated at

$500,000) (19)

calwts4 intercept, age/race/sex, region, urban×region,

age of reference person, housing tenure (23)

PP age/race/sex (56)

For this application, the population totals nec

evaluate X = (,, ) X X ¢ 1 K K were obtained mostly 

1990 Census figures projected to 1992 and the

Population Reports published by the U.S. Burea

Census.

3.1 Comparisons of Weights and Estimated CU

Counts

Alth h th d t d d
eplicate weights, nearly half the sets for each of regwts0,

egwts1 and regwts2 had some negative weights though

he maximum number of negative weights for any replicate

was 3. The negatives are a potential cause of inflated

tandard errors, since the negative weights will be offset by

arge positive weights in order for the fixed population

control totals to be met in every replicate. Calwts, which

estrict the deviation from the base weights by choosing

L and appropriately, (in this instance, U L=0.5 > 0)

naturally did not produce any negative weights.

On examining scatter plots (not shown here) comparing

ome of the different weights to each other, the PP and

egwts0, while being substantially different from each

other, exhibited final weights that can be considerably

different from the adjusted base weights. The adjustments

can be either up or down. A less variable set of

adjustments was apparent in regwts1, calwts0, and calwts1.

Calwts1 and calwts4 were quite similar and both were

close to regwts1. The two sets of weights that involve the

quantitative variable family income before taxes, calwts2

and calwts3, were closely related. Some CU’s had calwts2

values larger than 60,000 but had calwts0, calwts1, calwts4

< 30,000. These CU’s all had family incomes before taxes

of a quarter of a million dollars or more. Thus, the

nclusion of that variable in the calibrations did have a

ubstantial impact on some units. We did use a control

only on the grand total income; having controls by income

classes might have changed the weights on some of these

cases.

Figure 1. Four sets of weights plotted against adjusted

base weights. Reference lines correspond to L=.5 and

U=2.

indicate that the PP weights and regwts0 do not

to the restriction a wa i ii /2 2 £ £ .

Previous studies at BLS regarding ge

regression estimation in the CE had concluded

number of single person CU’s was under estimated

compared to the estimate produced by the PP met

found minimal evidence of that phenomenon here

indicated by the ratios shown in Table 2. It con

ratio of the estimated number of CU’s under the a

procedures to that of the PP estimation procedure

of CU.

Table 2. Estimated counts in thousands of CU’s b

size for PP weights and ratios of other estimated c

the PP weights estimates. Ratios greater than 1.02

than 0.98 are highlighted.

Weights CU Size

1 2 3 4 5+

PP 28,784 30,680 15,409 15,068 9,993

regwts0 0.96 0.99 1.01 0.99 1.02

regwts1 1.00 1.00 1.02 0.98 0.99

regwts2 1.00 1.01 1.01 1.01 0.97

calwts0 0.96 0.99 1.00 1.00 1.02

calwts1 1.00 1.00 1.02 0.98 0.99

calwts2 0.99 1.01 1.01 1.00 0.97

calwts3 0.98 1.02 1.01 1.01 0.96

calwts4 1.01 1.00 1.01 0.99 0.99

A similar table constructed by Composition of CU

that while regwts0 and calwts0 estimato

substantially different from PP for the category On

1+ children, for single person CUs they were not.

3.2 Precision of Estimates from the Different M

Although comparison of weights is instruc

methods must ultimately be judged based on the

estimated CU means and their precision. The 

errors of these estimators were computed via the

of balanced half sampling (BHS) using 44 repl

currently implemented in the CE for the PP estim

BHS estimator is constructed to reflect the stra

and the clustering that is used in the CE. 

expenditure estimates from the CE Survey are 

for various domains of interest, we computed th

and the standard errors for a few chosen domain

For each of these, the coefficient of variation 

computed and then its ratio to the cv of the PP

was calculated
expenditures, and for each of the following domains: Age

of Reference Person, Region, Size of CU, Composition of

Household, Household Tenure, and Race of Reference

Person.

Table 3. Ratios to CE cv to cv’s for the different

weighting methods. The minimum ratio is

highlighted in each row.

Expendit

ure

regwts calwts

012012

All Exp. 0.98 0.90 0.79 0.98 0.90 0.78

Shelter 0.93 0.85 0.75 0.93 0.85 0.74

Utilities 1.08 1.03 0.94 1.07 1.03 0.88

Furniture 1.08 1.21 3.52 1.06 1.21 2.58

Maj. ap. 1.08 1.06 1.04 1.06 1.08 1.09

All vehi. 0.90 0.89 0.98 0.91 0.89 0.98

Cars, (n) 0.95 0.91 1.01 0.96 0.91 1.02

Cars, (u) 0.98 0.94 0.96 0.97 0.94 0.97

Gasol., 1.17 1.11 1.03 1.12 1.10 0.99

Health 1.05 0.97 0.86 1.07 0.97 0.85

Educat. 0.92 0.93 1.04 0.91 0.93 1.06

Contrib. 1.01 1.02 1.28 1.01 1.02 1.30

Pers.

ns.

1.00 0.97 1.64 1.01 0.98 1.24

Life, Ins. 1.08 1.02 1.53 1.08 0.98 1.38

Pensions 1.00 0.99 1.75 1.01 0.99 1.34

n addition, ratios for all CU’s, i.e., the total across the

domains, were computed for each expenditure and those

or regwts 0, 1, 2 and calwts 0, 1, 2 are shown in Table 3.

For All Expenditures, regwts2 and calwts2 with ratios of

79 and .78 provide substantial reduction in cv compared to

PP. For less aggregated expenditures, regwts1 or calwts1

provide reasonably consistent improvements over PP

without the losses incurred by some of the other weights

or expenditures like Furniture, Personal insurance and

pensions, and its subcategory Pensions and social security.

A trellis plot (Cleveland 1993) of the cv and mean

atios for calwts0 and calwts1 by age of reference person is

given in Figure 2. Calwts0 is pictured because it is the

nearest calibration equivalent to the current method of

post-stratification. Calwts1 appears to be the best of the

alternatives we have examined in the sense of improving

he All Expenditures estimates while providing consistent

performance for individual expenditure groups. In each

panel of the plot a vertical reference line is drawn at 1, the

point of equality between the calibration results and those

or the PP method The lower tier in the plot presents

ratios tend to be less than 1, for most dom

expenditures, and calwts1 is somewhat better than

Calwts2 and calwts3, which used family incom

taxes as one of the auxiliaries, had somewha

performance for domains, sometimes makin

improvements over PP but occasionally showin

losses. This is connected to the nature of th

income variable which had a substantial number

with negative and zero values. These CU’s v

usefulness of this variable in predicting expenditur

Taking all of the above into consideration, 

calwts1 and calwts4 can be deemed a clear imp

over the PP estimator. Calwts1 has the advantag

negative weights over regwts1. Since calwts4 re

auxiliary variables as opposed to calwts1’s 

recommend calwts1 over all the other types of we

have considered.

4. CONCLUSION AND FUTURE RESEA

The objective of this study was to in

alternatives to the principal person method for

household weights that did not depend on the w

one single member of the household. Different

weights based on the regression estimation proced

presented and their relative merits evaluated. R

estimation incorporates the current surve

stratification methods in which the weighted su

persons in each post-stratum is forced to be equ

independent census count of that number. 

accomplished via auxiliary variables that are inco

into the regression model. It also automatically 

for each sample household a weight that does no

on any single one of its members. In order to elim

undesirable negative weights that can result from

least-squares regression estimation, calibration e

were adapted to the present problem. The c

estimation procedure has the flexibility to res

possible deviation of each final weight from its ba

while adhering to the properties discussed above

particular allows the constraint of positive weigh

calibration weights are easily computed via matrix

software like S-PlusTM.

Overall, the ordinary regression estimator 

calibration estimator both appeared to be an imp

over the Principal Person estimator in terms

coefficient of variation. For the future, the c

estimators can be further refined by using the prop

regression estimation to choose the auxiliary varia
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Figure 2. Ratios to CE of cv’s and means for two weighting methods by age of reference person.