Properties of Alternative Sample Design 

and Estimation Methods 

for the Consumer Expenditure Surveys 

John L. Eltinge and Jeffrey Gonzalez, Office of Survey Methods Research, Bureau of Labor Statistics 

OSMR, PSB 1950, 2 Massachusetts Avenue NE, Washington, DC 20212 

Eltinge.John@bls.gov Gonzalez.Jeffrey@bls.gov

Abstract: 

The U.S. Consumer Expenditure Survey (CE) is a large-scale household survey conducted to collect 

information on expenditures at a relatively fine level of classification. Data currently are collected through 

quarterly interviews (the Consumer Expenditure Interview Survey) and weekly diaries (the Consumer 

Expenditure Diary Survey). Due to interest in reduction of respondent burden, improvement of data 

quality, or possible reduction in the cost of data collection, the Bureau of Labor Statistics is carrying out 

some preliminary exploration of alternatives to the current designs of the diary and interview surveys. This 

paper provides an overview of some of the alternatives and outlines several areas of research that will be 

required to evaluate these alternatives. 

Key words: Adaptive sampling; Balanced repeated replication; Constraints; Cost weights; Efficient design 

of survey procedures; Generalized linear model; Global questions; Matrix sampling; Operational risk; 

Survey cost structures; Synthetic data; Two-phase sampling; Variance estimation. 

 

1. Introduction 

1.1 The Consumer Expenditure Survey 

The U.S. Consumer Expenditure Survey is a large-scale household survey carried out to collect data on 

consumer expenditures in the U.S. civilian noninstitutionalized population. The survey is sponsored and 

managed primarily by the Bureau of Labor Statistics, and data collection is carried out by the Bureau of the 

Census. For some general background on the U.S. Consumer Expenditure Survey, see Bureau of Labor 

Statistics (2007). 

 For the current paper, four features of the Consumer Expenditure Survey are of special interest. 

First, the survey uses a stratified multistage sample design, with strata and primary sample units defined on 

the basis of geography. Second, the sample elements are known as “consumer units” or CUs, roughly 

equivalent to households. Third, sample consumer units are assigned to one of two instruments known 

respectively as the “Interview” and “Diary” components. Fourth, consumer units selected the Interview 

component are asked to participate in a total of five interviews in which a field representative collects very 

detailed information about expenditures by the consumer unit. 

1.2 Alternatives to the Current Design of the Consumer Expenditure Survey 

Due to the length of the interview, and the associated potential for concerns related to respondent burden 

and data quality, the Bureau of Labor Statistics is carrying out some initial exploration of alternatives to the 

current CE design. Several of these alternatives involve the use of matrix sampling methods to assign 

subsets of the current instrument to sample consumer units selected for the interview survey. This paper 

develops some notation and simple estimators that could be used for the Consumer Expenditure Survey 

under a matrix-sampling design; and outlines several areas of research that would be required to evaluate 

the potential practical value of matrix sampling designs for the CE, including weighting methodology; 

variance estimation; estimation efficiency; nonsampling errors; and cost issues. 
2. Background, Notation and Simple Estimators for Matrix Sampling Designs 

2.1 Matrix Sample Designs 

To date, matrix sampling has been used extensively in the education-testing literature, and has also been 

used in some survey and administrative-data applications. For some general background on matrix 

sampling designs, see, e.g., Mislevy (1983), Hinkins (1983, 1984), Thomas and Gan (1997), zeger and 

Thomas (1997), Thomas et al. (2006), and references cited therein. In addition, Gonzalez and Eltinge 

(2007) provide a literature review and discuss the potential application of matrix sampling designs to the 

Consumer Expenditure Survey. 

2.2 Some Simple Weighting and Point Estimation Methods 

For the current Consumer Expenditure Quarterly Survey, in which each sample unit receives the full 

instrument, let Yhij equal the m -dimensional row vector of data that one would collect under full response 

from consumer unit hi = K,,1 Jj in PSU h = ,1 KIi , in stratum = K,,1 Hh . In addition, let Uh

be the population of Nh consumer units within stratum h ; let U

H

h

UU h

=1

= be the overall population of 

∑=

=

H

h

NN h

1

consumer units; let Sh be the full set of nh sample consumer units within stratum h ; let 

S be the full set of ∑==

H

h

nn h

1

sample consumer units; define the full-sample selection probabilities π hij 

for each consumer unit ),,( ∈Ujih ; and define the corresponding simple inverse-probability weights 

−1

whij = π hij . In addition, define the population total ∑==

H

h

YY h

1

 where ∑∈

= Uhij h

h YY hij

)(

; and define the 

corresponding population means NYY hhh = / and = / NYY . Also, to simplify notation, assume for 

this paper that selected sample units provide complete responses to all questions that they receive; i.e., that 

there is no “unplanned missing data.” 

Then simple estimators of the vectors of population totals at the stratum and full-population levels 

are, respectively, 

∑∈

= Shij hNY FhFh Yw hijhij

)(

)1,() ˆ , ˆ ( and ∑==

H

h

FF NYNY FhFh

1

) ˆ , ˆ () ˆ , ˆ (

The corresponding estimators of means are Fh NYY FhFh

ˆ / ˆ ˆ = and NYY FFFˆ / ˆ ˆ = . 

 

 Now consider a relatively simple matrix-sampling design under which some sample units (called 

“Group 0”) receive the full instrument; and all remaining sample units receive certain “core” questions as 

well as questions from one of K specialized sub-groups of questions. (In practice, one might consider 

more complex matrix sample designs, with a corresponding increase in the complexity of notation.). In 

parallel with the partition of the instrument, partition the vector of survey variables 

 ),,,( = 10 K YYYY K 
where Y0 is the m0 -dimensional row vector corresponding to the “core” questions; for each 

= K,,1 Kk , Yk is the mk -dimensional row vector corresponding to questions from section k of the 

instrument; and ∑=

=

K

k

mm k

0

. 

In addition, let S k )( be the subset of sample units that received questions from subgroup 

= K,,1 Kk . Thus, the original full sample S is partitioned into K +1 subsamples: U

K

k

SS k

0

)(

=

= . In 

addition, for k = K,,1,0 K define the selection indicators 

α hijk = { 1 if consumer unit (h,i,j) is contained in subsample k ; 0 otherwise } 

Thus, consumer unit (h,i,j) receives section k of the instrument if and only if α hijk = 1 or α hij0 = 1. 

 For the current discussion, assume that the assignment of sample consumer units to one of the 

subsamples K,,1,0 K is based on a random process such that 

= = )1( hijk Pp α hijk

where the probabilities phijk are assumed to be fixed. For k > 0 , define 

hijk hij hijk += ppp 0

* , 

note that *

hijk p is equal to the probability that consumer unit (h,i,j) receives section k of the instrument, 

and define the modified weights 

* * / hijk hij hijk = pww

Then under the matrix-sample design described above, simple design-based estimators for population totals 

and means at the stratum and full-population levels, based only on data from units that received section k 

of the instrument, are 

∑∈

= +

Shij h

NY khkh w hijhijk hijk Yhijk

)(

0

*

)()( ) ( )1,)( ˆ , ˆ ( αα and 

∑=

=

H

h

NY kk NY khkh

1

)()( )()( ) ˆ , ˆ () ˆ , ˆ (

The corresponding estimators of means are )()()( ˆ / ˆ ˆ = NYY khkhkh and )()()( ˆ / ˆ ˆ = NYY kkk . 

 

2.3 Some Simple Imputation-Based Point Estimation Methods 

The estimators in Section 2.2 used only data collected directly from Section k of the instrument. One can 

develop alternative estimators based on imputation of data for Section k . For instance, define 
Yhijk

~ = {the observed Yhijk if α hij0 = 1 or α hijk = 1; an imputed vector otherwise } 

and define the point estimators 

∑∈

= Shij hY kIh Yw hijkhij

)(

)(

~ ˆ , ∑=

=

H

h

kI YY kIh

1

)( )( ˆˆ , NYY FhkIhkIh

ˆ / ˆ ˆ = )()( and NYY FkIkIˆ / ˆ ˆ = )()( . 

 

Four examples of potentially applicable imputation methods are simple mean imputation within strata; 

mean imputation within more refined adjustment cells; regression or ratio-type imputation; or hot-deck 

imputation 

3. Research Issues in Evaluation of Alternative Sample Designs and Estimation Methods 

Detailed development and evaluation of the estimators considered in Section 2 would involve a wide range 

of theoretical and empirical issues, including the following. 

3.1 Adjustment of Calibration Weighting Methods to Account for Subsampling 

Current weighting for the CE involves several steps of nonresponse adjustment and incorporation of 

auxiliary information. For some general background on this methodology, see, e.g., Bureau of Labor 

Statistics (2007), Greenlees et al. (1982), Jayasuriya and Valliant (1996) and references cited therein. 

Extension of these methods to a matrix-sampling design would involve a number of adjustments. For 

example, one would need to adjust the weights to account for subsampling probabilities, as considered in 

the simple estimators in Section 2; and one would need to adjust the current CE weighting-cell structure to 

account for a potentially smaller number of sample units receiving a given instrument component k. 

 In addition, the simple estimators in section 2 above used only data collected directly for 

component k , or imputed for that component. One could develop an alternative set of point estimators for 

stratum and full-population means based on pattern-mixture models. For some general background on 

pattern-mixture models, see, e.g., Little (1993, 1994) and references cited therein. 

3.2 Adjustment of Standard Balanced Repeated Replication Methods of Variance Estimation 

At present, customary variance estimators for the Consumer Expenditure Survey area based on balanced 

repeated replication, in which the original sample is repeatedly partitioned into half-sample replicates, and 

each step in weighting adjustment and estimation is repeated for each half-sample replicate. In practice, 

under the current CE design, the weighting adjustment skips or simplifies some steps due to limitations on 

sample sizes within the half-sample replicate. Under matrix sampling, the issues with limited sample sizes 

are likely to become much more severe. 

3.3 Two-Phase and Adaptive Forms of Matrix Sampling Designs 

The development in Section 2 used a relatively simple form of matrix sampling in which the subsampling 

probabilities are fixed a priori. One could consider more complex forms of matrix sampling in which 

subsampling probabilities are determined adaptively, e.g., based on demographic data collected during the 

first CE interview. Under some conditions, the resulting design would lead to more efficient point 

estimators of expenditure means within specified sections. 
3.4 Estimation of Population Covariance Matrices, Regression Coefficients, and Related Analytic 

Parameters 

In addition to use in production of estimates of mean expenditures, data from the Consumer Expenditure 

Survey are also used in research on modeling of relationships among different types of expenditures, and 

on relationships between consumer-unit expenditures and incomes. One could continue to carry out this 

modeling research through use of data from sample units in Group 0. However, one could also consider 

modeling methods based on extensions of the literature on “synthetic datasets” e.g., Fienberg and McIntyre 

(2005), Reiter (2003, 2004, 2005a, 2005b) and references cited therein. The “synthetic datasets” literature 

developed originally to address issues with microdata disclosure risk, but much of the underlying 

mathematical structure would extend readily to the current matrix-sampling framework. In many of these 

cases, implementation would involve multiple imputation. 

3.5 Evaluation of Burden, Total Survey Error and Costs 

Finally, as noted in Section 1, consideration of matrix-sampling methods for the CE arose in part from 

concerns related to the perceived burden of the current interview component. Thus, evaluation of specific 

matrix-sampling options for the CE would require balanced consideration of empirical evidence related to 

perceived burden, as well as the impact of a matrix-sampling design on the total cost of data collection and 

the overall error components in the resulting data. 

 Evaluation of burden generally would involve both direct measures (e.g., total elapsed time for a 

given interview) and other measures (e.g., debriefings of field representatives or previous respondents). 

Evaluation of total survey error (e.g., sampling error, nonresponse and reporting error) would potentially 

involve classical sample-design approaches; laboratory and field testing of matrix-sampling instruments; 

and evaluation of current patterns of error components. 

 In addition, evaluation of costs would potentially include the relative costs of training field 

representatives on one or more forms of the instrument; survey initiation in wave 1; scheduling and recontact in each of waves 2 through 5; interviewing for incremental minutes during waves 2 through 5; and 

conversion of reluctant respondents and other work related to caseload management, attrition and perceived 

burden. 

Acknowledgements

The authors thank Jennifer Edgar, Karen Goldenberg, Sioux Groves, Mike Hidiroglou, Bill Mockovak and 

Clyde Tucker for helpful discussions of the application of matrix sampling and other design methods to the 

Consumer Expenditure Survey. The views expressed in this paper are those of the authors and do not 

necessarily reflect the policies of the Bureau of Labor Statistics. 

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