TEMPLATE DESIGN © 2008

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We are interested in drawing inferences about mean expenditures for various categories. The primary statistic of interest is . Suppose we draw a random sample, denoted by S, from the target population, then 

a design-based estimator would be: Since not all information is collected, we can either: (1) make an appropriate adjustment to the sampling weights (Gonzalez and Eltinge, 2008), (2) impute the non-observed information. Under (2), a reasonable estimator would be (where is the imputed value):A useful evaluative tool for judging the performance of this estimator is the variance. The variance of due to three sources of variation (initial sample selection, matrix subsampling, and imputation error) can be written as:

         

                         Ik Ik k

Ik Ik Ik

Ik Ik Ik

YEVE YVE YVYEVE YVE YEV

YEVYVEYV

ˆ

 S P, ˆ S P, ˆ

S ˆ S P, ˆ S P, ˆ

S ˆ S ˆ ˆ

321 32 1321 32 231

231 231

     

Yk

 

 

 

   



 Si

iki

Si

k i YwwY

1

ˆ

Imputation Methods for Adaptive Matrix Samplin

Jeffrey M. Gonzalez and John L. Eltinge

U.S. Bureau of Labor Statistics, Office of Survey Methods Research

Abstract Simulation Setup 

Simulation Results

The GolusesamThe not 

Matrix sampling methods involve dividing a lengthy questionnaire into subsets of questions and administering each subset to subsamples of a full sample. In a panel survey, information about a sample unit can be learned during the first interview and this information can be used both to assign questions and to impute missing quantities at later interviews. Previous research has considered estimators based onavailable cases and simple adjustments to the design weights (Gonzalez and Eltinge, 2008). Here we extend this research by developing an imputation procedure for recovering the data not collected from a sample unit at subsequent interviews. We use data from the Consumer Expenditure Quarterly Interview Survey to explore potential efficiency gains incurred from incorporating these imputation methods into the estimation procedures of an adaptive matrix sampling design.KEY WORDS: Adaptive design, Burden reduction, Multiple imputation, Panel survey, Sample survey, Variance estimation

Background and Motivation

Mathematical Details

U.SOff2 MWa

Je

Ma

92.54 596.87 78.95 (8.9) Utility expenses (e.g., electricity) Utilities65.86 262.86 541.72 (23.3) Various expenses (e.g., pet services, cash contributions) Miscellaneous60.90 277.00 318.72 (17.9) Medical expenses, including medical supplies Medical

Insurance Non-health (e.g., life, auto) 69.84 481.20 177.95 (13.3)Clothing For persons age 2 and over 70.06 259.74 82.94 (9.1)

Variance (SE)

of the Mean1

Mean 

($)

Reporting 

Rates (%) Brief Description Expenditure Category

Table 1: Population Description (N = 10412)

1: The corresponding standard deviations are listed in parentheses2: Ratio of the variance of the imputation-based mean relative to the variance of the full sample mean 

Two Simulation Scenarios• Common features: SRSWOR (n = 2500) from population, randomly allocate each sample unit to 1 of the 5 expenditure categories, 1000 iterations (computing and each iteration)• Scenario 1: Uses Option 1 to impute missing expenditure information• Scenario 2: Uses Option 2 to impute missing expenditure information

YIk

ˆ Yk

ˆ

 

 

 

   



 Si

iki

Si

Ik i YwwYˆ ~ 1

YIk

ˆ

The U.S. Consumer Expenditure Quarterly Interview Survey (CEQ) is an on-going rotating panel survey of a nationally representative sample of addresses that solicits information on the spending habits of American consumers. The data are also used in the calculation of the Consumer Price Index (CPI). Recently there has been increasing concern over (1) declining response rates; (2) high respondent burden; (3) potentially low data quality; and, (4) increasing data collection costs. To address these concerns we explore the use of matrix sampling methods under an adaptive design framework.Definitions• Matrix sampling: To divide a lengthy questionnaire into subsets of questions and, based on some probabilistic mechanism, administering each to subsamples of a full sample• Adaptive design (responsive design): To use information learned about a sample unit during the data collection process with the purpose of tailoring features of the survey administration to that particular unit• Adaptive matrix sampling: To use data collected in the first interview (e.g., expenditure and demographic characteristics) to (1) adjust matrix subsampling probabilities for subsequent interviews and (2) impute information not collected from a particular subsampleHere our focus is on the imputation procedure used when data are 

collected via an adaptive matrix sampling design. 

Interesting Feature of Expenditure DataResponses are often equal zero OR otherwise (approximately) follow 

a continuous distribution. Thus, the ability to impute “zero dollar”expenditure amounts is a desirable property of the chosen imputation procedure. We accomplish this with a two-stage imputation in which we (1) predict the presence or absence of an expenditure and then (2) conditional on that, impute the specific dollar amount. 

Data Source• Data collected from April 2006 to March 2008 using the full CEQCAPI survey instrument; represents 8 calendar quarters• Subset to sample units responding to BOTH interviews 1 and 2 • Identified 5 expenditure categories with varying interview 2 reporting rates, quarterly mean expenditures, and variances (see Table 1)• Demographic information on sample units: family type (describes the relationship among the persons living within the sample unit), housing tenure (own vs. rent), age, sex, and educational attainment of the respondent

where:• are the moments with respect to the original sample selection; • are the moments with respect to the matrix subsampling, conditional on the initial sample, S; and, • are the moments with respect to the imputation procedure, conditional on the matrix subsampling, P, and the initial sample.

   ,  11 VE 

   S ,S  23 23  VE 

   S P, ,S P,  3 3  VE 

Two Central Questions1. What effect will the predictive precision of the imputation procedurehave for the non-observed expenditure sections?2. What is the additional variance reduction obtained by assigningsections to sample units with unequal probabilities?The simulation study will begin to shed light on the first question byconsidering two imputation procedures hypothesized to have differingpredictive precisions. Step 1: Fit to all respondents (where is the probability of the ith unit having an expense for k. Estimate for all 

sample units who were not asked about that expenditure using therelationship: .Step 2: Fit a regression model to all respondents (see Handout), draw

  1,0~ , and impute via Option 1 or Option 2 below:  ik Uni

  k

T Xp iik logit   ik p

ik p

     k

T

k i

T pˆik exp1 Xi ˆ exp X ˆ 1 

Yik

Yik

~

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k ik

T

i

ik

ik k

ik

X p

p

SiY

Y

 if ˆ ˆ

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 if ~

ik

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 if ˆ 0 and 

ˆˆ

0 and ˆ if ˆ

 0 ˆ if 

 if 

~

,1 ik ,1

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ik ,1

ik

YikIntk ik ikInt

T

i

k ik ikInt

T

i

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YX Yp

X Yp

p

Y Si

Y

k

 

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Option 1: Option 2:

1: Theoretical variance (SE) of the full sample mean

Table 2: Scenario 1 Results

Utilities 596.64 75.83 (8.7) 595.05 433.55 (20.8) 1.14Miscellaneous 262.25 500.35 (22.4) 268.06 4509 (67.1) 1.80Medical 277.45 337.49 (18.4) 279.68 2533 (50.3) 1.50Insurance 480.90 184.66 (13.6) 480.71 1031 (32.1) 1.12Clothing 259.82 78.47 (8.9) 260.48 513.79 (22.7) 1.31

Variance 

Component Mean 1 Variance Component Mean 1

Variance 

Ratio2

Full Sample Matrix Sample Expenditure 

Category

Table 3: Scenario 2 Results

Utilities 596.42 83.23 (9.1) 595.11 469.85 (21.7) 1.13Miscellaneous 263.14 570.38 (23.9) 268.76 7217 (85.0) 2.53Medical 277.17 322.06 (17.9) 278.99 2302 (48.0) 1.43Insurance 479.87 171.83 (13.1) 494.88 1469 (38.3) 1.71Clothing 259.75 79.28 (8.9) 259.00 534.97 (23.1) 1.35

Variance 

Component Mean 1 Variance Component Mean 1

Variance 

Ratio2

Full Sample Matrix Sample Expenditure 

Category

1: The corresponding standard deviations are listed in parentheses2: Ratio of the variance of the imputation-based mean relative to the variance of the full sample mean 

Le• As (of 

masta• WexpOp• Dunbin t

Fu• Cord• Thof vcom• W
Imputation Methods for Adaptive Matrix Sampling

(Handout)

Jeffrey M. Gonzalez∗ and John L. Eltinge

U.S. Bureau of Labor Statistics, Office of Survey Methods Research

2009 Joint Statistical Meetings

August 4, 2009

Imputation Procedure Details1

To impute yik, the expenditure amount for the i

th sample unit on item k, we will implement the

following two-step procedure:

Step 1:

Fit the logistic regression model, logit(pik) = x

0

iγk, to all sample units receiving expenditure section

(or item) k. For this model, pik is the probability that the i

th sample unit reports an expense on item

k and x

0

i

is a vector of covariates (family type, housing tenure, and age, educational attainment,

and gender of the respondent).

Using the relationship pik = (1 + exp(x

0

iγk))−1

(exp(x

0

iγk)), estimate pik for all sample units not

receiving expenditure section k. We will denote the estimated probability as ˆpik.

Step 2:

? Option 1:

Fit the linear regression model, yik = x

0

iβk, to all sample units receiving expenditure section

k and reporting a positive expense (i.e., yik > 0). For this model, x

0

i

is the same vector of

covariates as in Step 1.

? Option 2:

Estimate the regression parameters from the following two linear regression models:

1. Fit yik = x

0

iβk to all sample units receiving expenditure section k, reporting a positive

expense (i.e., yik > 0) at the current interview, but a zero-dollar expense during the first

interview (i.e., yint1,ik = 0)

2. Fit yik = x

0

iβ

∗

k + yint1,ikβYk

to all sample units receiving expenditure section k, reporting

a positive expense (i.e., yik > 0) at the current interview, and a positive expense during

the first interview (i.e., yint1,ik > 0)

Now draw, θik ∼ Uni(0, 1). For all sample units not receiving expenditure section k, impute the

non-observed information using either Option 1 or 2 in the following manner:

∗Gonzalez.Jeffrey@bls.gov

1The views expressed here are entirely those of the authors and do not necessarily reflect policies of the U.S.

Bureau of Labor Statistics.

1
Table 1: Imputation Options

Option 1 Option 2

y ̃ik =

(

0 θik > pˆik

x

0

iβˆ

k θik ≤ pˆik

y ̃ik =







0 θik > pˆik

x

0

iβˆ

k θik ≤ pˆik, yint1,ik = 0

x

0

iβˆ∗

k + yint1,ikβˆ

Yk

θik ≤ pˆik, yint1,ik > 0

FCSM 2009

Sensitivity of Inference under Imputation: An Empirical Study

Jeffrey M. Gonzalez and John L. Eltinge

Abstract

Item nonresponse, a common problem in many surveys, occurs when a respondent fails to provide a

response for a survey question. Imputation models can be used to fill in the item missing information

with plausible values. These models are built on assumptions about the nature of the missing

information. Varying the assumptions on the imputation model would likely change the imputed

value. If the primary inferential goal was point prediction of the missing value, then an undesirable

result of the imputation procedure would be variation in the imputed values. Oftentimes, however,

the main analytic goal is estimation of aggregate values, such as population means. Thus, variation

in the individual imputed values is of lesser importance while variation in the final population

estimate moves to the forefront. Therefore, we examine to what extent, if any, the imputation

model assumptions affect the estimation of these aggregate values.

To investigate the sensitivity of inferences when using imputation models built on different assumptions, we provide a simulation study with historical data from the U.S. Consumer Expenditure

Interview Survey (CE). The CE allows an in-depth consideration of the impact of three features

on this potential sensitivity. They are (1) panel survey design; (2) range of expenditure dollar

amounts; and (3) prevalence of certain expenditures (i.e., rare vs. frequently incurred expenses).

The imputation models should account for these special features of the CE. Thus, we develop several imputation models for imputing a variety of expenditures. These expenditures vary in dollar

amount, proportion of item nonresponse, and proportion of respondents with true zero-dollar expenses. We then calculate and compare estimates of population means based on the imputed data.

Finally, we offer a commentary on imputation model parsimoniousness and implementation feasibility.

Key Words: Zero-inflated distribution; Panel survey; Missing data; Regression imputation; Twostage imputation

Reference Cited on Poster

Gonzalez, J.M. and Eltinge, J.L. (2008). Adaptive Matrix Sampling for the Consumer Expenditure

Quarterly Interview Survey. Proceedings of the Section on Survey Research Methods, American

Statistical Association, 2081-2088.

2