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Model-Assisted Domain Estimation 

Dan Liao and Phillip Kott 

RTI International 

Federal Committee on Statistical Methodology (FCSM) 

Research Conference 

November 4-6, 2013 

Washington Convention Center 

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Introduction 

 Domain Estimation: estimation of population quantities 

(e.g. totals or means) for the desired population 

subgroups in a descriptive survey 

 Context: Design-based estimation 

– the randomness is introduced by the sampling design

– mainly used for domains whose sample size is reasonably large 

(for small domains, small area estimation is often used) 

– references of design-based estimation for domains: Yates (1953,

1960), Durbin (1958), Hartley (1959), Lehtonen and Veijanen 

(2009) 

 Use of auxiliary information: model-assisted approach 

(Särndal et al., 1992) 
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Use of Auxiliary Data 

 With high-quality auxiliary information, it is possible to 

obtain better accuracy for domain estimates. 

– accurate 

– moderately or highly correlated with the domain variables

 Different types of auxiliary data 

– population-level aggregates (e.g. from population census, other 

official statistics) 

– unit-level auxiliary data (e.g. from administrative records) 

– domain-level aggregates (e.g. from State registers) 

– intermediate-level aggregates (e.g. from first-phase sample 

surveys) 

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Notations 

Let 
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Two Domain Estimators 

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We are interested in estimating the population mean in the 

domain, 

 U: Md = U dk yk / U dk
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Application 1: Combing Information from 

Administrative Records with Sample Surveys 

Sample Survey 

• X 

• y 

• Design Weight

Calibration Estimator 
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Bias Measure 
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Bias Measure 

If the model is correct in the domain (H0), the idealized test 

statistic: 

 T* = S wkdk

(yk xk

T )/ S

 wkdk 

has expectation (nearly) zero. 

 Estimated test statistic: 

T = S wkdk (yk xk

T b)/ S

 wkdk 

 = S wkdk qk

/ S

 wkdk 

This can be treated as a calibrated mean and the 

estimated variance be computed with WTADJUST in 

SUDAAN. 
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Variance Estimation 
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Example: 2010 Natality Data 

 Data File: 2010 Natality Public Use File 

– Excluding foreign residents 

– Excluding records with missing values in the following variables: 

 DBWT: Birth Weight 

 UBFACIL: Facility Type 

 UPREVUS: Number of Prenatal Visits 

 COMBGEST: Gestational Age 

 MAGER: Mother’s Age 

– Select 1 out of 100 records (to reduce the data size) 

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 Population Size: N= 38,358 

 Variable of Interest (yk): Baby’s Birth Weight 
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Sample Selection 

•14 Strata: 

 FACIL2 (2 facility types) 

 GEST7G (7 gestational age groups) 

n=500 for each stratum in hospital; n=50 for each stratum in the

other facility types

*FACIL2 

1=Hospital; 2=Others (e.g. Freestanding Birthing Center or 

Clinic/Doctor’s Office, Residence) 

*Gest7G 

1=18-36 weeks, 2=37 weeks, 3=38 weeks, 4=39 weeks, 5=40 weeks, 

6=41 weeks, 7=42+ weeks 
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Calibration 


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Domain Estimates: Mother’s Race

 Mother’s Race: Black 

(when domain variable is part of calibration variables) 

Estimator Mean SE

Calibration Estimator Variance estimation accounted for 

calibration (PROC WTADJUST) 

3125.86 45.14

 Variance estimation NOT 

accounted for calibration 

 45.22

Model-Assisted Estimator Proper Variance Estimation 3079.16 44.60

 Naïve Variance Estimation 

(treating ŷ as true value) 

 8.10 

Bias Measure of the ModelAssisted Estimate Variance estimation accounted for 

calibration (PROC WTADJUST) 

0 44.88

P-value of the bias measure: 1.000 
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Domain Estimates: Gestational Age 

 Gestational Age 

(when domain variable is NOT part of the calibration variables) 

Gestational 

Age 

Calibration 

Estimator 

Model-Assisted 

Estimator 

Bias 

Measure* 

P-Value of 

the Bias 

Measure 

Mean SE Mean SE

≤ 36 weeks 2573.59 60.26 2531.91 16.36 -706.15 0.000 

37-38 weeks 3205.85 28.91 3200.72 16.02 66.04 0.020 

39 weeks 3437.19 33.98 3391.67 15.94 149.96 0.000 

40 weeks 3418.95 34.42 3454.89 15.89 139.27 0.000 

41 weeks 3507.68 32.98 3517.14 16.09 233.26 0.000 

≥42 weeks 3490.92 42.46 3450.13 16.01 215.80 0.000 

* Bias Measure of the Model-Assisted Estimate 
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Domain Estimates: Mother’s Age

 Mother’s Age 

(when domain variable is correlated with the calibration variables) 

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Mother’s Age Calibration 

Estimator 

Model-Assisted 

Estimator 

Bias 

Measure* 

P-Value of 

the Bias 

Measure 

Mean SE Mean SE

≤ 19 3103.55 62.86 3139.02 34.91 -98.04 0.115 

20-24 3221.35 33.55 3216.27 21.47 -12.26 0.709 

25-29 3343.99 35.55 3298.23 16.37 65.75 0.062 

30-34 3318.32 35.05 3313.47 20.03 5.05 0.885 

≥35 3289.98 44.98 3279.74 31.90 -55.42 0.202 

* Bias Measure of the Model-Assisted Estimate 
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Conclusions 

 Design Consistency 

– When computing a domain estimate, a calibration 

estimator is design-consistent. 

– A model-assisted estimator is asymptotically designconsistent, only when domain variable is a component 

of the calibration variables. 

 Bias Measure for Model-Assisted Estimator 

– When the domain variable is NOT a component of the 

calibration vector, a proper test should be performed 

to assess the potential magnitude and significance of 

the bias of the model-assisted estimate. 
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Conclusions (continued) 

 Variance Estimation 

– When the domain variable is a component of the 

calibration variables, the calibration estimator 

performs similarly to the model-assisted estimator 

(both the estimates and SE of estimates are similar; 

both methods are asymptotically unbiased). 

– When the domain variable is NOT a component of the 

calibration variables, if the model-assisted estimate is 

NOT biased, then the model-assisted estimate has 

smaller SEs (i.e. more efficient) than the calibration 

estimate. We can test for a potential bias. 

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Application 2: Two-Phase Sample Survey 

2

nd Phase 

• X 

• y 

• 2nd phase weight

Calibration Estimator 

 
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Contact Information 

 Dan Liao 

dliao@rti.org

 

 Phillip Kott 

pkott@rti.org

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