A Simple Solution For Adjusting 

Weights in a Complex Sample

Quatracia Lucky 

Yeng Xiong, Ph.D.

Any opinions and conclusions expressed herein are those of the author(s) and do not reflect the views of the U.S. Census Bureau. The Census 

Bureau has reviewed this data product to ensure appropriate access, use, and disclosure avoidance protection of the confidential source data 

(Project No. P-7529180, Disclosure Review Board (DRB) approval number: CBDRB-FY23-ESMD010-041)
Acknowledgements 

Jacklyn Jonas 

Jacob Sandruck

Daniel Whitehead
Outline

• Survey background

• Research objective/requirements

• Considerations

• Simulation Study and recommendations

• Conclusion
Annual Integrated Economic Survey (AIES)

• Newly developed survey integrating seven industry-specific surveys to 

measure the economic activity in the US at the national and varying 

subnational levels

• Covers a wide range of economic industries, which are determined by the 

North American Industry Classification System (NAICS)

• National estimates at NAICS6 level 

• 445210 – Meat Markets

• 445220 – Fish and Seafood Markets

• 445291 – Baked Good Stores

• Subnational estimates at NAICS3 level

• 445 – Food and Beverage Stores

• Produces national, regional, divisional, and geographic estimates

• Geographic estimates = state or an assembly of states
Annual Economic Surveys (2022)

Survey Sampling Unit Estimation Levels

Annual Retail Trade Survey (ARTS) Company and EIN National: 2-through-6-digit NAICS

Annual Wholesale Trade Survey (AWTS) Company and EIN National: 2-through-5-digit NAICS by

Type of Operation

Services Annual Survey (SAS) Company and EIN National: 2-through-6-digit NAICS by 

Tax Exempt Status 

Annual Capital Expenditures Survey (ACES) Company National: 3- or 4-digit NAICS 

Report of Organization (COS) Company

M3 Unfilled Orders (M3UFO) Company National: M3 industry category

Annual Survey of Manufactures (ASM) Establishment National: 2-through-6-digit NAICS

Subnational: 2-through 4-digit NAICS 

(50 states plus DC)

Annual 

Integrated 

Economic 

Survey
Sectors sampled

Sector

Manufacturing 31-33

Wholesale Trade 42

Retail Trade 44-45

Transportation & Warehousing 48-49

Information 51

Finance & Insurance 52

Real Estate & Rental & Leasing 53

Professional, Scientific, & Technical 54

Management of Companies & Enterprises Excluded 55

Administrative & Support, & Waste Management & Remediation 56

Educational 61

Health Care & Social Assistance 62

Arts, Entertainment, & Recreation 71

Accommodations & Food Services 72

Other Services (except Public Administration) 81
Research Objective

• Poststratification to correct for the fact that we were not 

able to sample at the state or industry level

• Adjust national estimates to frame national total

• Adjust geographic estimates to geographic frame total

• Conduct empirical research to develop a recommendation 

for weight adjustment procedures to ensure AIES tabulations 

additivity requirements are met

7
Additivity Requirements

National Industry Estimates

sample-based national industry estimates of measure of size (MOS) = national frame totals

Geographic Estimates

sample-based geographic industry estimates of MOS = geographic frame totals

sample-based division industry estimates of MOS = divisional frame totals

sample-based region industry estimates of MOS = regional frame totals

National Level Summations (NAICS3)

NAICS3 aggregated national industry estimates = aggregated geographic estimates

8
Weight Adjustment Considerations

• HT estimator for a population total Y

෠

= σi=1

n diyi

• di = 1/πiis the inverse of the inclusion probability of company i

• diis the design weight as it is directly obtained from the sampling design

• Using known auxiliary information, such as population totals (T), 

modify the basic design weights such that

T෠ = ෍

i=1

n

wiyi

is equal to T.

• wiis the adjusted weight that satisfies the equation.
AIES Considerations 

• Have three administrative variables (payroll, employment, and 

receipt)

• Payroll is used as the measure of size (MOS) in the sample design

• Missing data for the other two

• All adjusted weights are non-negative
%SurveyCalibrate: Methods Tested

11

Method Pro(s) Con(s)

Linear • Runs quickly

• No errors

• Negative weights

Exponential • No negative weights • Seldom runs without 

errors in optimization

• Slow to run

Logistic • Could not get to run

• Errors in optimization

Truncated Linear • No negative weights • Seldom runs without 

errors in optimization

• Slow to run

• Weights less than 1
Ratio Adjustment

This ratio adjustment method modifies the company sample design 

weights, di, such that wi = R ⋅ di where the ratio, R, is the population 

(auxiliary) total divided by the sample total

R

෠

h

s(NC) =

σg∈s σi∈s σj∈i xhgij

σg∈s σi∈s σj∈i wixhgij

ratio of the noncertainty frame MOS total to the noncertainty sample estimates 

of the MOS total within s (s is the domain k for national estimates and s is the 

domain for subnational estimates)
Simulation

• 9 hand-selected industries at the NAICS3 level

• Less than 3,000 companies in the sampling stratum

• 5,000 random samples per industry

• Horvitz-Thompson estimates compared to ratio estimates

• RB =

1

5000

σi Y෠

l,i

m−Yl

Yl

• l is the industry estimate (NAICS6 or NAICS3 by geography)

• m is the estimation method (HT or ratio)

• i is the simulated sample

• Ylis the true value 
Findings

• Ratio estimates had low bias levels

• Comparable to Horvitz-Thompson estimator
Next Steps

• Include additional auxiliary variables

• Repeat the research

• Revisit after at least two years in production

• Include additional industries
Questions, suggestions, 

comments?

quatracia.lucky@census.gov

yeng.xiong@census.gov