Winsorizing Low Inclusion Probabilities 

from an Unequal Probability Sample in 

a Multi-Purpose Annual Survey

Nikki Czaplicki and Yeng Xiong

Economic Statistical Methods Division, U.S. Census Bureau

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 for unauthorized disclosure of confidential information and has approved the disclosure 

avoidance practices applied. (CBDRB-FY23-ESMD010-001).
Obtain

Allocations

Compute 

Inclusion 

Probabilities

Select 

Stratified 

Sequential 

Samples

Compute 

JOINT 

inclusion 

probabilities

Obtain 

adjusted 

sampling 

weights

Compute 

Estimates & 

Variance 

Estimates 

Assess 

Reliability

Stratification

Overview of AIES Sample Design Process

Sector Processing

Complexity 

Certainty 

Companies

Noncertainty 

Companies

Sector 

11

Sector 

21

•••

Sector 

72

Sector 

81

Sampling 

Frame
AIES sample inclusion probabilities 

• Recall that the key estimates for AIES will be industry totals at both 

the national and subnational level.

• Strata are defined by the cross classification of three-digit NAICS by 

state (or balance of region).

• Like most economic data, the AIES data come from a skewed 

distribution with most of the total coming from a relatively small 

number of companies. 

• Inclusion probabilities are assigned in a probability proportional to 

estimated size (PPES) design. 
Probability Proportional to Size

In PPS, the measure of 

size (MOS) is assumed to 

be linearly related to the 

survey variable. 
Probability Proportional to Size

At the lower end of the 

distribution, the linear 

relationship no longer holds.

These companies are 

essentially exchangeable on the 

survey variable. 
Probability Proportional to Size

Are unequal inclusion 

probabilities necessary for 

these companies?
Probability proportional to estimated size

• PPES design will assign unique inclusion probabilities to essentially 

exchangeable companies

• These inclusion probabilities can be extremely small for some companies

• Practically excluded from selection

• Large sample weights 1

ππ

, Max > 670,000!!!

• Large variance estimates 
Winsorization

• We want to apply a one-sided Winsorization procedure to equalize the 

inclusion probabilities of exchangeable companies. 

• Winsorization is a robust estimation technique that “replaces extreme 

values with less extreme values, effectively moving the original extreme 

values toward the center of the distribution” (Mulry et. al., 2014)
Winsorization of sample weights

• It is common practice for sample weights to be capped at some 

predetermined threshold after sample selection

• Sample weights ≠ 1

ππ - > estimation bias

• Threshold infrequently updated

• One-size fits all

• The proposed method modifies the sample weights by adjusting 

sampling probabilities before sample selection

• Sample weights = 1

ππ

• Data-dependent thresholds defined for each sampling strata
Objectives

1. Maintain unique inclusion probabilities for non-exchangeable units

2. Identify the set of exchangeable units in the lower tail

3. Equalize the inclusion probabilities of exchangeable units (reduce 

extreme sample weights)
Company

Industry 

AAAAAA

Industry 

AAAAAB

Probability of 

Selection in 

AAAAAA

Probability of 

Selection in 

AAAAAB

Probability of 

Selection in 

AAAAAA or 

AAAAAB

Company

Industry 

AAA, 

State 1

Industry 

AAA, 

State 2

Probability of 

Selection in 

AAA, State 1

Probability of 

Selection in 

AAA, State 2

Probability of 

Selection in 

AAA, 

State 1 or 2

National

Geographic

Select MAXIMUM

Company 

Probability 

of Selection

Company 

Probability 

of Inclusion

Company 

Probability 

of Inclusion

Initial Final

Perform “Winsorization” procedure
Initial inclusion probabilities by MOS
Clustering and Winsorization Procedure

Identify eligible 

companies

Winsorization of 

select inclusion 

probabilities

Cluster companies 

with similar 

characteristics
Identify eligible companies

Companies with inclusion 

probabilities ≤ nnh

NNh

Eligible for change

Companies with inclusion 

probabilities > nnh

NNh

Do not change

All noncertainty companies

Stratum h
Initial inclusion probabilities by MOS– lower tail

Sample weight = 

34,000 
Clustering – Two methods 

• K-means 

• 100 or more eligible companies

• Inclusion probability and MOS (standardized first with PROC STANDARD)

• PROC FASTCLUS 

• K=4 clusters

• Cumulative square root of the frequency (cum-root-f)

• Dalenius and Hodges Jr, 1959

• Less than 100 companies

• Only MOS

• Number of clusters (2-4) determined by the number of eligible companies in the stratum
Clustering – Two methods 

Companies with inclusion 

probabilities ≤ nnh

NNh

Eligible for change cum-root-f

Winsorization cluster

K-means 

Other clusters
Winsorization cluster 

Initial inclusion probabilities

ππ�h

WW ππ�hWW ππ�hWW ππ�hWW ππ�hWW ππ�hWW ππ�hWW ππ�hWW ππ�hWW ππ�hWW

ππh1

NN ππh2NN ... ππhccNN

Final inclusion probabilities

Replace inclusion probabilities 

with the cluster average ππ�h

WW =

∑cc∈h ππhcc

NN IIhccWW

∑cc∈h IIhcc

WW
Max sample weight 

= 34,000 

Initial inclusion probabilities by MOS

Companies eligible for clustering and Winsorization
Final inclusion probabilities by MOS

Companies eligible for clustering and Winsorization

Max sample weight 

= 468 


Results – Sample weights before and after

Min Q1 Med Q3 Max

Winsorized – k-means 13 152 200 260 950

Original Sample Weights 7 128 246 598 598,000

Min Q1 Med Q3 Max

Winsorized – cum-root-f 6 133 272 492 1,410

Original Sample Weights 4 127 319 753 670,000


Summary and future work

• The proposed procedure modifies extreme sample weights by 

equalizing the inclusion probabilities of exchangeable companies. 

• In a way, this marries the equal probability selection of similar sized 

units from previous surveys with the unequal probability selection of 

the AIES design and can be used for other PPS designs. 

• Future research includes investigating alternative clustering methods 

for strata with a small number of eligible companies to incorporate 

the inclusion probabilities into the clustering algorithm and 

developing a data-dependent method to determine the number of 

clusters
Acknowledgements

• Katherine Jenny Thompson

• Lucas Streng

• James Hunt

• Andrea Roberson

• Justin Nguyen 

• Shelby Plude

• Laura Bechtel
Questions? 

Nicole.Czaplicki@census.gov

Yeng.Xiong@census.gov