Incorporating Misclassification 

into Capture-Recapture 

Methodology in the 2012 

Census of Agriculture

Daniel W. Adrian

Andrea C. Lamas

Denise A. Abreu

Shu Wang

Linda J. Young

USDA

National Agricultural Statistics Service

Federal Committee on Statistical Methodology (FCSM)

November 6, 2013
 

 

Census of Agriculture: history

 Conducted in years ending in 2 and 7.

 Before 1997: U.S. Census Bureau

 1997-present: USDA/NASS 
 

 

 

Census of Agriculture: purpose

 Quantity of interest: number of farms

– State

– Farm type

– Race

– Gender 

 Factor in allocation of funds for Federal 

agriculture programs

– e.g. land grant universities
 

 

Farm definition

 Target population: operations that meet 

USDA farm definition.

 An agricultural operation that either

– Produces at least $1,000 of sales in a year,

– Normally produced $1,000 in sales,

– OR 1,000 points of agricultural items
 

Examples of Point Farms

 5 horses

 1 acre of Christmas trees

 100 acres of pasture land

 $1,000 in government payments
 

 

 

Documented farm

misclassification

 Farm Numbers Research Project (2009): 

tracts in June Area Survey (JAS) were 

incorrectly identified as non-agricultural when

agriculture was present.

 Classification Error Survey following 2007 

Census: classification errors made during 

both Census and JAS.

 June Area Survey (JAS): used as 

supplemental survey to Census for farm 

number estimation.
 

Purpose of talk

 Accounting for farm misclassification in 2012 

Census of Agriculture

 Adjusts traditional methodology

– Dual System Estimation (DSE)

– Capture-recapture 
 

Dual system estimation or 

capture-recapture methodology 

 Similar methodologies in different contexts 

(in this presentation, used interchangibly)

 DSE

– U.S. Census of Population 

– UK Census Coverage Assessment

 Capture-recapture

– fish and wildlife populations
 

 

Main ideas of DSE and capturerecapture

 DSE

– supplementary (independent) survey to quantify 

Census undercount

– i.e. What proportion of units from the supp. survey 

are counted by the Census?

 Capture-recapture

– Capture, tag, and return

– Recapture

– What proportion of animals captured in second 

catch are tagged from first catch?
 

 

Example: catching trout (taken 

from UK Census documentation)

 Day One: catch 100 trout. Tag each and 

release

 Day Two: catch 50 trout. 25 have tags.

 Estimate of total

200

50

25 100

ˆ

1

N
 

 

 

Analogy between DSE and 

capture-recapture

CapturerecaptureDSE

Day one catch Records counted by the Census

Day two catch Records counted by supp. survey

Tagging Matching of Census and survey 

process records

1

.

& . ˆ

Supp S

Census Supp S

Census N

N

N N
Supp. Survey: June Area Survey
 

DSE for Census of Agriculture 

 Challenge: traditional DSE does not deal with

misclassification errors

 Census of Agriculture has 4 types of 

enumeration errors, including

– under-coverage

– non-response 

– 2 types of misclassification.
 

 

 

 

Errors in Enumeration Process

 List under-coverage: the omission of farms 

from the Census Mailing List (CML)

 Non-response: The failure of farm operators

to return a completed Census questionnaire.

 Both conditioned on farms
 

 

 

Errors in Enumeration Process II

 Misclassification: Errors in Census reporting

cause two types of misclassification errors:

– Farms are classified as non-farms

– Non-farms are classified as farms
 

 

 

 

Summary of Enumeration Errors

1. CML under-coverage of farms

2. Farm non-response

3. Farms are misclassified as Census nonfarms

4. Non-farms are misclassified as Census

farms

1-3: undercount, 4: overcount
Venn Diagram

U.S. Farms

U.S. Agricultural Operations

Operations on CML

Census Respondents

Census Farms
 

Adjustments to traditional DSE

 Probability of farm imputed for unresolved

records: where Census and JAS disagree on 

farm status

 Three undercount errors are combined into 

“capture”

 Account for “differential catchability”

 Correct for misclassification overcount
 

 

 

Definition of Capture

 An operation is “captured” by the Census if it 

is

– on CML | Farm

– Responds | CML, Farm

– Classified as Census Farm | CML, Responds, 

Farm 
 

 

 

 

Product of probabilities

P(capture) = P(on CML | Farm)

x P(Responds | CML, Farm)

x P(Classified as Census Farm | 

CML, Responds, Farm)

Or 

pCap pCov pResp pCCF
Dual System Estimation

 Dual System Estimator using capture:

C

J

JC F

F

F

N

1

ˆ

– FJC: number of farms captured by JAS and 

Census

– FJ: number of farms captured by JAS

– FC: number of farms captured by Census
 

Two problems with first-adapted 

DSE

 Doesn’t account for heterogeneity in capture 

probabilities

 Doesn’t account for misclassification of nonfarms as Census farms (overcount)
 

 

 

 

 

Example: Heterogeneity in capture 

probabilities

 Catching trout, roach, and catfish.

First day

catch, 

#Tagged

Second day

catch, Fraction

tagged

Estimate of Total

Trout 100 25/50 200

Roach 50 5/20 200

Catfish 10 1/10 100

All fish 160 31/80 413

 Account for differential capture rates = 500

 Don’t account = 413
 

 

Heterogeneity in capture 

probabilities: estimator

 Partition farms into groups so that the 

probability of capture is about the same 

within each group.

 Example: by state, farm sales, farm type, 

race, gender

 Sum DSE’s for each group

n groups

i

C i

J i

JC i

F

F

F

N

_

1

,

1

,

,

ˆ
 

 

Logistic regression

 Logistic regression extends this approach: 

allows

– continuous variables 

– more complex models.

 Each Census record has its own capture 

probability.

n records

j

N pCap j

_

1

1

,

ˆ
 

 

 

Logistic regression

 The 0/1 capture indicators Yifollow a 

Bernoulli(πCi) distribution, where

logit ( ' )

1

Ci Ci C

x

 βCis estimated using the matched dataset (of 

JAS and Census)

 Then Census record probabilities of capture

are 

)

ˆ

logit ( '

-1

Cj Cj C p x
 

Two problems with traditional DSE

 Doesn’t account for heterogeneity in capture 

probabilities

 Doesn’t account for misclassification of nonfarms as Census farms (overcount)
 

 

Adjustment for Misclassification

 The probability of correct Census farm 

classification is

pCCFC = P(Farm | Census Farm)

 Multiplied by capture weights to correct for 

overcount.

n records

j Cap j

CCFC j

p

p

N

_

1 ,

, ˆ
 

Final estimator

 The final estimator is obtained after 

expanding the capture probability into its 

components.

n records

j Cov j sp j CCF j

CCFC j

p p p

p

N

_

1

, Re , ,

,

ˆ
 

 

 

Computing probabilities

 The 4 probabilities 

– pCov = P(on CML | Farm)

– pResp = P(Responds | CML, Farm)

– pCCF = P(Classified as Census farm | CML,

Responds, Farm)

– pCCFC = P(Farm | Census farm)

 use different subsets of the matched dataset 

depending on the conditions.
 

Wrap-up

 Census of Agriculture

 Adjusts traditional DSE/capture-recapture

methods for misclassification.
Thank you!