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Methods for Capture- Recapture Analysis 

When Cases May Not Be Identified Uniquely 

Betsy L. Cadwell†, Philip J. Smith‡, Andrew L. Baughman‡ 

† Center for Disease Control and Prevention, Division of Diabetes Translation, 3470 Buford Highway, NE, Atlanta, GA 30341; ‡ Center 

for Disease Control and Prevention, National Immunization Program, 1600 Clifton Road, NE, Atlanta, GA 30333. 

1. Introduction

Capture-recapture methods are used for estimating the unknown size of a closed population from overlapping lists of cases 

[1, 2, 3]. After the cases on each list are enumerated and cases recorded on both lists are identified, an estimate is obtained 

for the number of cases that are not on either list. This estimate of the number of unknown cases is combined with the 

number of matched and unmatched cases to yield an estimate of the size of the population. To estimate the number of 

unknown cases, Lincoln and Petersen developed the maximum likelihood estimator [4, 5]. Chapman later adjusted the 

estimator for use with small samples [6, 7]. 

To obtain a valid population estimate using capture-recapture methods with two lists, three conditions are assumed: for 

each list, the probability a case is recorded on the list is equal for all cases; the probability of being recorded on a list is 

independent of the probability of being recorded on the other list; and there is no immigration or emigration to the 

population during the study period. Even if these assumptions are violated, Hook and Regal suggest the estimates may still 

be valid [1]. Regardless of the validity of these assumptions, current methods require cases appearing on each list to be 

uniquely identifiable such that unique matches between lists can be determined. 

Epidemiological applications of capture-recapture methods utilize surveillance data or administrative lists. Administrative 

lists often do not have a sufficient amount of specific information on individual characteristics to insure cases are uniquely 

identifiable. In this paper we modify commonly used capture-recapture methods to overcome this problem. In our methods 

we create profiles, or a collection of characteristics, that describe one or more individuals on a list. For example, if profiles 

were defined by gender, birth month and birth year then a possible profile is males born in May of 1976. For profiles 

appearing on both lists, we account for potential matches of cases between the two lists. Two methods are developed to 

account for potential matches: (1) a weighted estimator, which considers all potential matches and (2) a bootstrap 

estimator, which considers potential matches by resampling profiles with replacement. We illustrate the use of the 

bootstrap estimator by estimating the number of pertussis hospitalizations reported during 1996 in New York State using 

two data lists without unique identifiers. 

2. Methods

2.1 The Chapman Estimator: Unique Matches

When two lists, A and B, the observed counts are: the number of uniquely matched cases on both list A and list B, X AB , the

number of cases on list A but not on list B, AB X , and the number of cases on list B but not on list A, X . The unknown AB 

number of cases on neither list A nor list B is estimated by 

X ˆ = ( X X ( X +1)) .

AB AB AB AB 

The modified Chapman Lincoln-Petersen estimate of the population total is 

Nˆ = X AB + X AB + X AB + Xˆ 

AB (1) 

and its estimated variance is approximately 

1 

( X + X +1) ( X AB + X AB ) X AB AB X AB AB VN ˆ 7 2 ( ) .

( X AB +1) ( X AB + 2)
 

 

 

 

 

 

 

 

 

2.2 The Weighted Estimator: Non-unique Matches 

When the identifiers on the two administrative lists do not allow individuals to be matched uniquely, potential matches can 

be identified by merging two or more characteristics to create a “profile” (e.g. males born in May of 1976). When more 

than one person on a list has the same profile, there are many ways to match cases between the two lists with respect to the 

profile. Further, when there are many profiles, each of which may be associated with more than one person, there are many 

different ways that individuals in list A could match individuals in list B. We refer to each of the possible ways in which 

individuals on list A could match individuals on list B as a “profile match configuration.” 

Letting Ai denote the number of individuals on list A with profile i, Bi denote the number of individuals on list B with 

profile i, S = min { Ai , Bi }, P denote the number of unique profiles, and ji

i the number of ways profile match 

configuration can occur for profile i, i= 1, ..., P is 

  A i   Bi

qji =   , (2) 

ji ji   

ji = 0,..., Si . Letting jj , , ..., j denote a combination of profile match configurations in which there are j matches on 1 2 p 1

profile 1, j matches on profile 2, ..., and j matches on profile P, the number of ways j j , , ..., j can occur is 2 P 1 2 p 

P 

qj1

, j2 ,..., jp 

= ∏qji 

i=1 

and the probability of j j , , ..., j is 1 2 p 

qj j , , ... j

1 2 p 

π jj , , ..., j = . (3) 1 2 p Sj Sj Sj 1 2 P 

∑ ∑... ∑ qjj , , ... j

1 2 p 

j1 =0 j2 =0 jP =0 

The modified Chapman Lincoln-Petersen estimator corresponding to j j , , ..., j is 1 2 p 



P P  P P 

A −

P P P P P ∑ ∑ ∑ j

 B − ∑ j

∑ ∑∑

 

∑ ∑  

i i

ˆ  i i

 N j j 12 ... jP 

= j

i +  Ai − j i 

+  Bi − j

i 

+

i=1 i=1 i=1 i=1 

P (4) 

i=1  i=1 i=1   i=1 i=1 

1+∑ j

i 

i=1 

whe ∑

P 

P P  P P 

re, j i is the number of matches, ∑Ai − ∑ j i 

 is the number of cases on list A without a match, ∑Bi − ∑ j

i 

i=1  i=1 i=1   i=1 i=1 



P P   P P 

∑ Ai − ∑ ∑ j

i   B − ∑ j 

   i i 

is the number of cases on list B without a match, and i=1 i=1 i=1 i=1 

P is the modified Chapman estimator 

1+∑ j

i 

i=1 

for the number of cases appearing on neither list. When there is only one profile match configuration, cases are uniquely 

identifiable and equations (4) and (1) are identical. The weighted estimator of the population total is 

Sj1 Sj2 SjP

N ˆ ˆ = ∑ ∑... ∑π j , j , ..., j Nj , j , ..., j

1 2 P 1 2 P 

j =0 j =0 j =0 . (5) 

1 2 p 

2.3 The Bootstrap Estimator: Non-unique Matches 

The weighted estimator (5) may be useful when the number of profiles (P) is small or when there are relatively few records 

existing in each profile, i.e. Ai and Bi

 are small. However, qj1

, j2 ,..., jp 

approaches infinity when there are either a large number 

of profiles (P) or when Ai

 and/or Bi are moderate or large for many profiles. In this case we use a bootstrap procedure to 

obtain a point estimate for the closed population size, N, and to derive percentile confidence intervals for the estimate. 

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I 

I 

In the r-th bootstrap replicate sample, r = 1,..., R1, we obtain a with-replacement sample of size 

P 

∑Ai from list A and of 

j=1 

P 

size ∑Bi from list B. From these samples we note the number of individuals belonging to each of the P profiles on each 

j=1 

list. For the r-th bootstrap replicate let Ai

′ denote the number of individuals sampled from list A that have profile i, 

  A ′

i 

′ Bi 

P 

Bi

′ denote the number of individuals on list B with profile i, S′i = min { Ai

′ , Bi′ }, q′

ji =   

, and q′

j , j2 ,..., j = ∏q′ji .

ji ji 

1 P

   i=1 

Then, we compute the probability of every profile match configuration j j , , ..., j ,

1 2 P 

q′jj , , ... j

1 2 p

π ′

1

, 2, j

p ′ ′ ′

jj ..., = .

S j S j S j 1 2 P 

∑ ∑...∑ q′j , j , ... j

1 2 p 

j1 =0 j2 =0 jP =0 

Next, we obtain R with-replacement samples from the profile match configurations jj , , ..., j from the r-th bootstrap 2 1 2 P 

replicate sample. In this step, profile match configuration j j , , ..., j is sampled with probability π′ j j , , ..., j . Each of the 1 2 P 1 2 p 

profile match configurations resampled in this step represent one of the ways in which individuals on list A could correctly 

r s match individuals on list B. For the s-th, resample from the data in the r-th replicate, X( , ) denote the number of uniquely AA 

(,) rs (,) matched cases on both list A and list B, X denote the number of cases on list A but not on list B, X

rs denote the AB AB 

number of cases on list B but not on list A, and 

ˆ

(,) (,) rs rs (,) r s (,) r s X = ( X X ( XAB +1))

denote the estimated number of individuals in the population on neither list A nor list B, and 

AB AB AB 

ˆ (,) rs ˆ

(,) rs (,) rs rs (,) ( , ) rs N = X AB + X + X + X AB AB AB 

R 

ˆ

() ˆ(r s ) 

2 

r ,

denote the modified Chapman Lincoln-Petersen estimator. The estimator for replicate r is N = ∑N R2 and the 

s=1 

mean bootstrap replicate estimator is 

R 

ˆ 

1 

ˆ

( )r Nboot = ∑N R1

. 

r=1 

Bootstrap percentile confidence intervals for the close population size can be obtained using quantiles of the replicate 

ˆ

( ) estimates, Nr 

, r = 1,..., R1 . It can be shown that the mean bootstrap replicate estimator is unbiased for the weighted 

estimator, and that the weighted estimator minimizes squared error loss. 

3. Example: Pertussis Hospitalizations During 1996 in the State of New York 

Reliable estimates of the burden of disease are needed to evaluate disease control policies. Similar to other nationally 

reportable diseases, evidence suggests the numbers of pertussis cases and hospitalizations in the United States are 

underestimated [9]. The combination of increasing disease and possible underestimation of reporting motivated a study to 

estimate the magnitude of pertussis hospitalizations by state and year in the United States using a two-sources capturerecapture analysis. We report here only the results for the state of New York during 1996. 

3.1 Data Sources 

Two surveillance lists were identified which both captured hospitalizations from the population of interest during 1996. 

The first list comes from the National Electronic Telecommunications System for Surveillance (NETSS); the second list 

was obtained from the Health Care Information Association (HCIA). 

NETSS is a concatenation of weekly reports submitted electronically from state health departments to the Centers for 

Disease Control and Prevention. Included in these reports are cases of diseases determined to be nationally notifiable by the 

Council of State and Territorial Epidemiologists [10]. The data are primarily used to rapidly identify disease epidemics. 

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The HCIA database contains acute-care hospital discharge records form both public and proprietary state data during 1992-

1996. The database contains reports from more than 2500 non-federal, self-selected, acute-care hospitals in the United 

States. These reports represent approximately 40% of the total US hospital discharges. 

3.2 Application of Bootstrap Estimator 

The attributes common to both surveillance lists were gender, birth month, birth year, year of illness and month of illness 

(hospitalization month from HCIA matched with cough onset month from NETSS). These common attributes do not 

identify individuals uniquely within a list and thus we cannot be certain that the individuals across lists are unique. 

Therefore, we used this set of attributes to define profiles. Because of the lack of unique identifiers and because the number 

of profiles was large, we implemented the bootstrap estimator to estimate the total number of pertussis hospitalizations in 

New York State during the year 1996. 

The HCIA database listed 200 records of hospitalizations due to pertussis while the NETSS database listed 123 records of 

pertussis hospitalizations in New York State, 1996. Each record was defined by one of 157 profiles determined by the 

above-mentioned variables. For HCIA there were 88 unique profiles and the average number of cases described by each 

profile was 1.40 (Range: 1-5). For NETSS there were 113 unique profile; the average number of cases described by each 

profile was 1.77 (Range: 1-6). However, 51 (41%) cases listed in NETSS matched more than one individual in HCIA. In 

HCIA, 85 (43%) cases matched more than one case in NETSS. The number of matching configurations exceeds 4 x 1015

. 

Thus, we used the bootstrap method with R1 = 500 times and R2 = 250 . Figure 1 shows a histogram of the bootstrap 

ˆ

( ) replicate estimates, N

r 

, r = 1,..., R1 . The estimate for the number of pertussis hospitalizations in New York State during 

ˆ

( ) ˆ Nr

the year 1996 is the mean of the replicate estimates, Nboot = 894 . The 2.5% and 97.5% quantiles of were used to 

give the 95% percentile confidence interval, (737, 1102). 

To examine the impact of profile definition on our results, we implemented two profile definitions for the 1996 New York 

State data. The first, presented above, defined the profile by gender, year of illness, month of illness, birth month and birth 

year. The second profile definition used only gender, year of illness, birth month, and birth year. Thus the second profile 

definition can be considered less specific then the first. As expected, with the less specific matching criterion the estimate 

decreased to 468 (95% CI: 421, 523). 

4. Discussion 

A fundamental requirement of capture-recapture methods for estimating population totals is cases can be matched using 

unique identifiers. In our research we account for the uncertainty of uniqueness by developing the weighted estimator, 

which incorporates all possible matching configurations when non-uniqueness exists. In addition, we developed the 

bootstrap estimator for use when a large number of profiles are present or when large numbers of individuals exist within 

profiles. The bootstrap estimator is approximately unbiased for the closed population size. 

Both the weighted and bootstrap estimators assume individuals within lists are not duplicated. That is, two or more cases 

defined by the same profile within a list are in fact unique. It is therefore important to carefully evaluate the administrative 

lists used to ensure that there is, at least theoretically, no duplication of individuals on any one list providing information 

for the capture recapture analysis. However, if duplicates do exist within one or both of the lists, the resulting estimates 

will be inflated. 

The methods presented here are not recommended as a substitute for use of unique identifiers when they do exist within and 

between lists. Yet, using the weighted or bootstrap estimator is a feasible alternative when assumptions of uniqueness 

cannot be met and allows for broader applications of capture-recapture methods. Although this work demonstrates that 

capture-recapture estimates can be obtained when cases are not uniquely identifiable, the numerical results suggest that 

estimates can depend sensitively on the specificity of the profile definition. 

5. Acknowledgements 

The authors would like to acknowledge the useful comments provided by Chima Ohuabunwo, Kristine Bisgard, Chuck 

Vitek and Ted Thompson. In addition, we thank Mary McCauley for her editorial assistance. 

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Figure 1: Distribution of Replicate Estimates 

NY 1996: Distribution of Bootstrap Replicate Estimates(R1=500; R2=250) 

150

100 Mean: 894 

95% CI: (737 , 1102) 

Frequency 

50 

0 

400 600 800 1000 1200 1400 

Bootstrap Replicate Estimate 

7. References

1. Hook EB, Regal RR. Capture-recapture methods in epidemiology: methods and limitations. Epidemiologic

Reviews 1995; 17(2):243-64 [correction appears in Epidemiologic Reviews 148: 1219].

2. International Working Group for Disease Monitoring and Forecasting. Capture-recapture and multiple-record

systems estimation I: history and theoretical development. American Journal of Epidemiology 1995; 142: 1047-1058.

3. International Working Group for Disease Monitoring and Forecasting. Capture-recapture and multiple-record

systems estimation II: applications in human diseases. American Journal of Epidemiology 1995; 142: 1059-1068.

4. Lincoln FC. Calculating waterfowl abundance on the basis of banding returns. Circular no. 118. Washington DC:

US Department of Agriculture, 1930: 1-4.

5. Petersen CGJ. The yearly immigration of young plaice into the Limfjord from the German sea. Rep Dan Biol Stat

1896; 6:1-48.

6. Chapman CJ. Some properties of the hypergeometric distribution with applications to zoological censuses. U

California Public Stat 1951; 1:131-60.

7. Wittes JT. On the bias and estimated variance of Chapman’s two-sample capture-recapture population estimate.

Biometrics 1972; 28: 592-7.

8. Efron B, Tibishriani RJ. An Introduction to the Bootstrap. Chapman & Hall/CRC: Boca Raton, 1993.

9. Sutter RS, Cochi SL. Pertussis hospitalizations and mortality in the United States, 1985-1988. JAMA 1992; 267(3):

386-91.

10. http://www.cdc.gov/epo/dphsi/netss.htm; Internet; accessed October 7, 2002.

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