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This article was downloaded by: [Van Pelt and Opie Library] On: 18 October 2014, At: 09:50 Publisher: Taylor & Francis Informa Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK Communications in Statistics - Theory and Methods Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/lsta20 Estimation of Covariate Distribution with Capture–Recapture Data Xiaogang Duan a , Liping Liu a & Ping Zhao a a LMAM, School of Mathematical Sciences , Peking University , Beijing, China Published online: 12 Oct 2009. To cite this article: Xiaogang Duan , Liping Liu & Ping Zhao (2009) Estimation of Covariate Distribution with Capture–Recapture Data, Communications in Statistics - Theory and Methods, 38:20, 3705-3712, DOI: 10.1080/03610920802645395 To link to this article: http://dx.doi.org/10.1080/03610920802645395 PLEASE SCROLL DOWN FOR ARTICLE Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) contained in the publications on our platform. However, Taylor & Francis, our agents, and our licensors make no representations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of the Content. Any opinions and views expressed in this publication are the opinions and views of the authors, and are not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon and should be independently verified with primary sources of information. Taylor and Francis shall not be liable for any losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoever or howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use of the Content. This article may be used for research, teaching, and private study purposes. Any substantial or systematic reproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in any form to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http:// www.tandfonline.com/page/terms-and-conditions

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Page 1: Estimation of Covariate Distribution with Capture–Recapture Data

This article was downloaded by: [Van Pelt and Opie Library]On: 18 October 2014, At: 09:50Publisher: Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House,37-41 Mortimer Street, London W1T 3JH, UK

Communications in Statistics - Theory and MethodsPublication details, including instructions for authors and subscription information:http://www.tandfonline.com/loi/lsta20

Estimation of Covariate Distribution withCapture–Recapture DataXiaogang Duan a , Liping Liu a & Ping Zhao aa LMAM, School of Mathematical Sciences , Peking University , Beijing, ChinaPublished online: 12 Oct 2009.

To cite this article: Xiaogang Duan , Liping Liu & Ping Zhao (2009) Estimation of Covariate Distributionwith Capture–Recapture Data, Communications in Statistics - Theory and Methods, 38:20, 3705-3712, DOI:10.1080/03610920802645395

To link to this article: http://dx.doi.org/10.1080/03610920802645395

PLEASE SCROLL DOWN FOR ARTICLE

Taylor & Francis makes every effort to ensure the accuracy of all the information (the “Content”) containedin the publications on our platform. However, Taylor & Francis, our agents, and our licensors make norepresentations or warranties whatsoever as to the accuracy, completeness, or suitability for any purpose of theContent. Any opinions and views expressed in this publication are the opinions and views of the authors, andare not the views of or endorsed by Taylor & Francis. The accuracy of the Content should not be relied upon andshould be independently verified with primary sources of information. Taylor and Francis shall not be liable forany losses, actions, claims, proceedings, demands, costs, expenses, damages, and other liabilities whatsoeveror howsoever caused arising directly or indirectly in connection with, in relation to or arising out of the use ofthe Content.

This article may be used for research, teaching, and private study purposes. Any substantial or systematicreproduction, redistribution, reselling, loan, sub-licensing, systematic supply, or distribution in anyform to anyone is expressly forbidden. Terms & Conditions of access and use can be found at http://www.tandfonline.com/page/terms-and-conditions

Page 2: Estimation of Covariate Distribution with Capture–Recapture Data

Communications in Statistics—Theory and Methods, 38: 3705–3712, 2009Copyright © Taylor & Francis Group, LLCISSN: 0361-0926 print/1532-415X onlineDOI: 10.1080/03610920802645395

Estimation of Covariate Distributionwith Capture–Recapture Data

XIAOGANG DUAN, LIPING LIU,AND PING ZHAO

LMAM, School of Mathematical Sciences, Peking University,Beijing, China

The main purpose of this article is to estimate the underlying covariate distributionwith a biased capture–recapture sample. Two procedures are proposed, andthe derived estimates are found to be consistent and asymptotically normal.The proposed methods are compared and are shown to perform well in mostcircumstances via simulation study, and are applied to a real example.

Keywords Conditional likelihood; Covariate distribution; Sampling bias.

Mathematics Subject Classification 62D05; 62F10.

1. Introduction

In the study of wild lives, capture–recapture methods are very useful in theestimation of population sizes (Chao, 2001; Pollock, 1991, 2000). Some biologicalcharacteristics, such as sex and weight of individuals, are often called covariatesor auxiliary variables (Pollock, 2002), and their observed values are included invarious parametric or semi-parametric regression models to improve the behaviorof the corresponding population size estimators (see, e.g., Huggins, 1989; Yip et al.,1996). Some researchers try to use certain information on covariate distributionto weaken model assumptions (Wolter, 1990) or to get more accurate estimates(Liu et al., 2003; You and Liu, 2006). However, until now, there is no researchwork to consider the estimation of covariate distribution using capture–recapturedata, which is also very important in wildlife investigations.

It is well known that heterogeneity in capture probabilities between differentindividuals almost always exists, therefore direct empirical estimation (EE) forcovariate distribution using observed covariate values from capture–recapture datais incorrect, because this sample is biased. For example, if more males are observedthan females, it does not necessarily mean that there are more males in thepopulation, but maybe because males are more active.

Received July 20, 2008; Accepted November 21, 2008Address correspondence to Liping Liu, LMAM, School of Mathematical Sciences,

Peking University, Beijing 100871, China; E-mail: [email protected]

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3706 Duan et al.

The capture probabilities of observed individuals can be estimated via existingmethod (Yip and Wang, 2002), resulting weight functions. Using all observedindividuals as a weighted sample, the distribution of covariate can be estimated.This two-step estimate (TSE) for covariate distribution parameters is shown to beasymptotically unbiased.

Alternatively, conditional likelihood function with the contribution fromcovariate probability density function is used to derive the conditional maximumlikelihood estimate (CMLE). Asymptotic properties for this one-step procedure areobtained via standard arguments, and the proposed estimate is also approximatelyunbiased.

In the next section, the model is introduced explicitly, and parameter estimatesbased on two procedures are obtained. Simulation results and comparisons of thetwo approaches are shown in Sec. 3, and the proposed methods are applied to a realdata set of Prinia subflava at the Mai Po Bird Sanctuary in Hong Kong in Sec. 4.The article concludes with a short discussion in Sec. 5.

2. Inference Procedures

Consider a continuous time capture–recapture experiment conducted in �0� ��,suppose that the population is closed with size �. Captured subjects are marked andare released back to the population immediately. Let Ni�t� be the number of timesthat subject i has been caught by time t, and assume that �N1�t�� N2�t�� � � � � N��t��

′ isa �-component counting process. Let Zi ≡ �Zi1� Zi2� � � � � Zip�

′ denote a p-dimensionalvector of time-independent covariates for subject i, and suppose that Zi’s areindependent and identically distributed from an underlying density function f�z �,where is the parameter to be estimated. For i= 1� 2� � � � � �, let �i indicate, by values1 versus 0, whether or not subject i be captured at least once during the experiment.If there are altogether n individuals observed, we relabel them as i = 1� 2� � � � � n forsimplicity. In the following, we assume that there is no behavioral response.

We adopt the most commonly used Cox regression model to fit the countingprocess Ni�t�, i.e., the intensity function of it is given by: ��t �Zi� = �0�t� exp�

′Zi�,where is a p-dimensional regression parameter vector and �0�t� is the baselinehazard function. We mainly discuss the case when �0�t� is constant.

2.1. Two-Step Procedure

There are many research works for the estimation of parameter , we take theestimate ̂ obtained in Yip and Wang (2002), which is relatively new and performswell. Given ̂, based on the conditional density that given an individual is observed,its covariate value is Zi, the likelihood function for is given by

LT�Z � ̂� =�∏

i=1

[f�Zi � ̂� � = 1�

]�i = n∏i=1

f�Zi � ̂� � = 1� =n∏

i=1

f�Zi ��i� ̂�

�� ̂� ��

where �� � � = P�� = 1� = 1− ∫exp�− ∫ �

0 ��t � z�dt�f�z� �dz is the expectedprobability that a generic individual be captured at least once during the experiment,and �i� � = P��i = 1 � zi� = 1− exp�− ∫ �

0 ��t � zi�dt� is that for the specific

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Estimation of Covariate Distribution 3707

individual i. By equating the partial derivative of the logarithm of this likelihoodto 0, and note that �i� � does not depend on , we have

UT�� ̂� =�∑

i=1

�i

[� log f�Zi �

�− ��� ̂� �/�

�� ̂� �

]= 0�

When the parametric form of f is known, the estimate of , ̂� ̂�, is then obtainedby solving the above equation.

Define

QT�z � = f�z � ̂� � = 1��

then the Fisher information matrix is

IT �0� = E

{[�� logQT�Z�

]2} ∣∣∣∣ =0�

It can be proved by standard method that, if the true value 0 is known,then

√��̂� 0�− 0� is asymptotically normal with asymptotic covariance matrix

I−1T �0�. Notice that under mild conditions, I−1

T �0� is positive definite, so byTaylor’s series expansion, the asymptotic covariance matrix of

√��̂� ̂�− 0� is

then

�T� 0� 0� =[�̂� �

]′I−10 � �

�̂� �

∣∣∣∣ = 0+ I−1

T �0��

where I−10 � � is given in Yip and Wang (2002), and �̂� �

� can be shown to be

�̂� �

� = −�UT�� �

� ·[�UT�� �

]−1

The corresponding confidence interval can be then constructed by using�T� ̂� ̂� ̂��.

2.2. Conditional Likelihood Approach

The two steps above can be combined together, so the full likelihood with theconsideration of the covariates distribution is given by

LC��� =�∏

i=1

[ ∏0≤t≤�

��t �Zi�dNi�t� exp

(−∫ �

0��t �Zi�dt

)· f�Zi �

]�

where � = � ′� ′�′. Because this likelihood function consists of the covariate valuesof unobserved subjects, which are not available, similar to Yip and Wang (2002),we decompose LC��� as LC��� = L1��� · L2���, with L1��� and L2��� being themarginal likelihood and conditional likelihood, respectively, i.e.,

L1��� =�∏

i=1

{[exp

(−∫ �

0��t �Zi�dt

)· f�Zi� �

]1−�i

· �����i}

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3708 Duan et al.

and

L2��� =�∏

i=1

[∏0≤t≤� ��t �Zi�

dNi�t� exp�− ∫ �

0 ��t �Zi�dt� · f�Zi �

����

]�i

where ���� = �� � �. Because L2��� depends on the observed individuals only,we make inference about � based on it, whose logarithm is

l��� =�∑

i=1

�i

[ ∫ �

0log ��t �Zi�dNi�t�−

∫ �

0��t �Zi�dt + log f�Zi �− log ����

]�

The numerical solution of the CMLE of the underlying parameters �, �̃ = � ̃′� ̃′�′,can be obtained by solving UC��� = �U1���

′� U2���′�′ = 0, where

U1��� =�l���

� =

�∑i=1

�i

[ ∫ �

0

� log ��t �Zi�

� dNi�t�−

∫ �

0

���t �Zi�

� dt − �����/�

����

]�

and

U2��� =�l���

�=

�∑i=1

�i

[� log f�Zi �

�− �����/�

����

]�

Population size estimate can be obtained by using the plugged-in Horvitz–Thompson estimator (Horvitz and Thompson, 1952), however, this is not our maintheme.

Define

QC�z �� =∏

0≤t≤�

�� exp� z��dN�t� · e−�� exp� z� · f�z� �/�����

then the Fisher information matrix is

IC��0� = E

{[�� logQC�Z ��

��

]2} ∣∣∣∣ �=�0�

Asymptotic normality can be proved by standard method and is omitted.The explicit expressions of the asymptotic covariance matrix I−1

C ��0� for somedistribution types of f are calculated and are available from the authors. Notethat because of the presence of �, i.e., an individual might be not observed,the information is less than usual situations, which is also true in the two-step case.

It can be proved that when there is only one Bernoulli distributed covariate,then the above two procedures give the same point estimate, but the varianceestimates are different. In other cases, the estimates are similar.

The main purpose of using the conditional likelihood approach (Huggins, 1989)in both procedures is to get rid of the unobserved covariate, so that estimation ofunknown parameters is possible. Compared with the full likelihood, there might becertain amount of information loss, but because the counting processes consideredin this article essentially belong to Poisson type, so according to Yip (1991),conditional inference can avoid computational problems, and it has been pointedout further in Fong and Yip (1995) that this loss is often negligible.

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Estimation of Covariate Distribution 3709

Table 1Probabilities that subjects be captured at least once

Z1 = 1 (M) Z1 = 0 (F)

Z2 = 18 Z2 = 22 Z2 = 18 Z2 = 22 ����

� = 2 0.738 0.453 0.593 0.332 0.528� = 4 0.932 0.700 0.834 0.554 0.756

3. Simulation

Simulation experiments are conducted to assess the performance of the proposedestimates. Four combinations of population size and capture period are consideredcorresponding to � = 200� 400 and � = 2� 4. For each combination, 5,000 replicationsare carried out so as to get stable results. The capture times are generated from themodel ��t �Z� = exp�2�8+ 0�4Z1 − 0�2Z2�, where Z1 is a discrete Bernoulli variablefrom B�1� p� representing sex, and Z2 is a continuous normal variable from N��� �2�

representing weight. The parameters are taken to be p = 0�5� � = 20, and � = 2.This implies that males are more catchable than females and the capture probabilitydeclines with weight. The capture probabilities in various cases are summarized inTable 1.

The point estimates, with standard error in parenthesis, and coverage rates (CR)of the constructed 95% confidence interval are displayed in Table 2. From the tableit can be seen that while direct empirical point estimate is biased, both the TSE andCMLE are approximately unbiased with reasonable standard error estimate, and

Table 2Simulation results for three estimation methods

EE TSE CMLE

� � p̂ CR(%) p̂ CR(%) p̂ CR(%)

2 200 0.564 (0.049) 72.44 0.499 (0.067) 94.52 0.499 (0.067) 93.96400 0.563 (0.034) 54.54 0.499 (0.047) 94.20 0.499 (0.047) 94.40

4 200 0.540 (0.040) 82.66 0.499 (0.044) 94.96 0.499 (0.044) 94.74400 0.541 (0.029) 70.60 0.500 (0.031) 94.94 0.500 (0.031) 94.84

� � �̂ CR(%) �̂ CR(%) �̂ CR(%)2 200 19.51 (0.192) 26.18 20.02 (0.290) 94.96 20.02 (0.292) 94.74

400 19.50 (0.135) 4.18 20.00 (0.200) 95.04 20.00 (0.200) 94.804 200 19.68 (0.159) 47.28 20.00 (0.192) 95.04 20.01 (0.192) 95.02

400 19.68 (0.112) 18.30 20.00 (0.133) 95.44 20.00 (0.134) 95.42

� � �̂ CR(%) �̂ CR(%) �̂ CR(%)2 200 1.950 (0.137) 91.48 1.990 (0.148) 94.20 1.991 (0.148) 94.30

400 1.952 (0.096) 90.24 1.994 (0.104) 94.62 1.994 (0.104) 94.684 200 1.940 (0.115) 88.76 1.990 (0.125) 93.90 1.990 (0.125) 93.86

400 1.942 (0.079) 87.44 1.995 (0.086) 94.80 1.995 (0.086) 94.74

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3710 Duan et al.

Table 3Simulation results when covariate information possibly missing with � = 4� � = 400

EE TSE CMLE

p̂ p̂ �̂ p̂ �̂

pm = 0 pf = 0 0.541 (0.029) 0.500 (0.031) 401.9 (18.15) 0.500 (0.031) 401.9 (18.03)pm = 0�1 pf = 0�1 0.541 (0.030) 0.500 (0.033) 362.1 (18.13) 0.500 (0.033) 362.1 (18.04)pm = 0�2 pf = 0�2 0.541 (0.032) 0.500 (0.034) 322.1 (17.83) 0.499 (0.034) 322.1 (17.73)pm = 0�1 pf = 0�2 0.570 (0.031) 0.529 (0.033) 342.0 (17.72) 0.529 (0.033) 342.0 (17.62)

the coverage rates of the 95% confidence intervals by the proposed procedures areall close to their nominal values.

Motivated by the problem of missing covariate values, which occurs in thereal data set in the next section, some simulation studies are also conducted.After generating capture–recapture data, sex information are missing randomly withprobabilities pm and pf for males and females, respectively, while weight informationare complete. The estimation procedures are based on those captured individualswith complete covariate records, the results are displayed in Table 3.

It can be seen that, if missing is completely at random, i.e., pm = pf , then theestimates for p given by the proposed methods are still approximately unbiased,although the variance is larger due to less information is used, but the estimates for� is negatively biased, which is reasonable because the experiment can be regardedas conducted on a smaller population. If missing mechanism depends on sex itself,i.e., pm �= pf , then both the estimates for p and � are biased. Because missing doesnot depend on weight, the estimates for � and � are almost unchanged and are notdisplayed for brevity.

4. Data from Mai Po Bird Sanctuary

To illustrate the proposed methods, we apply them to a capture–recapture experimentat the Mai Po Bird Sanctuary in Hong Kong where bird banding was conductedweekly. We only consider capture–recapture data from January 1–April 30, 1992,when the population can be assumed to be closed. Detailed description of thisexperiment can be found in Lin and Yip (1999) and Yip and Wang (2002). In total,131 birds were captured at least once during this period, of which 49 birds havespecific sex information (32 males and 17 females), 122 birds have reasonableweight records, and 45 birds have both sex and weight information. So due to theavailability of covariate’s information, we consider three modeling procedures to fitthe data mentioned above, which include sex only, weight only, or both, respectively.The estimation results are displayed in Table 4.

It is seen that whenever sex is included in the model, the estimates of p via theproposed methods differ significantly from that via EE. This suggests strongly thatcapture is highly selective in favor of males. Meanwhile, the fact that all obtainedestimates for � and � are similar indicates that weight has relatively less influence oncapture probabilities, although heavier birds are more catchable. These conclusionscoincide well with Lin and Yip (1999) and Yip and Wang (2002).

With the illustration of the analysis about missing covariate problem in theprevious section, it is not surprising that three population size estimates are

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Estimation of Covariate Distribution 3711

Table 4Results for Mai Po Bird Sanctuary data

Z1 only Z2 only Z1� Z2 together

p̂ �̂ �̂ �̂ �̂ p̂ �̂ �̂ �̂

EE 0.65 (0.07) – 7.28 (0.04) 0.46 (0.03) – 0.67 (0.07) 7.29 (0.06) 0.40 (0.04) –TSE 0.48 (0.14) 114.7 7.20 (0.07) 0.46 (0.03) 308.9 0.50 (0.14) 7.21 (0.09) 0.40 (0.04) 103.8CMLE 0.48 (0.14) 114.7 7.20 (0.07) 0.46 (0.03) 308.6 0.50 (0.14) 7.21 (0.09) 0.40 (0.04) 103.2

so different. The first and third estimates are based on such a small proportion ofavailable information, that the derived estimates are even smaller than the observednumber of birds. This implies that the missing mechanism is non ignorable. Thesecond estimate might be more reasonable, but it does not use the sex information,therefore the estimation of p, which is one of the primary purpose of this article,is not possible. The challenging task of using all observed information to derivebasically unbiased estimate of covariate distribution, as well as of the populationsize, is under investigation. On the other hand, we are fortunate that in all practicalexperiments, if some covariate values are partly missing, the fact is known to us;if some covariates are not recorded at all, the estimation procedure using frailtymodel can be adopted (Xu et al., 2007).

5. Concluding Remarks

In this article, we propose two procedures for estimating covariate distributionfrom biased observation, which is the first work in capture–recapture studies. Thederived estimates are found to be asymptotically normal and perform well, andthe application to a real example gives reasonable results. The problem of missingcovariate values is also discussed briefly.

We only consider the case in which both the hazard rate of capture and thedistribution of covariate have a parametric form. The TSE procedure can be easilygeneralized to the case where the distribution form of parameter is unknown, withthe help of kernel estimation method using reweighted sample.

Acknowledgments

This work is supported by the National Natural Science Foundation of China(10731010) and RFDP. The authors are grateful to the referees for valuablesuggestions.

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