Conditional likelihood methods are applied to data from removal experiments to both model capture probabilities in terms of auxiliary variables, corresponding to observable individual characteristics and environmental conditions, and to estimate the size of the population. Asymptotic properties of the resulting estimators are derived and the methods are applied to a data set which include time dependent covariates. A simulation study is also given.published_or_final_versio

Huggins, RM

Yip, PSF

Statistica Sinica

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Title Statistical analysis of removal experiments with the use ofauxillary variablesAuthor(s) Huggins, RM; Yip, PSFCitation Statistica Sinica, 1997, v. 7 n. 3, p. 705-712Issued Date 1997URL http://hdl.handle.net/10722/45345Rights Creative Commons: Attribution 3.0 Hong Kong LicenseStatistica Sinica 7(1997), 705-712STATISTICAL ANALYSIS OF REMOVAL EXPERIMENTSWITH THE USE OF AUXILLARY VARIABLESRichard M. Huggins and Paul S. F. YipLa Trobe University and University of Hong KongAbstract: Conditional likelihood methods are applied to data from removal exper-iments to both model capture probabilities in terms of auxillary variables, cor-responding to observable individual characteristics and environmental conditions,and to estimate the size of the population. Asymptotic properties of the resultingestimators are derived and the methods are applied to a data set which includetime dependent covariates. A simulation study is also given.Key words and phrases: Covariates, log-linear models, removal method.1. IntroductionThe use of auxillary variables in the analysis of removal experiments has beenpreviously considered in Pollock, Hines and Nichols (1984) who consider mod-els which include environmental conditions and catch effort. Their methods alsoallow the use of individual characteristics which are analysed by constructing cat-egories based on these variables and then estimating the size of each category. Forexample this allows varying capture probabilities for males and females or largeand small individuals. The categories can however become restrictive if there areseveral characteristics related to capture probabilities. A further difficulty withtheir method is the somewhat arbitrary rescaling of the variance estimates (seePollock et al. (1984), p.355). The conditional likelihood approach to capture-recapture experiments of Huggins (1989, 1991) enables the modelling of captureprobabilities in terms of observable characteristics of the captured individualsdirectly and of the environment which removes the need for the construction ofcategories and provides asymptotically valid estimates of variance. Here this ap-proach is extended to removal experiments by conditioning on the removal of anindividual and using information on which occasion the individual was removedto estimate the parameters in the model for the removal probabilities.In Section 2 the conditional likelihood is constructed and its use in estimatingthe population size is discussed in Section 3. The asymptotic properties of theresulting estimator is discussed in the Appendix. To illustrate these methodswe consider a banding recoveries data set collected at Mai Po Bird Sanctuary in706 RICHARD M. HUGGINS AND PAUL S. F. YIPHong Kong in 1992 and a recapture data set from East Stuart Gulch Coloradoin U.S.A. in Section 4. A simulation study to examine the performance of theestimator is given in Section 5.2. The Conditional LikelihoodSuppose the removal experiment consists of t distinct removal occasionswhere individuals are captured and then removed from a closed population. Theindividuals are supposed to behave independently of one another and the indi-vidual covariates are supposed to be independently and identically distributed sothat, in particular, the individual covariates are independent observations froma common distribution. Let β = (βT1 , βT2 )T be a vector of parameters. Take xito be the vector of characteristics corresponding to the ith individual and let zjbe a vector of environmental characteristics. The probability that individual i iscaptured and removed from the population on occasion j given he has not beenremoved before j is denoted by pij and let qij = 1− pij. We assume a log linearmodel, i.e.log(pij1− pij)= βT1 xi + βT2 zj , for i = 1, . . . , ν, j = 1, . . . , t,where ν denotes the population size, and assume that individuals act indepen-dently of one another.Let {1, . . . , n} denote the captured individuals and {n+1, . . . , ν} the uncap-tured individuals. Let cij = 1 if individual i is captured on occasion j and be 0otherwise and ci = 1 if individual i is captured in the course of the experimentand be 0 otherwise.The full likelihood is given byL =n∏i=1t∏j=1(qi1 . . . qi,j−1pij)cijν∏i=n+1(qi1 . . . qit)(1−ci)which may be rewritten asL =[ n∏i=1∏tj=1(qi1 . . . qi,j−1pij)cij(1− qi1 . . . qit)] ν∏i=1(qi1 . . . qit)1−ci(1− qi1 . . . qit)ci .Since no information about the covariates of the uncaught individual is available,inferences concerning β are based on the conditional likelihoodL∗ =n∏i=1∏tj=1(qi1 . . . qi,j−1pij)cij(1− qi1 . . . qit) =ν∏i=1∏tj=1(qi1 . . . qi,j−1pij)cij(1− qi1 . . . qit)ci , (1)REMOVAL EXPERIMENTS WITH COVARIATES 707where the later form of the conditional likelihood is used in the Appendix belowto derive the asymptotic properties of the estimators.The maximum conditional likelihood estimators of β are then obtained frommaximising the conditional likelihood in (1). The asymptotic properties of theseestimators are easily derived and are outlined in the Appendix. These propertiesallow the use of a likelihood ratio tests to test between models as in Section 2.4of Pollock et al. (1984).3. Estimating the Population SizeTo estimate the population size the approach of Huggins (1989) is followed.Essentially it is a Horvitz-Thompson estimator (1952). Let pi be the probabilitythat individual i is captured in the course of the removal experiment. Thenpi = 1− qi1qi2 . . . qit . If pi is known, an unbiased estimate of the population sizeis given byνˆ(β) =n∑i=1p−1iwhich has variance σ2 =∑νi=1(1 − pi)/pi and an unbiased estimator of σ2 iss2 =∑ni=1(1 − pi)/p2i . In practice we estimate ν by νˆ(βˆ) =∑ni=1pˆ−1i where pˆiare evaluated at the estimated β. The variance of νˆ(βˆ) includes a contributionfrom the variance of βˆ and in the Appendix the estimate of the asymptoticvariance of νˆ(βˆ) is shown to be s2+D(βˆ)T I(βˆ)−1D(βˆ), where I(βˆ) is the observedinformation matrix and D(βˆ) = dνˆ(β)/dβ evaluated at the estimated value β.Note that it is possible for an experiment to fail and valid estimation of ν not bepossible. Seber and Whale (1970) and Otis et al. (1978), p.113 give conditionsunder which the generalised removal model fails but for our models this is stillan open question.4. ExamplesExample 1. We consider bandings of a species Prinia inornate collected at theMai Po Bird Sanctuary in Hong Kong in 1992. During the year 105 birds werebanded and there were no recaptures after the first four months suggesting thepopulation size is very close to 105. In fact using the full data set the methodsof Huggins (1989, 1991) estimated the popualtion size as 105 with an estimatedstandard error of 10−4. We regard the first banding as a removal and consider thefirst four months data. Available covariates were the weight of each bird and anage variable indicating whether the bird was aged less than one year at capture.Usually the age of a bird is determined by its color and length of feathers. Inorder to introduce occasion effects we allowed different removal probabilities inthe last two months compared with the first two. The likelihood ratio test showedno significant relationship between the covariates and the removal probabilities708 RICHARD M. HUGGINS AND PAUL S. F. YIP(full model, -log-likelihood=104.9, 4 parameters: constant removal probabilities,-log-likelihood=105.9, 1 parameter) and the resulting estimate of the populationsize was 107.9 with an estimated standard error of 2.24.Table 1. Removals of peromyscus maniculatus.m y 12 1 m y 13 1 f y 5 2 m sa 16 4f y 15 1 f a 22 1 f a 20 2 f y 11 4m y 15 1 m y 14 1 m y 12 2 m y 14 5m y 15 1 m y 11 1 f y 6 3 f y 11 5m y 13 1 f y 10 1 f a 22 3 m a 24 5m a 21 1 f a 23 2 f y 10 3 m y 9 6m y 11 1 f y 7 2 f y 14 3 m sa 16 6m sa 15 1 m y 8 2 f a 19 3 f a 19 6m y 14 1 m a 19 2 f a 19 3m y 14 1 m y 13 2 f a 20 4The columns represent, respectively, the sex (m or f ), the ages (y = young,sa = semi-adult, a = adult), the weights in grams, and the occasion the in-dividual was removed over 6 trapping occasions.Example 2. The data of Table 1 is extracted from Appendix 1 of Huggins(1991) who reports a data set on the captures of peromyscus maniculatus collectedby V. Reid at East Stuart Gulch Colorado and distributed with the programCAPTURE of Otis et al. (1978). Here the first captures are treated as removalsand the capture probabilities are modelled as functions of the sex, age and weightof the individuals. Using this full model the population size was estimated at43.69 with a standard error of 3.04 which is quite close to the estimates givenby Otis et al. (1978) using their model Mb(νˆ = 41, se(νˆ) = 3.01). Howeverthe estimate and its standard error are somewhat less than that of 47.1 with astandard error of 7.2 obtained in Huggins (1991) using the full capture histories.A likelihood ratio test revealed that in the removal model, unlike the capture-recapture model based on the full capture histories, none of the covariates weresignificantly related to the first capture probability. Using this reduced model theestimate of the population size and the associated standard error model remainedessentially the same.The discrepancy between the estimate using the full capture histories andthe removal model which only uses the first capture occasion is of some interestas one would expect the estimate which uses the most information to have thelesser variance. The estimates of the population size and its associated standarderror of Huggins (1991) were inflated by contributions from two young femaleswith low weights which give them a leverage effect. Thus, it appears the use ofremoval methods applied to first captures may aid in the detection of influentialindividuals.REMOVAL EXPERIMENTS WITH COVARIATES 7095. SimulationsIn order to examine the performance of the estimators we conducted a smallsimulation study. We considered a population of 300 individuals whose removalprobabilities depended upon the sex (SEX) and weight (WT ) of the individualand allowed an environmental covariate (ENV ) for each occasion. The simu-lated individuals were assigned their sex with probability 12 and the weights werenormally distributed with mean 3 and variance 1. The environmental covariateswere normal with mean 2 and variance 1. All covariates were recomputed foreach simulation. We used the logistic model for the removal probabilities,ln(pij1− pij)= −7− 1× SEX + 1.5×WT + 1× ENV.We considered 4, 6 and 8 removal occasions. On several occasions the max-imum likelihood algorithm did not converge but stopped due to roundoff error,which we detected by the information matrix not being positive definite andthese simulations were excluded. Only 1% of the number of simulation for eachtrial is excluded. The percentage was even less when t is 6 or 8. Further, thesimulations did not include the modelling procedure which a statistician wouldemploy and we noted that the use of an over parameterized model can resultin grossly inflated estimates of the population size. This largely contributes tothe large tails observed in the simulations. The results of the 1000 simulationsare reported in Table 2 and Figure 1. It is clear from Table 2 and Figure 1that whilst the bulk of the estimates are close to the true population size, thedistribution of the estimator is not normal even for the moderately large popula-tion sizes considered in the simulations. Note that despite the non-normality ofthe estimators the coverage probabilities of the resulting confidence intervals arereasonable. However, in practice a bootstrap procedure to construct confidenceintervals may be preferable.Table 2. Simulation results for ν = 300.Number of occasions: 4 6 8Number of captures: 206.3 (27.4) 232.1 (20.2) 246.6 (16.2)νˆ+: 836.8 (6663.4) 340.5 (207.1) 319.2 (84.0)[νˆ < 1000+]: 336.8 (126.6) 324.3 (79.4) 316.6 (59.4)Median (νˆ): 304.8 279.5 286.2se(νˆ): 3188 112.1 49.5(< 1000): 137.7 72.0 45.6Coverage* (%): 88 90 92Note: + We give the mean over the simulations, with the standard deviationin parentheses.  The mean of the estimated standard errors. * Coverage isthe percentage of the 95% confidence intervals constructed using the asymp-totic distribution of the estimators which contains the true value of νˆ.710 RICHARD M. HUGGINS AND PAUL S. F. YIP200 400 600 800 1000020406080100120140160200 400 600 800 1000050100150200250200 400 600 800 1000050100150200250300t = 4t = 6t = 8Figure 1. Histograms of estimated population sizes for νˆ < 1000.AcknowledgementsThe authors would like to thank the associate editor and two referees forvery useful comments. This work is supported by Research Grant Council ofHong Kong Government. Part of the work was done when R.M. Huggins visitedthe Department of Statistics of the University of Hong Kong. The authors arealso grateful to the World Wild Fund for Nature Hong Kong for providing theMai Po data and David Melville for helpful discussion.Appendix. Asymptotic Properties of the EstimatesAs the methods are similar to those of Huggins (1989) we only briefly outlinethe derivation of our results. LetS(β) =d logL∗dβ=ν∑i=1[ t∑j=1cij(qi1 . . . qi,j−1pij)d(qi1 . . . qi,j−1pij)dβ+ci(1− qi1 . . . qit)d(qi1 . . . qit)dβ]REMOVAL EXPERIMENTS WITH COVARIATES 711be the derivative of the log-likelihood with respect to β. Then, under assump-tions that the individuals behave independently of one another and that theindividual covariates are independently and identically distributed, S(β) is thesum of independently and identically distributed random variables.Now E(cij) = qi1 . . . qi,j−1pij, E(ci) = 1− qi1 . . . qit and pij = 1− qij, and wemay writet∑j=1[d(qi1 . . . qi,j−1)dβ− d(qi1 . . . qij)dβ]= −d(qi1 . . . qit)dβ.Thus,E(S(β)) =ν∑i=1t∑j=1[d(qi1 . . . qi,j−1pij)dβ+d(qi1 . . . qit)dβ]= 0,Similar arguments show thatE(dS(β)dβ)= −ν∑i=1[ t∑j=1(qi1 . . . qi,j−1pij)−1[d(qi1 . . . qi,j−1pij)dβ]⊗2−(1− qi1 . . . qit)−1[d(qi1 . . . qit)dβ]⊗2],where for a vector a, a⊗2 = aaT and it is straightforward to show that E(S(β)⊗2)= −E(dS(β)/dβ) = I(β). To obtain asymptotic properties of our estimators weproceed as in Huggins (1989) and suppose we have an increasing sequence ofpopulations. Then as in Huggins (1989) after noting that for β+ between βˆand β, (βˆ − β) = −[dS(β)/dβ]−1β+S(β), we may conclude that the maximumconditional likelihood estimators, β are consistent and asymptotically normalwith variance and covariance matrix I(β)−1. Next, note thatCov (S(β), νˆ(β)) = E[ ν∑i=1t∑j=1(cicij(qi1 . . . qi,j−1pij)pid(qi1 . . . qi,j−1pij)dβ+c2i(1− qi1 . . . qit)pid(qi1 . . . qit)dβ)]=ν∑i=11pi[ t∑j=1(d(qi1 . . . qi,j−1pij)dβ+d(qi1 . . . qit)dβ)]= 0,arguing as above, where E(cicij) = E(cij) and E(c2i ) = E(ci). Hence S(β) andνˆ(β) are uncorrelated. Now νˆ(β) is the sum of independently and identicallydistributed random variables so that its asymptotic distribution is easily seen to712 RICHARD M. HUGGINS AND PAUL S. F. YIPbe normal with mean ν and variance σ2. Next, for some β∗ and β+ both betweenβ and βˆ we may writeνˆ(βˆ) = νˆ(β) +[dνˆ(β)dβ]Tβ∗(βˆ − β) = νˆ(β)−[dνˆ(β)dβ]Tβ∗[dS(β)dβ]−1β+S(β).We have shown that νˆ(β) and S(β) are uncorrelated and hence, we have asymp-totically, var(νˆ(βˆ)) = σ2 + D(β)T I−1(β)D(β), where D(β) = E(dνˆ(β)/dβ).In practice we estimate σ2 by s2 , I(β) by the observed information matrix−[dS(β)/dβ]βˆ and D(β) by [dνˆ(β)/dβ]βˆ .ReferencesHarding, C. M., Heathwood, A. W., Hunt, R. G., and Read, K. L. Q. (1984). The estimationof animal population size by the removal method. Appl. Statist. 33, 196-202.Horvitz, D. G. and Thompson, D. J. (1952). A generalization of sampling without replacementfrom a finite universe. J. Amer. Statist. Assoc. 47, 663–685.Huggins, R. M. (1989). On the statistical analysis of capture experiments. Biometrika 76,133-140.Huggins, R. M. (1991). Some practical aspects of a conditional likelihood approach to captureexperiments. Biometrics 47, 725–732.Otis, D. L., Burnham, K. P., White, G. C. and Anderson, D. L. (1978). Statistical inferencefrom capture data on closed animal populations. Wildlife Monograph 62, the WildlifeSociety. Washington, D. C.Pollock, K. H., Hines, J. E. and Nichols, J. D. (1984). The use of auxillary variables in capture-recapture and removal experiments. Biometrics 40, 329-340.Seber, G. A. F. (1982). The Estimation of Animal Abundance. Charles Griffin & Co. London.Seber, G. A. F. and Whale, J. F. (1970). The removal method for two and three samples.Biometrics 26, 393-400.School of Statistics, La Trobe University, Bundoora, Australia.E-mail: rhuggins@uranus.stats.latrobe.edu.auDepartment of Statistics, The University of Hong Kong, Hong Kong.E-mail: sepyip@hkucc.hku.hk(Received February 1995; accepted August 1996)

Statistical analysis of removal experiments with the use of auxillary variables

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Statistical analysis of removal experiments with the use of auxillary variables

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