Likelihood analysis of spatial capture-recapture models for stratified or class structured populations

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1 Likelihood analysis of spatial capture-recapture models for stratified or class structured populations J. ANDREW ROYLE, 1, CHRIS SUTHERLAND, 2 ANGELA K. FULLER, 3 AND CATHERINE C. SUN 2 1 USGS Patuxent Wildlife Research Center, Laurel, Maryland USA 2 New York Cooperative Fish and Wildlife Research Unit, Department of Natural Resources, Cornell University, Ithaca, New York USA 3 USGS and New York Cooperative Fish and Wildlife Research Unit, Department of Natural Resources, Cornell University, Ithaca, New York USA Citation: Royle, J. A., C. Sutherland, A. K. Fuller, and C. C. Sun Likelihood analysis of spatial capture-recapture models for stratified or class structured populations. Ecosphere 6(2):22. Abstract. We develop a likelihood analysis framework for fitting spatial capture-recapture (SCR) models to data collected on class structured or stratified populations. Our interest is motivated by the necessity of accommodating the problem of missing observations of individual class membership. This is particularly problematic in SCR data arising from DNA analysis of scat, hair or other material, which frequently yields individual identity but fails to identify the sex. Moreover, this can represent a large fraction of the data and, given the typically small sample sizes of many capture-recapture studies based on DNA information, utilization of the data with missing sex information is necessary. We develop the class structured likelihood for the case of missing covariate values, and then we address the scaling of the likelihood so that models with and without class structured parameters can be formally compared regardless of missing values. We apply our class structured model to black bear data collected in New York in which sex could be determined for only 62 of 169 uniquely identified individuals. The models containing sex-specificity of both the intercept of the SCR encounter probability model and the distance coefficient, and including a behavioral response are strongly favored by log-likelihood. Estimated population sex ratio is strongly influenced by sex structure in model parameters illustrating the importance of rigorous modeling of sex differences in capture-recapture models. Key words: capture-recapture; categorical covariates; class structure; density; missing data; sex assignment; spatial capture-recapture; stratified populations. Received 12 May 2014; revised 22 October 2014; accepted 3 November 2014; final version received 20 December 2014; published 12 February Corresponding Editor: R. R. Parmenter. Copyright: Ó 2015 Royle et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. aroyle@usgs.gov INTRODUCTION Spatial capture-recapture (SCR) models are a relatively recent development in ecological statistics (Efford 2004, Borchers and Efford 2008, Royle and Young 2008, Borchers 2011) and have become widely adopted for the analysis of capturerecapture data used to study animal populations, especially those based on DNA methods, camera trapping, and other new technologies. In many situations, and especially in SCR studies of carnivores, it is necessary to consider models with parameters that vary by population class structure (e.g., age or sex) to account for differences in encounter rate or space usage by individuals (Efford and Mowat 2014). However, in many studies of animal populations using capturerecapture methods, the class identity of individuals may be difficult to determine. This is especially true in studies with non-invasively v 1 February 2015 v Volume 6(2) v Article 22

2 collected samples of tissue, hair, or scat in which genetic analyses can fail to determine individual sex even though individual identity of the sample can be determined (Waits and Paetkau 2005). In studies where sex of individuals cannot be determined with certainty, it has been common to either ignore sex and not estimate sex-specific parameters (e.g., Mowat and Strobeck 2000, Immell and Anthony 2008, Frary et al. 2011), or to discard the individuals with unknown sex and proceed with sex-specific models (Belant et al. 2005, McCall et al. 2013). Utilizing only the known-sex individuals can result in a biased sample when sex determination is not independent of encounter probability. This is a common situation because there is a greater likelihood of determining sex from individuals that are captured more frequently. Moreover, if the accuracy in sex determination varies by characteristics of the individual (e.g., age), samples can be biased toward individuals who do not permanently disperse (Nichols et al. 2004), or those for which local movement rates are greater. In this paper we consider likelihood analysis of SCR models for class structured or stratified populations in which there is an individual-level covariate, say C i, which is categorical. We develop a likelihood framework for inference using what is classically referred to as the full likelihood (Borchers et al. 2002). In this formulation of the problem, the parameter N, population size, appears as an explicit parameter of the likelihood (this is in contrast to the conditional likelihood). We focus specifically on spatial capture-recapture models (SCR, or spatiallyexplicit capture-recapture, SECR) because such models have become widely used for analysis of capture-recapture data based on individuality obtained from DNA data. Further, the existence of missing sex information is pervasive with such data, and sex-specificity of SCR model parameters has direct biological interpretation (Efford and Mowat 2014). A general formulation of likelihood analysis of SCR models was developed by Borchers and Efford (2008). A key technical issue in the analysis of class structured capture-recapture models is that the covariate C i is unobserved for individuals that are not captured. Therefore, a model for the covariate is required (Borchers et al. 1998, Pollock 2002, Royle 2009). This has been suggested as a limitation of the full likelihood approach to analysis of individual covariate models, but the full likelihood approach has distinct benefits in some cases. For example, sometimes there is a natural innocuous model for the covariate, the model itself may be biologically relevant, or there may be missing values of the covariate even for some individuals that are captured. The canonical example of a categorical covariate that satisfies all of these conditions is when C i is the sex of individual i. In this case the natural model is Bernoulli, i.e., C i ; BernoulliðwÞ where the parameter w is the probability that an individual is male or female depending on the coding of C. We define C ¼ 0 as a female and C ¼ 1 as a male so w is the probability that an individual in the population is a male. In many studies that use DNA methods, it is not uncommon to have large sample sizes of individuals with unknown sex. For example, we provide an analysis of SCR data with missing sex information from a study of black bears in New York. For this study, sex could not be determined for 107 of 169 uniquely identified individuals. Therefore, it is clearly advantageous, even necessary, to devise methods which allow information on unique individuality to be used, even though sex is missing. Effective treatment of missing covariate data has previously been noted as a benefit of using Bayesian methods, e.g., as implemented in the software WinBUGS or JAGS (Gardner et al. 2010, Russell et al. 2012, Royle and Converse 2014), because missing covariate data are handled seamlessly by MCMC methods. Indeed, the original analyses of the black bear data we use here (Sun 2014) used a Bayesian analysis of the SCR model in order to deal with this problem. However, likelihood formulations of SCR models have potential advantages such as the development of general models for ecological distance (Royle et al. 2013a; Sutherland et al., in press), which motivate us to pursue a full likelihood formulation of models that explicitly account for data with missing sex information. CAPTURE-RECAPTURE FOR CLASS-STRUCTURED POPULATIONS Let N be the size of a closed population and v 2 February 2015 v Volume 6(2) v Article 22

3 denote by n the number of encountered individuals and by n 0 the number not encountered, so that N ¼ n 0 þ n. We will parameterize the likelihood in terms of n 0 because optimization over a parameter space in which log(n 0 ) is unconstrained is preferred to a parameter space in which N must be constrained N n. To introduce the notation for class structured populations in a simple case, we consider an ordinary closed population sampling problem in which the encounter frequencies y i for individual i are binomial outcomes based on a sample of size K sample occasions. Then, for each individual in the population the encounter frequencies have a binomial distribution: y i ; BinomialðK; pþ: We will use standard bracket notation to refer to probability distributions. For example, [y] will denote the binomial distribution of y in this case, and a conditional distribution is specified such as [yjx] and so on. Many of the standard closed population models differ in the way the parameter p depends on individual, time or other effects (Otis et al. 1978). We shortly consider the case where p is class structured according to a categorical covariate C taking on discrete values g ¼ 1, 2,..., G (G or groups ). The full likelihood of the parameters p and n 0 has contributions from each of the observed encounter frequencies y i and includes a component for the number of uncaptured individuals n 0 : Lðp; n 0 jyþ ¼ N! n!n 0! ( ) Y n ½y i Š p n 0 0 ð1þ where y ¼ (y 1,..., y n ) is the vector of encounter frequencies for the n encountered individuals, and the term p 0 ¼ Pr(y ¼ 0) is the probability that an individual is not encountered. The form of p 0 depends on the specific model being considered. For the case of model M 0, p 0 ¼ 1 (1 p) K. To formulate the full likelihood for a class structured population we assume that C i is the class membership of individual i. We express class structure in the model parameters by assuming that p varies by individual depending on its value of the covariate G and we denote this by: p i ¼ p g if C i ¼ g: The class probabilities are w g ¼ Pr(C i ¼ g). We recognize the class structure in our capturerecapture model by noting that the encounter model is conditional on C: [y i jc i ¼ g] ¼ Binomial(K, p g ) and the likelihood in Eq. 1 is therefore conditional on the parameters p g. However, in addition to encounter frequency data y i for each individual, we also regard the covariate C i as data. Thus the joint likelihood of y i and C i is simply the product of probabilities: ½y i jc i ¼ gšw g : In the case in which the covariate C is binary (e.g., sex) then the contributions to the likelihood of the data for individual i look like: [y i jc i ]w if C i ¼ 1 or [y i jc i ](1 w) if C i ¼ 0. For the y ¼ 0 individuals which have no observed value of C i, or for individuals that have a missing value of the covariate, we have to do a marginalization over the different values of the covariate: ½y i Š [ PrðyÞ ¼ X PrðyjC ¼ gþw g ð2þ g where the summation is over all possible values of the covariate C. The likelihood from Eq. 1 is extended to include n 1 individuals with nonmissing covariate data and n 2 individuals with missing covariate values as follows: ( )( ) Lðp; w; n 0 jyþ ¼ N! Y n 1 Y n 2 ½y i jc i ¼ gšw n!n 0! g ½y i Š p n 0 0 ð3þ where n ¼ n 1 þ n 2, p 0 ¼ Pr(y ¼ 0) is computed by the marginalization Pr(y ¼ 0) ¼ R c Pr(y ¼ 0jC ¼ g)w g. For individuals with missing values of the covariate, the marginal probability [y i ] depends on the various class specific parameter values p g as well as the probabilities w g via Eq. 2. To obtain estimates of class-specific population size, the population size N can be partitioned into the different classes by noting that the classspecific population sizes N g are multinomial based on N trials and probabilities w g. For the case where the classes are (male, female), N male ; BinomialðN; wþ and a point estimate of the population size of males is ˆN male ¼ ˆNŵ whereas N female ¼ N N male, and ˆN female ¼ ˆN ˆN male. v 3 February 2015 v Volume 6(2) v Article 22

4 Comparing likelihoods with and without class structure When the likelihood has parameters that depend on the class variable C, it is necessary to include the observed values of C i in the likelihood. However, even if no parameters of the likelihood depend on C, we might still retain the covariate values as data, and thus the Pr(C ) contributions to the likelihood, so that the parameters w g are estimated as part of the model. There are two reasons for retaining this portion of the likelihood: (1) There is no statistical cost to doing this because all of the information about w g comes from the observed covariate information. For example, in a sex-structured model, in the absence of sex-specific encounter probabilities, the additional parameter w male is estimated by ŵ male ¼ n male /n; (2) Moreover, in retaining the component of the likelihood involving C (and parameter w), we can compare directly the Akaike Information Criterion (AIC) values of models with sex-specificity to the AIC values of models without sex-specificity. If, on the other hand, we omit the component of the likelihood involving C and w for models without sex specificity, the likelihood will not be scaled consistently with sex-specific models and the models without sex specificity will always have a higher likelihood. It works out that the difference in log-likelihood is, for the binary covariate sex : X n C i logðŵþþð1 C iþlogð1 ŵþ ð4þ where ŵ is the MLE of w, i.e., w ¼ C. Therefore, post hoc adjustment of the log-likelihoods for models without sex could be done if they are fitted without the observed sex data C i, but it seems easier to retain the more general formulation within which all models (with or without class specificity of model parameters) are nested. In this case, the null model of no sex specificity of parameters includes the parameter w in addition to model parameters that are not sex specific. LIKELIHOOD ANALYSIS OF THE SCR MODEL Spatial capture-recapture models are an extension of ordinary capture-recapture to accommodate the spatial organization of individuals in a population and their capture locations. This is done by introducing an individual covariate, the home range or activity center of each individual, s i, for individual i ¼ 1, 2,..., N. The standard assumption is that the activity centers s i are distributed uniformly over the state-space S although covariates can be used to model non-uniform distributions (Borchers and Efford 2008, Royle et al. 2013b). SCR data are individual-, trap- and occasion-specific encounters y ijk for individual i, trap (or spatial location) j ¼ 1, 2,..., J, and occasion k ¼ 1, 2,..., K. Without loss of generality we consider an SCR model based on individual and trap-specific encounter frequencies y ij that are binomial outcomes based on a sample of size K (sample occasions). For example, if hair snares for bears are set for K ¼ 8weeks,theny ij would be the number of times individual i was encountered at hair snare j, having coordinate x j. The observation model is y ij ; BinomialðK; pðx j ; s i ÞÞ where the encounter probability p(x j,s i )depends on the distance between trap location x j and individual activity center s i.severalmodelsfor encounter probability are widely used (e.g., see Buckland et al. 2005) but the different models have no operational effect on the mechanics of computing the likelihood so we focus on a standard model which is proportional to the kernel of a bivariate normal density, centered at s i, for the coordinate x j : pðx j ; s i Þ¼logit 1 ða 0 Þexpð a 1 distðx j ; s i Þ 2 Þ; ð5þ where logit 1 (a 0 ) ¼ exp(a 0 )/(1 þ exp(a 0 )), and parameters a ¼ (a 0 a 1 ) are to be estimated along with population size N. For this model the parameter a 1 is usually expressed a 1 ¼ 1/(2r 2 ). Note if a 1 ¼ 0thentheSCRmodelreducestoan ordinary capture-recapture model but with traps serving as replicate samples. Therefore, SCR models can be viewed as ordinary CR models but with an individual covariate distance between trap and activity center. Let y i be the vector of trap-specific encounter frequencies (the space-time encounter history) for individual i, y i ¼ (y i1, y i2,..., y ij ). The joint distribution of the trap-specific encounter frequencies for individual i, conditional on s i, is the product of J binomial terms: v 4 February 2015 v Volume 6(2) v Article 22

5 ½y i js i ; aš ¼ YJ j¼1 K! y ij!ðk y ij Þ! pðx j; s i Þ y ij 3ð1 pðx j ; s i ÞÞ K y ij : The basic calculation required to evaluate the full likelihood is computing the marginal probability of the encounter frequencies y i for each individual i, which requires an integral over the state-space S: Z ½y i jaš ¼ ½y i js i ; aš½s i Šds i ð6þ s and the marginal probability of an all-zero encounter history: Z p 0 ¼½y ¼ 0jaŠ ¼ Binomialð0js; aþ½sšds: S Finally, we require the combinatorial term N under the binomial observation model for n n. Putting these three pieces together, the full likelihood for the SCR model has this form: ( ) Lða; n 0 jyþ ¼ N! Y n ½y n!n 0! i jaš p n 0 0 ð7þ which is implemented by the function intlik2 (Supplement; modified from Royle et al. 2014, chapter 6). The full likelihood is mentioned on page 379 of Borchers and Efford (2008) although they focus on a Poisson integrated version of this likelihood in which N is removed from the likelihood by integrating over a Poisson prior distribution. An R function for computing the SCR likelihood for a basic model without sex specificity is given in the Supplement. For the SCR model having class structure, the joint likelihood of the vector of trap-specific encounter frequencies y i and C i is, as before, the product of the likelihood conditional on the class variable C and the class probabilities: Pr(y i jc i )Pr(C i ). The first part, Pr(y i jc i ), comes from Eq. 6 but allowing for one or both parameters a to depend on C. For example, the intercept a 0 or the coefficient a 1 (related to the scale parameter r) may either or both depend on C. In the case where the covariate C is sex (say, C ¼ 0 for females and C ¼ 1 for males) then the contributions to the likelihood of the data for individual i look like: [y i jc i ]w if C i ¼ 1or[y i jc i ](1 w) ifc i ¼ 0. For the y ¼ 0 individuals which have no observed covariate, or for individuals that have a missing value of the covariate C i, we have to do a marginalization over the different values of the covariate: PrðyÞ ¼ X PrðyjC ¼ gþw g ð8þ g where the summation is over all possible values of the covariate C. The likelihood from Eq. 7 is extended to include n 1 individuals with nonmissing covariate data and n 2 individuals with missing covariate values as follows: ( Lða; n 0 jyþ ¼ N! Y ) ( ) n 1 Y n ½y n!n 0! i ja; C i Š½C i Š ½y i jaš p n 0 0 i¼n 1 þ1 ð9þ where n ¼ n 1 þ n 2, p 0 ¼ Pr(y ¼ 0) is computed by the marginalization Pr(y ¼ 0) ¼ R g Pr(y ¼ 0jC ¼ g)w g and, for individuals with missing values of the covariate, [y i ja] ¼ R g [y i ja, C ¼ g]w g. Describing the more general likelihood in the R language (we provide this in the Supplement) requires only a slight modification of the code from the model that does not contain class structure. Using this full likelihood which includes the individual covariate data C i, we may use AIC to compare models having class structure on either or both of the parameters a 0 and a 1 and models without class structure. We emphasize that, because this is an SCR model, we are in fact simultaneously marginalizing over two variables: the discrete class variable and the continuous covariate s being the activity center. But instead of making that explicit in the expression above, we just recognize that the marginalization over s is being done before the summation. Formally, the integral over s can be moved inside the summation implying that the integral can be done separately for each value of C. SIMULATION STUDY We have developed a formulation of the full likelihood that accommodates missing sex information and this allows for a comparison (e.g., by AIC) of models with and without sex-specificity of covariates. Without this formal framework for analysis of models containing sex structure, there v 5 February 2015 v Volume 6(2) v Article 22

6 are at least two ways to approach the problem of estimating population size or density from data sets with missing sex information. The first approach is to ignore sex specificity of model parameters and fit the null model with constant parameters ( null ). Intuitively, this should induce a bias in estimating N that depends on the true degree of sex specificity because there will be heterogeneity in encounter probability. Thus, there is induced bias analogous to misspecification of standard model Mh by a model without heterogeneity (Dorazio and Royle 2003). Larger differences in encounter probability among classes will lead to more heterogeneity in encounter probability and therefore more bias in estimating N. The second approach is that one could analyze the data set by discarding the encounter histories of individuals with missing sex information. While this has been done in practice (e.g., Boulanger et al. 2004, McCall et al. 2013), it is not advisable because it will necessarily bias estimates of population size. The expected value of ˆN in this case is not the true population size but, rather, the size of the population that is expected to produce non-missing sex information if it were exhaustively sampled. So if sex determination occurs randomly with probability u, and ˆN is the estimator of N under a model based only on known-sex individuals, then Eð ˆNÞ un. Therefore, some attention must be paid to estimating u if only known-sex individuals are used for estimation of N. This is not always so straightforward, especially in spatial capturerecapture problems where u is not constant but will depend on where individuals live and how many times they are captured. We conducted a small simulation study with two objectives: (1) to evaluate estimators of N under our full likelihood allowing for sex specificity of parameters and missing sex information and, especially, to study bias and variance as the proportion of missing values increases; (2) to evaluate estimators of N under models that disregard sex specificity of model parameters (i.e., the first approach to dealing with missing sex information mentioned in the preceding paragraph). We simulated a number of specific scenarios that varied by nature and strength of sex specificity on model parameters. In all cases we simulated population sizes of N ¼ 200 and the sex ratio was assumed to be 1:1 (w sex ¼ 0.50). To assess objective 1 we simulated data sets to have 0, 10%, 20%, 30% and 50% missing sex information. We simulated these levels of missingness by setting the per encounter probability of obtaining sex, h. Therefore, individuals that are captured more often will have a higher likelihood of obtaining sex information. We determined the value of h by simulation. For each value of h ¼ (0, 0.1,..., 0.9) we simulated an extremely large data set (22000 individuals). For each simulated data set the proportion of individuals with sex, p sex, was calculated. Then for these simulated data we interpolated the relationship between p sex and h to predict h for any desired level p sex (i.e., 50%, 70%, 80%, 90%, 100%) for the simulation study. Note that the relationship between h and p sex is specific to the state-space of the SCR model, the trap configuration, as well as the parameters a 0 and r. Populations of size N ¼ 200 were subjected to sampling for K ¼ 20 sample occasions by a array of traps having unit spacing. We used a half-normal encounter probability model of the form: pðx j ; s i Þ¼logit 1 ða 0 Þexpð ð1=ð2r 2 ÞÞdistðx j ; s i Þ 2 Þ with parameters defined as in Eq. 5. The statespace was defined by the square [ 1, 9] 3 [ 1, 9]. For each case 200 simulated data sets were used. We varied the parameters a 0 and r to obtain three levels of sex specificity in each parameter (none, low and moderate) (Table 1). Simulation results The results for each of the nine simulation scenarios are summarized in Table 2 which shows the mean and standard deviation of the MLE of N for each case. With respect to the first objective of our simulation study, we see that ˆN is essentially unbiased, as it should be, and we see that the standard deviation of ˆN hardly increases with the proportion of individuals with missing sex information except for cases 4 6 which show up to 5% increases in SD. These cases all have what we called moderate heterogeneity in r, corresponding to 0.4 trap spacing for females and 0.8 trap spacing for males. The precision of ˆN seems less affected by the level of heterogeneity in a 0 that we simulated (see cases 1 3). With respect to the second objective of the simulation, we see that there is clear bias in ˆN if we fit a model without sex-specificity. The bias is low v 6 February 2015 v Volume 6(2) v Article 22

7 Table 1. Description of nine scenarios used in the simulation study to evaluate the MLEs of N in the presence of missing sex information. For each case the proportion of encountered individuals with missing sex was varied between 0% and 50%. Each parameter is assigned distinct values for each sex as indicated in the table. Thus, if both values are the same there is no sex difference, denoted by none in column 2. The parameter definitions are: a 0 is the logit-scale baseline encounter probability, r is the scale parameter of a half-normal encounter probability model. Case degree of heterogeneity in a a 0 /r a 0 r Case 1 none/none ( 2.15, 2.15) (0.6, 0.6) Case 2 moderate/none ( 2.50, 1.80) (0.6, 0.6) Case 3 low/none ( 2.30, 2.00) (0.6, 0.6) Case 4 none/moderate ( 2.15, 2.15) (0.8, 0.4) Case 5 moderate/moderate ( 2.50, 1.80) (0.8, 0.4) Case 6 low/moderate ( 2.30, 2.00) (0.8, 0.4) Case 7 none/low ( 2.15, 2.15) (0.7, 0.5) Case 8 moderate/low ( 2.50, 1.80) (0.7, 0.5) Case 9 low/low ( 2.30, 2.00) (0.7, 0.5) (,1%) for cases 1, 3 and 8, somewhat larger but still modest (2 5%) for cases 2, 7, and 9 and considerably larger (between 7% and 15%) for cases 4 6. The results clearly suggest that larger sex differences will lead to larger biases in estimating N when the null model is used. Finally, we see that sex differences in the parameters a 0 and r can have antagonistic effects. Opposing effects in the parameters essentially cancel out. The reason for this is because both parameters interact to affect the typical area of space usage by individuals and they have a compensatory effect on space usage (Efford and Mowat 2014). If males have larger values of r but lower intercept a 0 then this reduces heterogeneity in net encounter probability and thus the bias is diminished (compare case 4 having roughly 14% bias with case 5 having roughly 7% bias). APPLICATION: BLACK BEARS IN NEW YORK We applied the SCR likelihood to data collected from a black bear study designed to investigate spatial patterns in bear density related to population growth and range expansion. The Table 2. This table shows the mean and SD of ˆN estimated using the full likelihood under various scenarios of missing sex information. The last row of each group of three ( null ) are results for fitting a model with no sex specificity of parameters. The model is only correct for Case 1 where we see the summaries of the MLE are the same for all rows, because there is no sex specificity in parameters and so the fraction of missing sex data has no effect on estimates. Percentage missing sex Case 1 Case 2 Case 3 Case 4 Case 5 Case 6 Case 7 Case 8 Case Mean SD Mean SD Mean SD Mean SD Mean SD Null Mean SD v 7 February 2015 v Volume 6(2) v Article 22

8 study was conducted in a c km 2 area in southern New York. The study utilized a noninvasive genetic sampling approach using barbed-wire hair snares (J ¼ 199) that were set from 4 June to 10 August 2012 to collect DNA samples to identify individual bears (Woods et al. 1999). Hair snares were relocated halfway through the 10-week sampling season to systematically sample the large study area, increase detection probability, and minimize snare habituation due to baiting (Boulanger et al. 2006). Hair samples were collected weekly and analyzed at seven microsatellite markers (G1D, G10L, G1A, G10B, G10H, G10O and Mu59) and one sexing marker (AMEL) using polymerase chain reactions (PCR)(Sun 2014; Sun et al., unpublished manuscript). The study identified 169 individual bears an average 2.8 times at 1.9 different snares. Genetic analysis for this study identified 31 males, 31 females, and 107 individuals of undetermined sex. The AMEL marker on the X and Y sex chromosomes have different nucleotide base-pair (bp) lengths; female (XX) black bears are distinguished by having only AMEL sequences of 244 bp while males (XY) have sequences of both 244 bp and 190 bp (Yamamoto et al. 2002). Sex was assigned by visual inspection of the AMEL amplicons on agarose gels. If no bands were present, we scored the sample as unknown sex and replicated the samples up to two additional times. The sex of an individual may be inconclusive or remain undetermined because of low DNA quantity or quality in the hair sample. The sex identified for one hair sample can be applied to other samples with the same microsatellite genotype, with the assumption that each individual has a unique genotype based on the suite of microsatellite markers used. Therefore, the likelihood of determining the sex of an individual increases with the number of hair samples collected from that individual. Genotypes are confirmed using a process of replicate PCR and analysis of genotyping errors. Nevertheless, missing data such as sex remain a possibility and is common in noninvasive studies that rely on samples with low quantity or quality DNA, such as hair or scat. We fitted a set of eight models which included a model with no behavioral response and sexspecificity on a 0, a 1 or both (four models total) and the same set of four models with a behavioral response. Results are summarized in Table 3. The models with behavioral response are favored by more than 400 AIC units across all models and, within the set of models containing a behavioral response, there is very strong evidence to support sex-specificity of both a 0 and a 1. The AIC of that model was about 14 units better than the model with sex-specificity of the intercept alone. We emphasize that the models with and without sex-specific parameters may be compared by AIC (or log-likelihood) here because estimates are based on the full likelihood, including the contribution of the observed sex data. The estimated sex ratio under the best model is w male ¼ 0.61 and naturally this estimate is sensitive to the form of sex-specificity of model parameters because the sex structure of the model determines the effective bias in observed proportions of males and females in the sample. The results indicate much lower baseline encounter probability for males than females: logit 1 ( 3.459) ¼ (females) and logit 1 ( 4.993) ¼ (males). Conversely, males are suggested to use more space with r m ¼ 8.20 km and r f ¼ 5.42 km. DISCUSSION Spatial capture-recapture models are increasingly being used for analyzing spatial encounter history data (Royle et al. 2014). We provided a general and explicit formulation of the likelihood for a binomial SCR model in the presence of missing categorical covariates such as sex. We used a formulation based on the full likelihood which we prefer over the conditional likelihood because it retains the population size parameter N in the likelihood to be estimated. Further, along with the class membership (sex) parameter w, class specific population size estimates may be obtained. An alternative formulation of class structured models in which N is removed from the full likelihood by integrating over a Poisson prior distribution for N has been implemented in the R package secr (Efford 2014; this has recently been described in the appendix of the documentation on finite mixture models: otago.ac.nz/density/pdfs/secr-finitemixtures. pdf ). Conditional formulations of the model v 8 February 2015 v Volume 6(2) v Article 22

9 Table 3. Model results of fitting sex-structured models to the black bear data set, ordered by AIC. The parameter N ¼ exp(log(n 0 )) þ n where n ¼ observed number of individuals. We defined w sex to be the probability that an individual in the population is a male, b is the behavioral response parameter and other parameters are as defined by Eq. 5. The models are: null (no sex effects), p(sex) (sex effect on the intercept only), r(sex) (sex effect on the coefficient of distance in the SCR model, both (sex effect on both the intercept and distance coefficient). The parameter r is related to the coefficient on distance in the encounter probability model by a 1 ¼ 1/(2r 2 ). Model-specific estimates are sorted by AIC for models without (top 4 rows) and with (bottom 4 rows) the behavioral response. Model a 0 or a 0,f a 0,m r or r f r m log(n 0 ) N w male b AIC D No behavioral response Both p(sex) r(sex) Null With behavioral response Both p(sex) Null r(sex) (e.g., Huggins 1989, Alho 1990) would not provide such information. And, such conditional approaches cannot accommodate missing values of the covariate. A benefit of our general formulation of the likelihood is that models with and without class structured parameters can be compared directly by AIC (or log-likelihood) with or without missing sex information. Given the prevalence of missing data, especially of individual sex, this is extremely important in practice. While our applied motivation for this work was based on development of models with sexspecificity, the general problem has relevance to other class structured situations such as group size, spatial strata, or multiple years, age or life stages, or other stratified populations. For example, it would be straightforward to analyze a small mammal trapping study involving multiple trapping grids, by defining C i to be the grid population to which individual i belongs (Royle and Converse 2013). If there are G such trapping grids then the likelihood sum given by Eq. 2 involves G terms and G 1 population membership probabilities to be estimated. The class-structured model allocates the total population size N (a super-population size among all G grids) among the G sub-populations. We note that the development we provided in this paper is precisely a likelihood formulation of the model described by Royle and Converse (2014). Imperfect assessment of sex has been considered in the development of Cormack-Jolly-Seber (CJS) models by a number of authors, where the general type of model is recognized as a multistate capture-recapture model with state uncertainty (Nichols et al. 2004, Pradel et al. 2008). Formulation of the SCR model is slightly complicated from a technical standpoint only because there are two marginalizations happening: the uncertain discrete state of the categorical covariate, and the uncertain continuous activity center variable s. In essence, the state model involves the joint distribution of discrete and continuous state variables. In the analysis of class structured populations, one has to take care in model selection based on AIC when comparing models with and without class structured parameters. In particular, the likelihood of the model without class structured parameters is not scaled properly unless the likelihood component involving the observed covariate values is retained in the model, (i.e., Eq. 4), along with the extra parameter w. For the reduced model lacking class structure of the parameters, there is no cost in estimating this extra parameter because the information comes from the data on the observed covariate values. However, the likelihood contribution of the covariate data must be accounted for in order to compare models with and without class structure based on likelihood (i.e., AIC). The use of non-invasive genetic methods has exploded over the last decade. These methods v 9 February 2015 v Volume 6(2) v Article 22

10 often produce samples for which individuality can be obtained, but sex information cannot. At the same time, sex-specificity of model parameters is necessary in many types of models, including SCR models, based purely on biological considerations. In the presence of missing sex information, the applicability or interpretation of model parameters can be problematic due to biasing mechanisms leading to the observation of the covariate sample. For example, in SCR studies the more often an individual is encountered the more likely is its sex to be determined. Thus individuals with more exposure to trapping have a higher likelihood of appearing in the known sex sample. As such, methods that account for unknown sex are extremely important. We applied the estimation method to a black bear data set from New York in which sex could not be ascertained for a majority of the individuals (107 out of 169 individuals had missing sex information). Discarding these individuals would not only represent an enormous loss of data, but, as noted previously, will produce a biased sample of individuals. ACKNOWLEDGMENTS We thank Melanie Moss for converting tables from Latex to Word. Any use of trade, firm, or product names is for descriptive purposes only and does not imply endorsement by the U.S. Government. LITERATURE CITED Alho, J. M Logistic regression in capturerecapture models. Biometrics 46: Belant, J. L., J. F. V. Stappen, and D. Paetkau American black bear population size and genetic diversity at Apostle Islands National Lakeshore. Ursus 16: Borchers, D. L A non-technical overview of spatially explicit capture-recapture models. Journal of Ornithology 152: Borchers, D. L., S. T. Buckland, and W. Zucchini Estimating animal abundance: closed populations. Springer, London, UK. Borchers, D. L., and M. G. Efford Spatially explicit maximum likelihood methods for capturerecapture studies. Biometrics 64: Borchers, D. L., W. Zucchini, and R. M. Fewster Mark-recapture models for line transect surveys. Biometrics 54: Boulanger, J., B. N. McLellan, J. G. Woods, M. F. Proctor, and C. Strobeck Sampling design and bias in DNA-based capture-mark-recapture population and density estimates of grizzly bears. Journal of Wildlife Management 68: Boulanger, J., M. F. Proctor, S. Himmer, G. Stenhouse, D. Paetkau, and J. Cranston An empirical test of DNA mark-recapture sampling strategies for grizzly bears. Ursus 17: Buckland, S. T., D. R. Anderson, K. P. Burnham, and J. L. Laake Distance sampling. John Wiley & Sons, New York, New York, USA. Dorazio, R. M., and J. A. Royle Mixture models for estimating the size of a closed population when capture rates vary among individuals. Biometrics 59: Efford, M. G Density estimation in live-trapping studies. Oikos 106: Efford, M. G secr: spatially explicit capturerecapture in R. pdfs/secr-overview.pdf Efford, M. G., and G. Mowat Compensatory heterogeneity in spatially explicit capture-recapture data. Ecology 95: Frary, V. J., J. Duchamp, D. S. Maehr, and J. L. Larkin Density and distribution of a colonizing front of the American black bear Ursus americanus. Wildlife Biology 17: Gardner, B., J. A. Royle, M. T. Wegan, R. E. Rainbolt, and P. D. Curtis Estimating black bear density using DNA data from hair snares. Journal of Wildlife Management 74: Huggins, R. M On the statistical analysis of capture experiments. Biometrika 76: Immell, D., and R. G. Anthony Estimation of black bear abundance using a discrete DNA sampling device. Journal of Wildlife Management 72: McCall, B. S., et al Combined use of markrecapture and genetic analyses reveals response of a black bear population to changes in food productivity. Journal of Wildlife Management 77: Mowat, G., and C. Strobeck Estimating population size of grizzly bears using hair capture, DNA profiling, and mark-recapture analysis. Journal of Wildlife Management 64: Nichols, J. D., W. L. Kendall, J. E. Hines, and J. E. Spendelow Estimation of sex-specific survival from capture-recapture data when sex is not always known. Ecology 85: Otis, D. L., K. P. Burnham, G. C. White, and D. R. Anderson Statistical inference from capture data on closed animal populations. Wildlife Monographs 62: Pollock, K. H The use of auxiliary variables in capture-recapture modelling: an overview. Journal of Applied Statistics 29: Pradel, R., L. Maurin-Bernier, O. Gimenez, M. Genov 10 February 2015 v Volume 6(2) v Article 22

11 vart, R. Choquet, and D. Oro Estimation of sex-specific survival with uncertainty in sex assessment. Canadian Journal of Statistics 36:1 14. Royle, J. A Analysis of capture-recapture models with individual covariates using data augmentation. Biometrics 65: Royle, J. A., R. B. Chandler, K. D. Gazenski, and T. A. Graves. 2013a. Spatial capture-recapture models for jointly estimating population density and landscape connectivity. Ecology 94: Royle, J. A., R. B. Chandler, R. Sollmann, and B. Gardner Spatial capture-recapture. Academic Press, Waltham, Massachusetts, USA. Royle, J. A., R. B. Chandler, C. C. Sun, and A. K. Fuller. 2013b. Integrating resource selection information with spatial capture-recapture. Methods in Ecology and Evolution 4: Royle, J. A., and S. J. Converse Hierarchical spatial capture-recapture models: modelling population density in stratified populations. Methods in Ecology and Evolution 5: Royle, J. A., and K. V. Young A hierarchical model for spatial capture-recapture data. Ecology 89: Russell, R. E., J. A. Royle, R. Desimone, M. K. Schwartz, V. L. Edwards, K. P. Pilgrim, and K. S. Mckelvey Estimating abundance of mountain lions from unstructured spatial sampling. Journal of Wildlife Management 76: Sun, C Estimating black bear population density in the southern black bear range of New York state with a non-invasive, genetic, spatial capture-recapture study. Thesis. Cornell University, Ithaca, New York, USA. Sutherland, C., A. K. Fuller, and J. A. Royle. In press. Modelling non-euclidean movement and landscape connectivity in highly structured ecological networks. Methods in Ecology and Evolution. doi: / X Waits, L. P., and D. Paetkau Noninvasive genetic sampling tools for wildlife biologists: a review of applications and recommendations for accurate data collection. Journal of Wildlife Management 69: Woods, J. G., et al Genetic tagging of freeranging black and brown bears. Wildlife Society Bulletin 27: Yamamoto, K., T. Tsubota, T. Komatsu, A. Katayama, T. Murase, I. Kita, and T. Kudo Sex identification of Japanese black bear, Ursus thibetanus japonicas, by PCR based on Amelogenin gene. Journal of Medical Veterinary Science 64: SUPPLEMENTAL MATERIAL SUPPLEMENT R code and scripts for computing the likelihood, simulating data and obtaining MLEs for the sexstructured SCR model (Ecological Archives, v 11 February 2015 v Volume 6(2) v Article 22