MULTISTATE MODELS FOR ESTIMATION OF SURVIVAL AND ...

The Auk 126(1):77–88, 2009  The American Ornithologists’ Union, 2009. Printed in USA.

Multistate Models for Estimation of Survival and Reproduction in the Grey-headed Albatross (Thalassarche chrysostoma ) S ar ah J. C onverse ,1,2,5 William L. K endall ,1 Paul F. D oherty, J r ., 3 2

and

P eter G. R yan 4

1 U.S. Geological Survey Patuxent Wildlife Research Center, Laurel, Maryland 20708, USA; Colorado Cooperative Fish and Wildlife Research Unit, Department of Fish, Wildlife, and Conservation Biology, Colorado State University, Fort Collins, Colorado 80523, USA; 3 Department of Fish, Wildlife, and Conservation Biology, Colorado State University, Fort Collins, Colorado 80523, USA; and 4 University of Cape Town, Percy FitzPatrick Institute of African Ornithology, Rondebosch 7701, South Africa

Abstract.—Reliable information on demography is necessary for conservation of albatrosses, the most threatened family of pelagic birds. Albatross survival has been estimated using mark–recapture data and the Cormack-Jolly-Seber (CJS) model. However, albatross exhibit skipped breeding, violating assumptions of the CJS model. Multistate modeling integrating unobservable states is a promising tool for such situations. We applied multistate models to data on Grey-headed Albatross (Thalassarche chrysostoma) to evaluate model performance and describe demographic patterns. These included a multistate equivalent of the CJS model (MS-2), including successful and failed breeding states and ignoring temporary emigration, and three versions of a four-state multistate model that accounts for temporary emigration by integrating unobservable states: a model (MS-4) with one sample per breeding season, a robust design model (RDMS-4) with multiple samples per season and geographic closure within the season, and an open robust design model (ORDMS-4) with multiple samples per season and staggered entry and exit of animals within the season. Survival estimates from the MS-2 model were higher than those from the MS-4 model, which resulted in apparent percent relative bias averaging 2.2%. The ORDMS-4 model was more appropriate than the RDMS-4 model, given that staggered entry and exit occurred. Annual survival probability for Greyheaded Albatross at Marion Island was 0.951 ± 0.006 (SE), and the probability of skipped breeding in a subsequent year averaged 0.938 for successful and 0.163 for failed breeders. We recommend that multistate models with unobservable states, combined with robust-design sampling, be used in studies of species that exhibit temporary emigration. Received 13 November 2007, accepted 25 June 2008. Key words: Grey-headed Albatross, mark–recapture, open robust design, temporary emigration, Thalassarche chrysostoma, unobservable state.

Des modèles multi-états pour estimer la survie et la reproduction de Thalassarche chrysostoma Résumé.—Les informations fiables sur la démographie sont déterminantes pour la conservation des albatros, la plus menacée des familles d’oiseaux pélagiques. Leur survie a été estimée à l’aide des données de marquage-recapture et le modèle Cormack-Jolly-Seber (CJS). Toutefois, les albatros peuvent sauter une saison de reproduction, dérogeant ainsi aux prémisses du modèle CJS. La modélisation multi-états intégrant les états inobservables constitue un outil prometteur pour de telles situations. Nous avons appliqué les modèles multi-états aux données sur Thalassarche chrysostoma pour évaluer la performance des modèles et décrire les patrons démographiques. Ces derniers incluaient un équivalent multi-état du modèle CJS (MS-2), qui tenait compte du succès de reproduction mais non de l’émigration temporaire, et trois versions d’un modèle multi-états à quatre états qui prend en compte l’émigration temporaire en intégrant les états inobservables : un modèle (MS-4) avec un échantillon par saison de reproduction, un modèle à conception robuste (RDMS-4) avec fermeture géographique saisonnière et échantillonnage multiple par saison et un modèle ouvert à conception robuste (ORDMS-4) avec un échantillonnage multiple par saison et des entrées et sorties des animaux du système pendant la saison. Les estimations de la survie à partir du modèle MS-2 s’avéraient plus élevées que celles du modèle MS-4, qui présentait un biais relatif apparent moyen de 2,2%. Le modèle ORDMS-4 était plus approprié que le modèle RDMS-4, en raison des entrées et sorties du système. La probabilité de survie annuelle de T. chrysostoma à l’île Marion correspondait à 0,951 ± 0,006 (erreur type) et la probabilité de ne pas se reproduire dans une année subséquente était en moyenne de 0,938 pour les reproducteurs ayant eu du succès et de 0,163 pour ceux ayant échoué. Nous recommandons que les modèles multi-états avec des états inobservables, combinés à un échantillonnage de conception robuste, soient utilisés dans des études sur des espèces qui émigrent temporairement. 5

E-mail: [email protected]

The Auk, Vol. 126, Number 1, pages 77–88. ISSN 0004-8038, electronic ISSN 1938-4254.  2009 by The American Ornithologists’ Union. All rights reserved. Please direct all requests for permission to photocopy or reproduce article content through the University of California Press’s Rights and Permissions website, http://www.ucpressjournals. com/reprintInfo.asp. DOI: 10.1525/auk.2009.07189

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Albatrosses (Diomedeidae) constitute a family of pelagic birds that is of great conservation concern, with 18 of 22 species threatened and the other four near-threatened under IUCN criteria (see Acknowledgments). Their life histories make albatross particularly vulnerable to anthropogenic threats: individuals are long-lived, delay breeding, and produce a maximum of one fledgling per year, and many breed only intermittently. In addition, the foraging behavior of albatross frequently brings them into contact with fishing vessels, resulting in bycatch mortality, the leading hypothesized cause of declines (Gales 1998). Given the threats to this group, demographic information is crucial for monitoring populations, investigating declines, and evaluating management effectiveness. Of the three basic demographic processes (reproduction, dispersal, and survival), the nest-success component of reproduction is the least difficult to estimate, because nests can be located and monitored with relative ease on breeding islands. Breeding probability has not been estimated as successfully. Dispersal of albatross between breeding colonies has received little attention because of perceived high fidelity to colonies. Estimation of survival has received the most attention, and mark–recapture methods have predominated (e.g., Weimerskirch et al. 1987, Waugh et al. 1999). However, estimating survival with mark–recapture models can be challenging, given the sampling situation for albatross. Albatross nest primarily on remote oceanic islands and are most easily observed by researchers during the breeding season. However, a variable proportion of individuals do not breed every year, and most nonbreeding individuals either do not visit the breeding colony or visit only sporadically. This “skipping” behavior constitutes what is referred to in the mark–recapture literature as “temporary emigration” (Kendall et al. 1997, Fujiwara and Caswell 2002, Kendall and Nichols 2002, Schaub et al. 2004). Typical mark–recapture models, such as the Cormack-Jolly-Seber model (CJS; Cormack 1964, Jolly 1965, Seber 1965), produce biased survival estimates if nonrandom (Markovian) temporary emigration occurs (Kendall et al. 1997). The multistate mark–recapture model (Schwarz et al. 1993), which permits individual birds to transition probabilistically between life-history states over time, has been proposed as a solution for estimating survival in the presence of temporary emigration (Fujiwara and Caswell 2002, Kendall and Nichols 2002, Schaub et al. 2004) by incorporating observable and unobservable states. In a case applicable to albatross, observable birds are those that attempt to breed, whereas unobservable birds are nonbreeders that are absent from the colony. States can be more finely defined, such as “failed breeder” and “successful breeder,” and estimated transitions between states can then be interpreted as estimated breeding attempt or success probabilities. Accounting for temporary emigration can be further enhanced by collecting data under Pollock’s robust design (Pollock 1982, Kendall et al. 1997), where detection probabilities are estimated from within-season data by repeated sampling of individuals over a period when a population can be assumed to be closed (e.g., during a breeding season). This model can yield unbiased survival estimates in the presence of temporary emigration, as well as estimates of emigration rates. An assumption of the original robust design models is that all observable individuals are available for sampling from the beginning to the end of the field


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season. In other words, the population is both demographically and geographically closed within a period, though Kendall (1999) noted some ways to relax this assumption. To generalize the robust design model, Schwarz and Stobo (1997) and Kendall and Bjorkland (2001) relaxed the geographicclosure assumption of the within-season model, referring to this formulation as the “open robust design model.” This model would permit individuals to arrive at and leave the breeding colony at different times within a breeding season and, thus, would not assume that every breeding individual is present for the entire breeding season. This may be a useful model for application to albatross, because individuals typically leave the breeding colony upon nest failure, and they also arrive at staggered times. However, both the closed and open robust design models described so far are restricted to one observable and one unobservable state. A useful extension of these models, which we use here, combines the multistate and robust design structures to maximize the ability to estimate parameters (e.g., Nichols and Coffman 1999). Kendall and Nichols (2002) explored parameter identifiability under the multistate temporary-emigration model (i.e., using an observable and unobservable state), including use of the robust design with this model, and concluded that the robust design improved both identifiability and precision of parameters. L. L. Bailey et al. (unpubl. data) and Converse et al. (2008) also found that the robust design offered significant improvements in the precision of parameters in multistate models, primarily in the parameters describing transitions between states. Here, we consider mark–recapture modeling of a data set on Grey-headed Albatross (Thalassarche chrysostoma). This species is listed as vulnerable on the IUCN Red List, with a declining population trend, and fisheries bycatch is thought to be a major culprit in declines. Some basic information exists on the demography of Grey-headed Albatross. For example, it is known that these albatross are not obligate “skippers” (i.e., it is possible for a bird to breed successfully in consecutive breeding seasons), though the large majority of successful breeders skip breeding in the subsequent year (Prince et al. 1994, Waugh et al. 1999, Ryan et al. 2007). Annual survival rates in this species have been estimated as 0.947 ± 0.005 [SE] (Prince et al. 1994) and 0.953 ± 0.009 (Waugh et al. 1999), and, as with many long-lived species, there is little evidence of strong interannual variation in survival. However, existing survival estimates for Grey-headed Albatross pre-date the availability of multistate models to account for the effects of temporary emigration. Furthermore, estimates of breeding probability have not been acquired with the benefit of multistate models. Improved information on the demography of Grey-headed Albatross is useful in understanding current status and threats and evaluating the effectiveness of ongoing conservation initiatives (e.g., Agreement on the Conservation of Albatross and Petrels; see Acknowledgments). We consider the application of four mark–recapture models to a data set on Grey-headed Albatross from Marion Island, Prince Edward Islands, in the southern Indian Ocean. Three of these models involve four life-history states: successful breeder, failed breeder, post-successful nonbreeder, and post-failed nonbreeder. The latter two states are considered unobservable and, therefore, detection probability is set to 0. Model MS-4 assumes that there is only one capture period per field season. Model RDMS-4

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incorporates multiple capture periods per field season and assumes that the breeding population is static for the entire season (i.e., closed robust design). Model ORDMS-4 also includes multiple capture periods per season but permits birds to arrive at and depart from the breeding colony at staggered times (i.e., open robust design). Finally, to emphasize the importance of acknowledging temporary emigration, we include a model (MS-2) that ignores unobservable states and acknowledges just the successful and failed breeding states. Our goals were (1) to evaluate the performance of the mark– recapture models considered and (2) to describe basic patterns in demographic parameters for Grey-headed Albatross at Marion Island, including survival and transitions between breeding states. M ethods Study area.—Marion Island is the larger of the South African Prince Edward Islands in the southern Indian Ocean (46°50′S, 37°45′E). Some 6,000 pairs of Grey-headed Albatross breed in colonies on steep slopes and sea cliffs along the southern coast of Marion Island each year (Nel et al. 2002a). Adults arrive in September and lay in October, and chicks fledge in April–May. Numbers have remained relatively constant over recent decades (Nel et al. 2002a), despite significant adult mortality in Patagonian Toothfish (Dissostichus eleginoides) longline operations around the islands in the late 1990s (Nel et al. 2002b). A study colony of 100–120 nests was established in 1997, at the onset of longline fishing for Patagonian Toothfish (Ryan et al. 2007). This study colony is separated by >200 m from nearby colonies; adjacent areas were checked once or twice each season for emigrants from the study colony, and very little emigration of breeders was observed. Grey-headed Albatross at Marion Island had not been studied in detail before, because the closest colony is about a 6-h walk from the island’s research station. Given the difficulty of regular access, checks of breeding birds were made at roughly two-week intervals during incubation and early chick stages to identify both partners at each nest (Ryan et al. 2007). Data collection.—We focus on breeding adults identified from the 1999 through 2006 breeding seasons. We use the year when eggs were laid (e.g., “2004”) to reference breeding seasons (e.g., October 2004–April 2005). An interval, such as “2004–2005,” refers to periods between breeding seasons. Breeding adults were individually banded with field-readable plastic bands as well as metal bands to reduce disturbance during subsequent monitoring. Nests were visited five to eight times during the early stages of each season to ascertain the social parents of each nest. Subsequent checks were made to determine the success of each breeding attempt, with chicks banded in early April before fledging. Each time a nest site was checked, the identity of the bird at the nest site was recorded. Thus, the data consist of a record of individuals both within and across breeding seasons, and whether they were detected or not on a particular sampling date, as well as their breeding success in a particular year. Data were collected from 469 individuals at the breeding colony (Appendix 1). Multistate structure.—In the MS-2 model, individuals were assumed to occupy one of two observable states—successful breeder (SB) or failed breeder (FB)—during each breeding season (Fig. 1, solid lines only). Two types of parameters describe


Fig. 1. Diagram of two multistate model structures used to model demography of Grey-headed Albatross: a two-state structure that does not account for temporary emigration and a four-state structure that does. The two-state model structure includes the states and state transitions represented by solid lines. The four-state structure adds to those the additional states and state transitions represented by dashed lines. States are successful breeder = SB, failed breeder = FB, post-successful nonbreeder = pSB, and post-failed nonbreeder = pFB, where “success” and “failure” refer to the breeding fate of birds breeding in the current year, and “post-success” and “post-failure” refer to the breeding fate of nonbreeding birds in the last year that they bred. Birds are unobservable, or “temporary emigrants,” when in states pSB and pFB.

between-season dynamics, including survival for each state in each interval t (StSB, StFB) and transition probabilities between states: ΨtSB-SB, ΨtSB-FB, ΨtFB-SB, ΨtFB-FB where Ψtx-y indicates a transition from state x at breeding season t to state y at breeding season t + 1, conditional on survival. In each case, one of these transition parameters (in our case, the probability of remaining in the current state) is derived by subtraction because of the multinomial nature of the transition probabilities (e.g., ΨtSB-SB = 1 - ΨtSB-FB). In addition to these two types of parameters, ptSB and ptFB describe the probability of detection during season t. In the MS-4, RDMS-4, and ORDMS-4 models, birds were assumed to occupy one of four states (Fig. 1, solid and dashed lines); two of these states are the observable states of the twostate model, including successful breeder (SB) and failed breeder (FB), whereas two additional states are unobservable, including post-successful nonbreeder (pSB) and post-failed nonbreeder (pFB). That is, the unobservable states correspond to birds that skip breeding after breeding successfully or unsuccessfully, respectively. In the four-state models, not every transition between states is possible. For example, transitions from SB or pSB to pFB are nonsensical. In the four-state models, a larger suite of survival and transition parameters (Fig. 1) are needed to describe between-season dynamics. Again we derived the probability of remaining in the current state by subtraction. In addition to survival and state-transition parameters, additional parameters appear with each model (MS-4, RDMS-4, and ORDMS-4) to account for within-season

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detectability and dynamics (described below). These within-season parameters apply only to observable states. Hunter and Caswell (2008) and L. L. Bailey et al. (unpubl. data) determined the set of nonredundant parameterizations of this four-state structure (sensu Gimenez et al. 2003). All of the basic parameterizations of interest to us were found to be identifiable given certain constraints, as described below. Data analysis.—For the two-state model, we were interested in four between-season parameterizations (i.e., parameterizations of survival and state transitions), and we formed these by varying structures on S and Ψ and considering combinations of these. The two structures on S were (1) S constant, S(.); and (2) S varying by breeding season, S(t). The two structures on Ψ were (1) Ψ constant, Ψ(.); and (2) Ψ varying by breeding season, Ψ(t). For the four-state model, we considered eight total betweenseason parameterizations by expanding the number of survival structures to four, including (1) S constant, S(.); (2) S varying by whether a bird was a breeder or not (i.e., was observable or unobservable), S(O,U); (3) S varying by breeding season, S(t); and (4) S varying by both breeding season and breeder–nonbreeder, including interactions, S(O,U * t). Previous simulation work (e.g., L. L. Bailey et al. unpubl. data) has indicated that, with unobservable states in this four-state multistate structure, different statedependent structures on S (e.g., StSB = StFB ≠ StpSB = StpFB vs. StSB ≠ StFB = StpSB = StpFB) cannot be reliably distinguished on the basis of model-selection metrics, so the state-dependent structure on S must be assumed. Therefore, we assumed that survival would be similar among breeding birds and similar among nonbreeding birds (e.g., StSB = StFB ≠ StpSB = StpFB). Under the MS-2 model, multiple within-season detection probabilities are not estimated, so multiple encounter occasions within a season in the Grey-headed Albatross data set were collapsed to a single encounter. This results in a maximum of t – 1 estimates of each of S, Ψ, and p, where p is the probability of detecting an individual during the breeding season, conditional on the bird being a breeder (i.e., being present in the breeding colony during the season). There is no p estimated for the first season, because the model conditions on first capture. Under the MS-2 model, we considered parameterizations wherein detection probabilities (1) were constant, p(.); (2) varied only by season, p(t); and (3) varied both by season and whether an individual bred successfully or not, p(t*success). The “success” effect allowed successful and failed breeders (states SB and FB) to have different detection probabilities. In all, we considered 12 parameterizations of the MS-2 model (i.e., four between-season models crossed with three within-season models). Similarly, in the MS-4 model, only one detection probability is estimated in each season, so we collapsed the multiple encounter histories as for the MS-2 model and considered the same suite of within-season parameterizations: (1) p(.); (2) p(t); and (3) p(t*success). We considered 24 parameterizations of the MS-4 model (i.e., 8 between-season models crossed with 3 within-season models). Using the robust design model, we estimated separate detection probabilities for each of the multiple sampling occasions during the season, over which we assumed the population to be closed. However, for the RDMS-4 model, we suspected a priori that the data did not meet the assumption of closure over the entire study period. Given failure of nests during the season, we suspected that, at a minimum, egress occurred (i.e., birds left the


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area before the end of the breeding season). If only egress occurs, unbiased estimates can be obtained from a model assuming geographic and demographic closure by collapsing all occasions after the first occasion into one, thereby creating a Lincoln-Petersen model (Kendall 1999) within a season. Therefore, we collapsed occasions 2 through j in each season into one detection occasion. Each season then had two sampling occasions: an individual could be detected on the first occasion and on any of the remaining occasions. Parameters in this model included t – 1 estimates of S and Ψ, in addition to two estimates of p for each (t) season. In this model, however, unlike in the multistate models, p refers to the two within-season detection probabilities. A metric of interest, and one analogous to the p estimate from the multistate models is p *, which is the probability that an individual is captured at least once in a breeding season and is computed as

p * = 1 – (1 – p1) * (1 – p2).

(1)

We used the delta method (Seber 2002) to calculate the variance on p*. We considered parameterizations of the RDMS-4 model (1) where detection probabilities were constant across breeding seasons and occasions within breeding seasons, p(.,.); (2) varying both by breeding season and occasion within breeding season, p(t,j); and (3) varying by breeding season, occasion within breeding season, and breeding success, p(t,j*success). In all, we considered 24 parameterizations of the RDMS-4 model type. The ORDMS-4 model (Kendall and Bjorkland 2001) uses the open model of Schwarz and Arnason (1996) to model within-season dynamics and allows both staggered entry and staggered departure dates from the sampling area (the RDMS-4 model with pooling, as described above, allows only staggered departure). We began the ORDMS-4 analysis by first analyzing each of the withinseason data sets (i.e., only observed individuals, or breeders) separately under the Schwarz-Arnason model to better understand within-season dynamics and to inform modeling of within-season dynamics in the ORDMS-4 model. The Schwarz-Arnason model includes three types of parameters: ϕj —the probability of being present in the sampling area at sampling occasion j, given present at occasion j – 1; pj – the probability of being detected at occasion j, given present in the sampling area at occasion j; and βj – the proportion of the population using the study area during the sampling period that arrives between occasion j and occasion j + 1. For these analyses, we put individuals into two groups corresponding to the two observable states, SB and FB. We considered five parameterizations of ϕ, including (1) fixed to 1 (i.e., allowing no egress, ϕ = 1); (2) varying between successful and failed breeders, ϕ(SB,FB); (3) varying between successful and failed breeders and over sampling occasions, ϕ(SB,FB *j); (4) fixed to 1 for successful breeders, constant for failed breeders, therefore assuming that all successful breeders are in the study area until the end of sampling but allowing failed breeders to exit, ϕSB = 1, ϕFB(.); and (5) fixed to 1 for successful breeders and varying by occasion for failed breeders, ϕSB = 1, ϕFB(j). We considered two parameterizations on p, including (1) constant over occasions, p(.); and (2) varying by occasion, p(j). We considered five parameterizations on β, including (1) no ingress (i.e., all individuals enter the colony before the first sampling occasion, β0 = 1; (2) constant, β(.); (3) varying between successful and failed breeders, β(SB,FB); (4) varying over occasion, β(j); and (5) varying between successful

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and failed breeders and over occasions, β(SB,FB *j). In all, we considered 50 parameterizations for each of the within-season data sets under the Schwarz-Arnason model. Considering all the possible within-season parameterizations described above in concert with the eight between-season parameterizations of interest under the four-state models would have yielded an overly large set of 50 * 8 = 400 model parameterizations for the ORDMS-4 model analysis. Instead, we considered three different parameterizations of the within-season dynamics, resulting in a more tractable set of 24 models. In one parameterization, we used the top-ranked structure (based on Akaike’s information criterion, corrected for small sample size [AICc]) from each of the separate within-season analyses under the SchwarzArnason model, ϕ(top) p(top) β(top). We also considered two parameterizations that were analogous to the RDMS-4 model, in which egress but no ingress was allowed, and with detection probabilities either constant or varying by breeding season, giving us ϕ(.) p(.) β0 = 1 and ϕ(.) p(t) β0 = 1. Again as for the RDMS-4 model, we report overall detection probability, p *, for each breeding season (see Kendall and Bjorkland [2001] for estimator). Several constraints were imposed to ensure that the models did not contain redundant parameters. In the MS-4, RDMS-4, and ORDMS-4, we constrained ΨT = ΨT – 1 for all states in parameterization Ψ(t) to avoid bias associated with estimating these parameters separately (Kendall and Nichols 2002). In the MS-2 and MS-4 models, we set pT = pT – 1 in parameterizations with season effects in both S and p; otherwise, the last S and p would be confounded. In the MS-4 model, in parameterizations with time dependence in both transitions and detections, it was necessary to fix p1 = p2 to eliminate redundant parameters, and in parameterizations with seasonal variation in all three parameters, it was necessary also to set ST – 1 = ST (L. L. Bailey et al. unpubl. data). An omnibus goodness-of-fit method for the models examined here does not currently exist. We used the most general available procedure, the median ĉ approach (Cooch and White 2007) implemented in Program MARK (White and Burnham 1999), to estimate overdispersion under the MS-2, MS-4, and RDMS-4 models. We assumed that the estimate of overdispersion, ĉ, that we obtained for the RDMS-4 model also applied to the ORDMS-4 model. The median ĉ approach attempts to, heuristically, “translate” the estimate of ĉ obtained by dividing model deviance by model degrees of freedom, which is known to be positively biased, to an unbiased estimate of ĉ through a combination of data simulation and logistic regression; we direct readers to Cooch and White (2007) for details on this procedure. For the MS-2 model, we used model structure S(t) Ψ(t) p(t*success) as the global model for goodness-of-fit testing. For the MS-4 model, we used S(O,U * t) Ψ(t) p(t*success). It is not possible to perform the median ĉ procedure in Program MARK with robust design data, so we used a 16-period multistate model, in which we fixed survival to 1 and transitions to 0 between the closed-capture period; we assumed a global model of S(O,U * t) Ψ(t) p(t,j*success) for the RDMS-4 model. We based inference on information-theoretic model selection, using quasi-likelihood AICc (QAICc; Akaike 1973, Hurvich and Tsai 1989, Burnham and Anderson 2002) for model selection; we also report Akaike weights (Burnham and Anderson 2002). All data analyses were completed in Program MARK, version 4.3 (White and Burnham 1999).


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R esults On the basis of the median ĉ procedure, we estimated an overdispersion parameter of 1.74 for the MS-2 model, 1.36 for the MS-4 model, and 1.15 for the RDMS-4 model (also assumed to apply to the ORDMS-4 model). We used these overdispersion parameters in calculating QAICc for model selection. In the MS-2 model analysis, only one parameterization, S(t) Ψ(t) p(t*success), was supported, with 98% Akaike weight (Appendix 2), which indicates that both survival and transitions were temporally variable and that detection probabilities varied by time and state (i.e., successful versus failed breeders). In the MS-4 model analysis, the parameterization with the most support was S(.) Ψ(t) p(t), with 48% Akaike weight (Appendix 3). Parameterizations with seasonal effects in survival had little support. Parameterizations without seasonal effects in transition probabilities had virtually no support, which indicates high yearto-year variation in transition probabilities. Comparison of survival estimates for equivalent parameterizations, S(t) Ψ(t) p(.), of the MS-2 and MS-4 models indicate that, assuming that the estimates from the MS-4 model are correct (i.e., assuming that temporary emigration should be accounted for in the estimation model), the MS-2 model provided survival estimates that are, in general, positively biased (5 of 7 estimates with positive bias; Table 1), with percent relative bias (PRB) ranging from –2.0% to 6.5% and averaging 2.2%. A similar pattern is evident when comparing estimates from the parameterization S(.) Ψ(t) p(t), where the MS-2 model estimates survival as 0.970 ± 0.007 (SE) and the MS-4 model estimates survival at 0.953 ± 0.007, with PRB = 1.8%. Furthermore, state-transition parameter estimates from the MS-4 model indicate that temporary emigration occurs to a large degree. For example, on the basis of estimates from the top-ranked model with time-constant transition probabilities, S(.) Ψ(.) p(t), on average, 0.938 of successful breeders skip breeding in the subsequent season, as do 0.163 of failed breeders. The RDMS-4 model analysis indicated strongest support (Akaike weight = 61%; Appendix 4) for the same between-season structure, S(.) Ψ(t), as the most supported MS-4 model parameterization. Again, there was no support under the RDMS-4 model Table 1. Survival estimates (S) and standard errors (SE, in parentheses) for Grey-headed Albatross at Marion Island, from the two-state (MS-2) and four-state (MS-4) multistate models where the models differ in whether they do (MS-4) or do not (MS-2) account for temporary emigration. Percent relative bias (PRB), associated with the MS-2 estimates, is presented assuming that the estimates from the MS-4 model are correct. The common model parameterization upon which estimates are based is S(t) Ψ(t) p(.). Interval 1999–2000 2000–2001 2001–2002 2002–2003 2003–2004 2004–2005 2005–2006

MS-4

MS-2

PRB

0.930 (0.027) 0.948 (0.020) 0.959 (0.021) 0.975 (0.028) 0.912 (0.035) 0.939 (0.052) 1.000 (0.000)

0.934 (0.031) 0.971 (0.027) 0.981 (0.031) 0.955 (0.037) 0.966 (0.040) 1.000 (0.000) 0.996 (0.076)

0.340 2.437 2.329 –2.044 5.909 6.490 –0.371

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Table 2. Range of estimates and average coefficient of variation (CV) on sets of season-specific state-transition probabilities based, in each case, on the top-ranked parameterization of between-season structure S(.) Ψ(t), from each of the four-state multistate models (MS-4, RDMS-4, and ORDMS4) used to model Grey-headed Albatross population dynamics. MS-4 Model ΨtSB-SB ΨtSB-FB ΨtSB- pSB ΨtFB-SB ΨtFB-FB ΨtSB-pFB ΨtpSB-SB ΨtpSB-FB ΨtpSB-pSB ΨtpFB-SB ΨtpFB-FB ΨtpFB-pFB

RDMS-4

ORDMS-4

Range

CV

Range

CV

Range

CV

0.01–0.10 0.00–0.08 0.82–0.99 0.29–0.77 0.16–0.51 0.08–0.21 0.64–0.77 0.14–0.30 0.04–0.15 0.26–0.87 0.13–0.66 0.00–0.14

73.1% 64.9% a 2.5% 13.0% 20.3% 38.9% 7.7% 22.5% 46.1% 35.5% 49.6% 84.3% a

0.01–0.09 0.00–0.09 0.82–0.99 0.29–0.72 0.18–0.54 0.10–0.19 0.60–0.76 0.15–0.34 0.01–0.20 0.19–0.75 0.21–0.73 0.00–0.48

67.1% 60.0% a 2.5% 12.5% 18.8% 40.7% 7.2% 21.7% 38.7% a 51.0% 57.2% 59.5% a

0.01–0.10 0.00–0.09 0.81–0.99 0.30–0.75 0.18–0.54 0.07–0.16 0.61–0.77 0.14–0.33 0.01–0.16 0.19–0.90 0.10–0.74 0.00–0.30

67.1% 59.6% a 2.5% 12.5% 18.5% 46.8% 7.1% 21.6% 43.0% a 62.1% 106.3% 283.9% a

a

One or more of the annual estimates was 0 or very near 0, so the CV could not be reliably computed. Reported averages exclude these cases.

for parameterizations without seasonal effects in transition probabilities. The detection model associated with all the top-ranked models was p(t,j*success), which indicates that detection probabilities were variable within seasons, across seasons, and between breeding states. The results of the eight within-season analyses under the Schwarz-Arnason model supported a few basic parameterizations of the within-season data (Appendix 5). Top-ranking models across all eight breeding seasons had parameterizations of β, the proportion entering the breeding colony, that were constant for all β (constant entry), or varied by occasion for all β (time-varying entry), or were fixed to 1 for β0 (no entry after start of sampling). There was generally no support for models where β varied by breeding outcome (i.e., arrival time was unrelated to breeding success). All supported parameterizations of the ϕ parameter had ϕ dependent on breeding success, including ϕ varying by breeding success and over occasions (exit probability depending on both breeding success and occasion), ϕ fixed to 1 for successful breeders and constant for failed breeders (no early exit for successful breeders and a constant probability of exit for failed breeders), or ϕ fixed to 1 for successful breeders and varying by occasion for failed breeders (no early exit for successful breeders and exit probability varying by occasion for failed breeders). Constant, p(.), and occasion-dependent, p(j), detection probabilities were supported in various seasons. For the ORDMS-4 model, the top-ranked between-season parameterization was again S(.) Ψ(t), with Akaike weight = 61% (Appendix 6). Again, all parameterizations without seasonal effects in transition probabilities had little support. Also, all the parameterizations with ≥ 0.001 weight had the open-model within-season structure, ϕ(top) p(top) β(top), rather than the simplified, closed-model within-season structure that was analogous to the RDMS-4 model. Because the top-ranked between-season parameterization for the three different four-state models was consistent—that is, S(.) Ψ(t)—we thought it most useful to directly compare the estimates from these top-ranked parameterizations in terms


of survival, transition, and detection probability estimates. The survival estimate from the MS-4 model was 0.953 ± 0.007 (coefficient of variation [CV] = 0.72%), that from the RDMS-4 model was 0.949 ± 0.006 (CV = 0.66%), and that from the ORDMS-4 model was 0.951 ± 0.006 (CV = 0.65%). The range of time-dependent transition-probability estimates and average CVs are presented in Table 2 for the MS-4, RDMS-4, and ORDMS-4 models. In most but not all cases, the average CV on transition probabilities was lower, as expected, for the RDMS4 and ORDMS-4 models than for the MS-4 model. However, because the variance inflation factor was smaller (1.15 versus 1.36) for the RDMS-4 and ORDMS-4 models, much of the gain in precision can be attributed to lower variance inflation. If the variance inflation factor is held constant across all three models, there is little gain in precision. Annual detection probabilities (p*) are provided in Table 3 for the MS-4, RDMS-4, and ORDMS-4 models. There is evidence from the RDMS-4 and ORDMS-4 models that failed breeders had lower detection probabilities than successful breeders, likely because of early departure from the breeding colony; this was not evident in the MS-4 model (i.e., the model selection did not support different detection probabilities for the SB and FB states). Although overall detection probabilities were generally 0.70, in the 2002 season (i.e., many birds transitioned into the successful-breeder state between 2001 and 2002; Fig. 3, solid line). Post-successful nonbreeders were unlikely to skip breeding for more than one year (Fig. 4, dotted line). Overall, post-successful nonbreeders were most likely to become successful breeders in the subsequent season (Fig. 4, solid line), and, again, this transition appeared to be particularly high in 2002. The estimated proportion of post-failed nonbreeders skipping in a subsequent year was small, frequently zero (Fig. 5, dotted line). Little information is available to estimate transitions out of this unobservable state, given that few birds enter this state

(i.e., most failed breeders breed again rather than skip). Therefore, these estimates have very low precision. Again, estimates indicate that a particularly high number of birds transitioned into the successful state between 2001 and 2002 (Fig. 5, solid line).

Fig. 2. Estimated state-transition parameters, and 95% Cl, describing movements from the successful-breeder state (SB), including transitions ΨtSB-SB (solid line; remaining in the successful-breeder state), ΨtSB-FB (dashed line; transition to the failed-breeder state), and ΨtSB-pSB (dotted line; transition to the nonbreeder state). Estimates are based on the ORDMS-4 model analysis.

Fig. 3. Estimated state-transition parameters, and 95% Cl, describing movements from the failed breeder state, including transitions ΨtFB-SB (solid line; transition to the successful-breeder state), ΨtFB-FB (dashed line; remaining in the failed-breeder state), and ΨtFB-pFB (dotted line; transition to the nonbreeder state). Estimates are based on the ORDMS-4 model analysis.


D iscussion Model performance.—The MS-2 model, because it fails to account for unobservable states, estimated higher survival probabilities than the MS-4 model. Kendall et al. (1997) found that if the probability of moving into an unobservable state is greater than the probability of staying in an unobservable state, there is a positive bias in survival estimates when the unobservable state is unaccounted for. Our findings agree with this result: according to our state-transition estimates, breeders are generally more likely than

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— Converse

et al.

— Auk , Vol . 126

Fig. 4. Estimated state-transition parameters, and 95% Cl, describing movements from the post-successful-nonbreeder state, including transitions ΨtpSB-SB (solid line; transition to the successful-breeder state), ΨtpSB-FB (dashed line; transition to the failed-breeder state), and ΨtpSB-pSB (dotted line; remaining in the nonbreeder state). Estimates are based on the ORDMS-4 model analysis.

Fig. 5. Estimated state-transition parameters, and 95% Cl, describing movements from the post-failed nonbreeder state, including transitions ΨtpFB-SB (solid line; transition to the successful-breeder state), ΨtpFB-FB (dashed line; transition to the failed breeder state), and ΨtpFB-pFB (dotted line; remaining in the nonbreeder state). Estimates are based on the ORDMS-4 model analysis.

skipped breeders to skip breeding in the following year. This is likely a common pattern among species of albatross that exhibit skipped breeding. Another indication of the effect of failing to account for temporary emigration can be seen in the model-selection results. Unlike for the four-state models, the MS-2 modelselection results gave greatest support to a parameterization with annual variation in survival probabilities. This annual variation was likely induced by the failure to account for annual variation in the degree of temporary emigration (i.e., the variation in the number of birds that skip breeding from year to year). The percent relative bias for a common parameterization of the MS-2 and MS-4 models with time-varying survival, assuming that the MS-4 model estimates (which account for temporary emigration) are correct, averaged 2.2% but was as large as 6.5%. With a time-constant survival parameterization, the percent relative bias was 1.7%. Compare this with the variation in survival estimates among the time-constant parameterizations of MS-4, RDMS-4, and ORDMS-4 models, where the variation was on the order of 0 to 0.4%. The influence of the apparently positively biased survival estimates from the MS-2 model, if used in a population-viability analysis, would likely be substantial, because population growth rates for long-lived species are generally most sensitive to adult survival (e.g., Croxall et al. 1990). Although the data probably met the assumptions of the MS-4 model, this model gave us less flexibility in modeling. We had to make more restrictions in parameters in time-dependent models (i.e., the constraints on detection probabilities in the first and last occasions and survival rates in the last occasion). Additionally, robust design data have the potential to give much more precision, especially on transition parameters (Kendall and Nichols 2002, Converse et al. 2008, L. L. Bailey et al. unpubl. data). Although this was not strongly borne out by our results—the overall increase in precision in the RDMS-4 and ORDMS-4 models as compared

with the MS-4 model was primarily attributable to the smaller variance-inflation factor—that is probably because the detection probabilities were quite high, and gains in precision with robust design sampling will diminish as p * approaches 1. The ORDMS-4 model has the most liberal assumptions of the two robust design-type models we applied to the data set. Given the within-season analyses and the model-selection results from the ORDMS-4 analysis, the data apparently did not meet the assumptions of the standard robust design, at least in some years. Although the RDMS-4 model may be preferred because it is simpler to use and may have greater precision (fewer parameters to estimate), having both staggered entry and exit is problematic with this model. Using the Schwarz-Arnason model (Schwarz and Arnason 1996) to evaluate within-season dynamics is, we believe, a valuable exercise, and it can be used to evaluate whether robust design data can be analyzed under the standard robust design model or whether the open robust design model is more appropriate. Staggered exit probably occurs in most albatross colonies, when failed breeders leave the colony before the end of the breeding season (i.e., ϕ ≠ 1). Given this, if within-season data supported parameterizations with β0 fixed to 1, indicating that no staggered entry occurs, this would allow use of the robust design model with pooling across the 2-t detection occasions. Otherwise, the open robust design model is more appropriate. However, the open robust design model will not be usable for robust design data that consist of only a few (i.e., 2–3) within-season sampling occasions, because these data will likely be inadequate for estimating the larger suite of parameters associated with the open robust design model. Demographic patterns.—Consistent with the findings of Prince et al. (1994) and Waugh et al. (1999), our analysis indicated low variability in survival across seasons, a pattern to be expected in a long-lived species. Prince et al. (1994) estimated annual survival at 0.947 ± 0.005 over the years 1975–1991 for Grey-headed


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J anuary 2009

— Demography

of


Albatross at South Georgia. This was a period when the annual nesting population was declining at an estimated average of 1.8% per year. Waugh et al. (1999) estimated Grey-headed Albatross survival at 0.953 ± 0.007 at Campbell Island, New Zealand, over the period 1984–1996, also during a period of population decline. Given that the findings of Nel et al. (2002a) and Ryan et al. (2007) suggest that the population of Grey-Headed Albatross at Marion Island is currently stable and has been since the early 1990s, it seems possible that our survival estimate of 0.951 ± 0.006 (ORDMS-4 model) is sufficient for population stability. Given the population declines noted by Prince et al. (1994) and Waugh et al. (1999), their estimates of survival, which are similar to ours but calculated from models that did not account for temporary emigration, may be positively biased. However, the importance of other demographic factors, such as prebreeder survival and reproductive success, cannot be ruled out, nor can overall island-to-island variation. These previous demographic analyses were conducted without the benefit of the newer tools available to demographers—specifically, tools to account for temporary emigration. Breeding probabilities traditionally have been estimated using return rates and are expected to be negatively biased unless detection is perfect. More recent estimates have not been produced for Grey-headed Albatross since the availability of multistate models. The ability to identify important demographic patterns directly from state-transition parameters is a benefit of these methods. For example, we estimated higher skipping probabilities in successful than in failed breeders, an expected pattern given the large energy requirement for successfully fledging a chick, and one that has been previously documented. In fact, Grey-headed Albatross have been viewed as near-obligate biennial breeders; Waugh et al. (1999) found that only 2% of successful breeders nested the following season at Campbell Island, and Prince et al. (1994) found that number to be