3 The Wilson Journal ofOrnithology 118(4):513-526, 2006 UTILITY OF OPEN POPULATION MODELS: LIMITATIONS POSED BY PARAMETER ESTIMABILITY 2IN THE STUDY OF MIGRATORY STOPOVER SARA R. MORRIS,1 AMANDA M. LARRACUENTE,1 KRISTEN M. COVINO,1 MELISSA S. MUSTILLO,1 KATHRYN E. MATTERN,1 DAVID A. LIEBNER,1 AND H. DAVID SHEETS — ABSTRACT. Open population models using capture-mark-recapture (CMR) data have a wide range ofuses in ecological and evolutionary contexts, including modeling of stopover duration by migratory passerines. In usingCMRapproachesinnovelcontextsthereisaneedtodeterminetheconditionsunderwhichopenpopulation models may be employedeffectively. Ourgoal wastodetermine whethertherewas asimpleapriorimechanism ofdeterminingthe conditions underwhich CMR modelscouldbe usedeffectively inthe studyofavian stopover ecology. Using banding data (n = 188 capture histories), we examined the challenges of using CMR-based models due to parameter inestimability, adequacy ofdescriptive power (Goodness-of-Fit, GOF), and parameter uncertainty. These issues become more apparent in studies with limited observations in a capture history, as is often the case in studies of avian stopover duration. Limited sample size and sampling intensity require an approach to reducing the number of fitted parameters in the model. Parameter estimability posed the greatest restriction on the utility ofopen population models, with high parameter uncertainty posing a lesserchallenge. Results from our study also indicate the need for >10 observations per estimated parameter (approximately 3 birds captured or recaptured per day) to provide a reasonable chance of successfully estimating all model parameters. Received 13 July 2005, accepted20 May 2006. Migratory birds frequently use stopovers to Species-specific stopover patterns may reflect complete migration successfully between their both intrinsic characteristics and ecological breeding and wintering grounds. Stopover factors associated with individual stopover sites provide refuge from predators, protection sites (Kaiser 1999). Schaub et al. (2001) argue against inclement weather, and food resources for accurate estimates of stopover duration to to allow fatdeposition to fuel migratory flight. test models ofoptimal migration strategy, spe- It is thought that many migrating passerines cifically the trade-off between time spent in cannot store enough fat to complete their mi- flight or at stopovers. gration in a single transit, but must refuel by Although the importance of en route mi- foraging at stopover sites along their routes gratory stopover sites is well recognized (Dunn 2001, Schwilch and Jenni 2001). Pro- (Moore 2000, Petit 2000, Sillett and Holmes viding evidence for the use of stopover sites 2002, Heglund and Skagen 2005), all sites are for refueling, Moore and Abom (2000) doc- not equal. Mehlman et al. (2005) recommend umented increased activity patterns and dif- that important stopover sites be identified ferential habitat use by lean versus fat mi- based on the relative migrant abundance, the grants. Lean migrants needing to refuel may availability ofresources that allow birds to re- stay longer at stopover sites than fat migrants plenish fat reserves, and the location of the (Moore and Kerlinger 1987, Yong and Moore site relative to other sites and ecological bar- 1997), and the rate of mass gain also may af- riers. However, specific criteria for assessing, fect stopover duration. The length oftime that and statistical approaches for comparing, sites migrants stay at stopover sites will affect the have not been established. Furthermore, there total duration of migration and may affect the is a recognized need for research on how sites ability of birds to obtain quality territories. differ by season, species, and species demog- raphy (Mehlman et al. 2005, Partners in Flight 1 Dept, ofBiology, CanisiusCollege, 2001 MainSt., Research Working Group 2002). Buffalo, NY 14208, USA. Since the mid-1980s, numerous researchers 2Dept, ofPhysics, CanisiusCollege, 2001 MainSt., Buffalo, NY 14208, USA. have described the basics ofthe stopoverecol- 3Corresponding author; e-mail: ogy of migratory landbirds at individual sites morriss@canisius.edu along the northern coast ofthe GulfofMexico 513 514 THE WILSON JOURNAL OF ORNITHOLOGY • Vol. 118, No. 4, December2006 (Moore and Kerlinger 1987, Moore et al. hood methods; characteristics of the capture 1990, Kuenzi et al. 1991), the New England history and the model’s mathematical struc- coast (Morris et al. 1994, 1996; Parrish 2000), ture will determine the number of parameters the Great Lakes coasts (Jones et al. 2002, that can be reliably estimated. Parameters that Bonter 2003), and in western states (Winker are inestimable due to limitations of a given et al. 1992, Finch and Yong 2000). Most of capture history are extrinsically non-identifi- these studies provide simple analyses of stop- able (McCullagh and Nelder 1989, Viallefont over duration based on recapturing banded et al. 1998). Capture histories that involve birds. Calculating the amount of time lapsing long periods of time, particularly those with between the first capture and the last recapture relatively few captures and/or recaptures, of- (Cherry 1982) has been the traditional method ten prevent successful estimation ofall param- ofestimating stopover duration at a given site; eter values; the resulting extrinsic non-identi- however, including only recaptured birds pro- fiability ofparameters either precludes the use vides conservative estimates of stopover du- of open population models or requires reduc- ration because birds not recaptured have not ing the number of parameters. necessarily left the field site. If only recap- One approach to reducing the number of tured birds are used in analyses (regularly parameters that must be fitted for a given <5% of all banded migrants are recaptured), model is to pool observations over several this simple approach might provide a biased consecutive observation periods (e.g., Schaub view of site use because >95% of migrants and Jenni 2001, Schaub et al. 2001). However, are excluded from analyses. pooling may bias the parameter estimates and The limitations of the minimum stopover preclude comparing models with different approach have resulted in the suggestion that pooling intervals (Hargrove and Borland open population models based on capture- 1994, Morris et al. 2005b). The difficulty as- mark-recapture (CMR) data be used to esti- sociated with the need to establish this basic mate stopover duration (Lavee et al. 1991, temporal interval has been recognized in the Holmgren et al. 1993, Kaiser 1995, Schaub et paleontological literature (Connolly and Mil- al. 2001). The Pradel (1996) extension of the ler 2001, Xu et al. 2005), where it has been Cormack-Jolly-Seber (CJS) models allows for addressed by determining whether ornot anal- a range of models of the probabilities of ani- ysis results remain consistent as the pooling mal capture, arrival, and departure within each interval is changed. Additional detailed dis- interval of a given study period. A number of cussion of pooling and its effects appears to useful statistics may be derived from the sto- be lacking in both the statistical and ecologi- chastic models, including mean time animals cal literature. An alternative to pooling is to are present in the study area, mean capture use multiple-day constancy (MDC; Fig. 1), probability, and temporal patterns of arrival, which holds parameter values fixed over a departure, and population size. These models given “constancy” interval, thus reducing the also could allow meaningful comparisons of number of parameters while retaining all in- several stopover characteristics among sites. formation in the capture history (Morris et al. Although the assumptions used in deriving 2005a). Regardless of the method used to re- open population models are widely known duce the number ofparameters, decreasing the (e.g., Pollock et al. 1990, Cooch and White number ofparameters in a model will increase 2005), the conditions under which these mod- the likelihood that all parameters can be suc- els can be used are rarely discussed. Charac- cessfully estimated, by reducing the incidence teristic—s ofthe data (i.e., capture/recapture his- of extrinsic non-identifiability. tories) especially sample size, number of When using open population models, good- temporal sampling intervals avai—lable, recap- ness-of-fit (GOF) tests must be applied to de- ture/resighting/recovery rate, etc. may great- termine whether the models have adequate de- ly impact the potential usefulness of these scriptive power prior to biological applica- models. To use a given open population mod- tions. Two distinct approaches (analytical tests el, first all the model parameters must be es- based on contingency tables and numerical timated. Typically, parameter estimates are tests based on comparing observed model obtained using numerical maximum likeli- misfit or deviance to estimates of misfit de- Morris et al. • UTILITY OF OPEN POPULATION MODELS 515 A Day Day Day Day Day Day Day Day Day 2 3 4 5 6 7 8 9 1 Pi P2 Pi P4 Ps Ps Pv Ps Ps m | 0; I I >2 I I foi I I 04 I I §5 I I <)>6 1 1 4*7 11 08 | | 4*9 • • • ~T1 Yi Y> YS Y7 YS 79 . | | | | | | | | 1 | | 1 B Interval 1 Interval 2 Interval 3 (Days 1-3) (Days 4-6) (Days 7-9) Pi Pi Pi 2 Yi Yi C FIG. 1. Open population models maybe usedtoestimate stopoverduration by migratorybirdsbyestimating daily rates of capture, arrival, and departure. Large numbers of parameters are required to work with (A) raw data, while both (B) pooled data (3-day pooling interval) and (C) multiple-day constancy (MDC, 3-day MDC interval) provide a reduction in the number ofparameters in the open population models fitted to bird banding data. Since limitedsample sizesmakeparameterestimationdifficult, somereduction inthe numberofparameters may allow use ofthese models with smaller data sets. Both pooling and MDC approaches reduce the number offitted parameters: p = probability ofcapture; 4> = probability that a bird captured on one day remained until the following day (i.e., survival); and y = probability that a bird captured on one day was there the day before (i.e., seniority). Pooling, however, loses information from multiple captures in the same interval, whereas MDC retains information on all captures. Figure adapted from Morris et al. (2005a). 516 THE WILSON JOURNAL OF ORNITHOLOGY • Vol. 118, No. 4, December2006 rived from simulations) have been used to de- tured or recaptured were transported to a cen- termine whether open population models fit tral location for banding and data collection. the data. Once the mostcomplex model passes For species with a sample size >50 indi- the GOF test, selection of the most appropri- viduals in a single season, we created a cap- ate model (of those nested within this most ture history that indicated whether any one in- complex model) for the data using Akaike’s dividual was captured on a given day. Using Information Criterion (AIC) can occur. Even this capture history, we calculated minimum when models can be chosen and fit, the vari- stopover by subtracting the date of first cap- ances of parameter estimates obtained from ture from the date of final capture, following open population models may be too large for Cherry (1982). Additionally, we calculated a the estimates to be useful. The coefficient of variety of descriptive statistics that were used variation (CV; the standard deviation of the for discriminant function analyses (see be- estimate/the value ofthe estimate X 100) may low). — be used to assess the potential utility of stop- Capture-mark-recapture. The first step in over estimates. A low CV is necessary for ef- the analysis was to determine the most com- fective comparison of statistical measures plex model for which all parameters could be among species, locations, and/or time periods. estimated. Numerical maximum likelihood However, little attention has been paid to the methods were used to fit Pradel’s (1996) ex- dependence of the CV on the characteristics tension of the CJS open population models to of the capture history. each capture history. Pradel’s model requires In this study, we examined capture histories estimation ofsighting (p = probability ofcap- from migration banding data to determine the ture), seniority (y = probability that the bird utility ofopen population models for estimating was present at a stopover site during the pre- avian stopover duration. We used a large num- vious day), and survival (4> = probability of = ber of field capture histories (n 188) from remaining at a stopover site until the next migration banding datasets rather than relying day). We considered time-dependent open MDC on computer simulations. Whereas computer population models with intervals (Mor- simulations would provide greater control over ris et al. 2005a) ranging from 1 to 7 days. In parameters, we wanted to be sure to cover a the MDC approach to time-varying parame- MDC wide range of natural conditions represented by ters, the parameters are fixed over the empirical data. Specifically, we were interested interval. However, all captures and recaptures in determining how data characteristics affect within and between MDC intervals have an parameter estimability (through extrinsic non- influence on the likelihood function and, identifiability), the ability of models to pass hence, the parameter estimates. Each of these GOF tests, and the CV of stopover duration es- time-dependent models (in which sighting, timates. Estimating the range of sample sizes survival, and seniority probabilities were all and recapture rates to which open population free to vary from one constancy interval to models can be fitted may help us determine the next) was fitted to the capture history, and whether these approaches are appropriate for a the number of extrinsically non-identifiable particular capture history. To that end, our re- parameters was identified using an estimate of sults indicate the conditions under which open the rank of the Hessian matrix (Viallefont et population models can be used effectively with al. 1998). Rank deficiency in the Hessian ma- banding data. trix was estimated by using finite-difference methods, and then tested using the singular METHODS value decomposition method (Viallefont et al. — Data collection Migrating birds were 1998). Rank deficiency was taken as indicat- . captured in mist nets at Appledore Island, ing extrinsic parameter non-identifiability in a Maine (1996-2002); Star Island, New Hamp- model. While some parameters in Pradel’s ex- shire (1999 and 2000); and Hamlin Beach tension of the CJS model are non-identifiable State Park, near Rochester, New York (1999 due the model’s structure (i.e., intrinsic ines- and 2000). Mist nets were operated daily dur- timability), this form of inestimability is part ing the spring and fall migration seasons ex- of the model, and does not negatively impact cept during inclement weather. All birds cap- its further use. We are concerned here with Morris et al. • UTILITY OF OPEN POPULATION MODELS 517 extrinsically inestimable parameters in band- category (>10 observations per parameter) ing data. Inestimability makes it difficult to roughly corresponds with three birds of that use eitherthe Schaub et al. (2001) formulation species captured or recaptured per day. This of the stopover duration or the more recent categorization allowed us to examine the de- estimate put forward by Efford (2005). Al- pendence of estimability on the ratio of ob- though Efford’s approach appears simpler servations to parameters, and does not require than that of Schaub et al. (2001), it still re- that the sampling intervals used in a study be quires an estimate ofthe distribution ofarrival in units of days. times, thus necessitating the estimation of the Capture histories were tested for GOF by same number of parameters (See Efford’s assessing the ability of time-dependent (i.e., equation 5 and discussion). To be useful in the most complex) models to fit the data. Both estimating stopover duration (Schaub and Jen- analytical tests (based on contingency tables) ni 2001, Schaub et al. 2001), all intrinsically and numerical tests (based on parametric estimable parameters in a model had to be bootstrap procedures) have been used in con- CMR completely identifiable, so those capture his- junction with models. The first approach tories with non-identifiable parameters due to is to use contingency tables to test whether MDC the structure of the data in all intervals assumptions of the open population models tested werejudged unusable forfurtheranalysis. are violated. Specifically, contingency tables We used software written by HDS and DAL are used to test the assumptions that each using MATLAB (The MathWorks, Inc. 1992) marked animal in the population at time t has to implement Pradel’s population growth rate (1) the same probability of recapture, and (2) (PGR) method (Pradel 1996). We compared the same probability of survival (Pollock et the performance of our software to that of al. 1990). Several variations on these tests MARK (White and Burnham 1999, Cooch have been incorporated into the programs RE- and White 2005) and SURGE (Lebreton et al. LEASE (Lebreton et al. 1992, Burnham et al. MARK 1992, Pradel and Lebreton 1993, Cooch et al. 1987), (White and Burnham 1999), 1997); it produced identical results for a num- and U-CARE (Choquet et al. 2005). The con- ber of capture histories, both from our data tingency tables can be pooled to produce an and from example files distributed with overall chi-square statistic for the capture his- MARK. When using very sparse data, our tory as a whole, as well as testing specific software and SURGE had similarconvergence hypotheses about violations ofmodel assump- properties, with results depending less on tions. When faced with sparse data, the con- sample size than they did in MARK, which tingency tables may be pooled to improve may be attributable to differences in the par- their performance, particularly when the num- ticular link function (the default choice) we ber of expected outcomes in one or more cat- MARK used in (Cooch and White 2005); this egories of the contingency table is very low. particular difference in performance was not Pooling contingency tables, however, does not investigated in depth. always result in tables with enough entries in Since capture histories included a range of each cell to be useful. All of our capture his- MDC sample sizes and durations, comparing capture tories that had estimable models for in- histories required a time-invariant measure of tervals of <7 days were submitted to GOF sampling intensity. We used the numberofob- testing using the contingency table methods in servations (sum of all capture and recapture U-CARE (Choquet et al. 2005). events) per estimated parameter in a 7-day, The second alternative is to use numerical MDC time-dependent model as the measure simulations to determine whether the ob- of observations per parameter. The 7-day served model deviance is consistent with the MDC model had the lowest number ofparam- deviance distribution obtained by using the eters of any model used in the estimability model in a parametric bootstrap procedure determination procedure discussed above. We (also called a Monte Carlo simulation). The divided the capture histories into three cate- model deviance is the difference between the gories, based on the number of observations observed log-likelihood and the log-likelihood (1)2<#<5, (#) per estimated parameter: for a “saturated” model, and it serves as a (2) 5 < # < 10, and (3) # > 10. Our highest model’s measure of fit. In such a procedure 518 THE WILSON JOURNAL OF ORNITHOLOGY • Vol. 118, No. 4, December2006 MARK (as implemented in and our software), reasonable models). The lowest AIC value in- c the model is used to generate a series of sim- dicated the most appropriate model foragiven ulated capture histories ofthe same size as the capture history, thus determining the appro- MDC original capture history. The model is fit to priate interval and whether each param- each ofthe simulated capture histories in turn, eter was constant or time-dependent. In addi- and a confidence interval for the deviances tion to determining which model was the most observed over the simulated data is obtained. appropriate, the AICc score was used to assign If the observed deviance is high (above the a relative AIC weight (w) to each model, c 95% upper bound of the simulation devianc- which reflects the relative probability that es), then it may be possible to continue the each model is correct. If the AICc weight of analysis by computing an estimated variance the chosen model was <0.95, we also includ- inflation factor (c) and using this to adjust the ed additional models with relatively high AIC c statistics ofmodel choice (White 2002, Cooch weights. Thus, the number ofmodels included and White 2005). Data sparseness also affects was determined by a cumulative AIC weight this parametric bootstrap approach to GOF of 0.95, so that all models with a reacsonable testing because the model must be fit to the chance of being correct were considered. We simulation data during the estimation of the used a bootstrapping procedure to determine range of deviances. Each capture history was the total stopover duration estimate and the GOF MDC tested for at the lowest interval standard deviation of this estimate (following for which the model parameters were identi- Schaub et al. 2001). fiable, using software written by HDS and Schaub et al. (2001) present a derivation of DAL. Parameter identifiability was monitored the expected total stopoverduration calculated during the GOF testing procedure, as it also as a daily value; we report the average total poses a problem when conducting Monte Car- stopover duration over the migration season. lo simulations. Similarly, capture histories ex- In our method, the daily stopover is weighted hibiting evidence of a lack-of-fit (i.e., those by the estimated probability of arrival times, with deviances outside the 95% confidence in- using the estimated population growth rate as tervals from the simulations) over all seven presented by Pradel (1996). Efford (2005) ar- intervals were not subjected to further analy- gues that the total stopover duration (Schaub sis. We did not make use of the c estimation et al. 2001) produces an overestimate of the procedure (White 2002, Cooch and White actual duration. Efford (2005) advocates using 2005), as it turned out that only two capture a weighted average of Schaub et al.’s “stop- histories fell into this category of results. over-after” estimate using a weighting derived After a time-dependent model was shown from Schwarz and Arnason’s (1996) estimates to exhibit GOF, we compared competing mod- of the distribution of arrival times (Equation els to determine which model was optimal for 5 in Efford 2005). We also present the stop- producing stopover estimates. Model selection over-after statistic, again weighted using the compared all prospective models over several estimated population growth rate as derived MDC intervals for each capture history, be- from Pradel (1996). Conceptually, this ap- MDC ginning with the smallest interval that proach is the same as that presentedby Efford, passed GOF. We excluded prospective models although the computations may differ slightly, that had both constant seniority and survival as the Pradel (1996) parameterization of the because they predict a population size that is problem differs from that used by Schwarz constant or monotonically increasing or de- and Amason (1996). creasing. Based on field observations, we In addition to having adequate descriptive know that during the migration period the power and being estimable, the chosen model population present at a stopover site increases must yield a useful statistic for comparisons. to a maximum value and then declines to zero, The coefficient of variation (CV) was used to making any model predicting constant popu- determine usefulness ofthe total stopover sta- lation size or a monotonic pattern of change tistic estimated for each species in each sea- in population size biologically unreasonable son. CV was calculated by dividing the stan- (see Burnham and Anderson 1998 for a dis- dard error of the total stopover estimate by its cussion of the exclusion of biologically un- mean and multiplying by 100. In this study. Morris et al. • UTILITY OF OPEN POPULATION MODELS 519 TABLE 1. Summary ofthe utility ofopen opulation models in three categories representing the numberof observations (#) per estimated parameter for a given capture history from avian banding data. To be applicable, models had to have estimable parameters and pass goodness-of-fit (GOF) testing. Asthe numberofobservations per parameter increased, the number ofcapture histories that could be analyzed using open population models also increased. Parameter inestimability in both model fitting and GOF testing poses the greatest impediment to the use of open population models at these sample sizes. Bird banding data were collected during spring and fall migration on Appledore Island, Maine (1996-2002); Star Island, New Hampshire (1999-2000); and Hamlin Beach StatePark, New York(1999-2000). The bandingdatawereusedtocreatecapturehistories,whichindicate whether and individual bird was captured on a particular day; a separate capture history was created for each bird species for which there were >50 captures at a single location during a specific season. No.observationsperestimatedparameter 2<#<5 5 <#< 10 Capturehistoriesthat: (n =42) (« = 81) <10(n = 65) Had inestimable parameters 24 (57%) 29 (36%) 16 (25%) Were inestimable in simulation GOF 15 (36%) 30 (37%) 6 (9%) Failed simulation GOF 0 (0%) 0 (0%) 2 (3%) Failed U-CARE “transients" test 0 (0%) 1 (1%) 4 (6%) Had an applicable model 3 (7%) 21 (26%) 37 (57%) Had a CV <50% in total stopover duration3 1 (2%) 7 (9%) 15 (23%) Had a CV >50% in total stopover duration3 2 (5%) 14 (17%) 22 (34%) Had a CV <50% in stopover-afterb 1 (2%) 9 (11%) 18 (28%) Had a CV >50% in stopover-afterb 2 (5%) 12 (15%) 19 (29%) 3Totalstopoverestimatesarebasedonopenpopulationmodelsandestimatesfromstopoverdurationanalysis(SODA)describedinSchaubetal.(2001); CV(coefficientofvariation)= (SE/mean)X 100. bStopover-afterestimatesarebasedonopenpopulationmodelsandestimatesusingequation5fromEfford(2005). only CV values <50% were considered useful percentage of days on which there was >1 because comparing different stopover esti- capture event, while “completeness two” re- mates is impossible when CV values are sub- fers to the percentage ofdays with >2 capture stantially >50%. CV values could, of course, events. “Recapture completeness” and “re- be determined for any estimated parameters in capture completeness two” refer to the per- the model; we focus here on the derived sta- centage of days with >1 or >2 recaptures, tistic (stopover duration) relevant to the study respectively. Backwards stepwise discriminant of migration ecology. analyses were performed in SYSTAT 10.2 — Discriminant function analyses. We used (SYSTAT Software, Inc. 2002). discriminant function analyses to examine RESULTS which conditions led to estimability ofparameters in the original capture history and during We examined the parameter estimability of GOF testing. We used a range of simple sta- 1- to 7-day MDC models applied to 188 cap- tistics that could be calculated without em- ture histories representing 34 different species CMR ploying the complex models. The vari- (97 capture histories from fall and 91 from ables included in these analyses were the spring migration). Of these, we were able to number of individuals captured, number of obtain estimable parameters of a completely MDC days sampled, percent of individuals recap- time-dependent model for 119 capture MDC tured at least once, total number of captures histories. The interval at which models and recaptures, total number of recaptures, could be estimated varied among capture his- number of captures per day, median captures tories. The shortest interval that could be used per day, recaptures per day, number of days ranged from 3 to 7 days (3-day n = 15, 4-day with no captures orrecaptures, minimum stop- n = 22, 5-day n = 40, 6-day n — 21, 7-day over estimate, standard deviation of the min- n = 21). Parameter estimability was strongly imum stopover estimate, standard deviation in dependent on the number of observations per the number of captures per day, and several parameter (Table 1). Estimability also played measures of capture consistency, which we a large role in the outcome of GOF testing. term “completeness.” Completeness is the Relatively few capture histories failed GOF 520 THE WILSON JOURNAL OF ORNITHOLOGY • Vol. 118. No. 4. December2006 MDC testing in an absolute sense. Five capture his- of time-dependent parameters and in the tories showed evidence of differences in cap- interval used in the models. When the AIC c ture probabilities of previously recaptured in- was used to compare the estimable candidate dividuals relative to new captures (the tran- models, regardless of the number of observa- sience test) in U-CARE. and two had excess tions per parameter. 88 viable models were deviance in the parametric bootstrap test (sim- identified for the 61 capture histories. The total ulation GOF). The remaining capture histories number of models exceeded the number of that “failed" GOF did so because of param- capture histories, as multiple models were con- eter inestimability in the bootstrap procedure. sidered for some capture histories. For 46 of In these instances, the models could not be fit the 61 capture histories, a single model had an rperloibalbleymstowtihtehseismtuilmaatbeidlidtaytain(i.>e.1.0th%ereofwetrhee othvaetrawhuenlimqiunegmAoIdeClt. wweaisghitde(n>t0i.fi9e5d)..Tiwndoicaalttienrg- simulated capture histories). The ability of native models were identified for seven capture models to satisfy the GOF criteria was sub- histories, three alternative models were identi- stantially greater for capture histories in our fied for six capture histories, and four and six highest category (>10 observations per pa- models were identified for one capture history rameter) than in those in the other two cate- each. Parameters that were time-dependentalso gories (2 < # < 5 and 5 < # < 10 observa- varied among the chosen models. All three pa- tions per parameter; Table 1). Data sparseness rameters were time-dependent in 14 capture also affected the contingency tests imple- histories, two parameters were time-dependent mented in U-CARE; 42% (n — 119) of the in 38 capture histories (p and 6: 15; p and y: capture histories with estimable parameters 17; d> and y: 6), and a single parameter was pthreodpuecrecdenutsaegfeulvacroinetdinagmeoncnygtoabulrest,hraelethcoautgeh- t1i3;mey-:de2p3)e.ndTehnet iMnD36Cctaipmtuereinhtiesrtvoarliecsho(pse: n0;f10: 61%, n = 49). (3-day n = 5; 4-day n = 2; 5-day n = 18; 6- A discriminant function analysis of all cap- day n — 19; 7-day n = 44). Although 52% of ture histories with >10 observations per pa- our original capture histories were collected rameter produced a moderately effective, sta- during the fall. 75% of the capture histories tistically significant discriminant function de- with applicable models were collected during = scribing parameter estimability (Wilks' A. the fall. 0.53, F559 = 10.41. P < 0.001) with positive Estimated total stopover duration values loadings on duration, recapture completeness, ranged from 0.76 to 17.08 days (Table 2). and and median captures per day. There were neg- the CV values were highly variable (ranging ative loadings on number of recaptured birds from 13% to 274%). Of the 61 capture his- and minimum stopover. To extract biological tories that were useable after GOF testing. 23 information from discriminant function load- had a total stopover CV of <50% (Table 1). ings. we examined a range of bivariate plots Stopover-after estimates ranged from 0.38 to depicting the various loadings. The plots 10.13 days, which were shorter than the esti- yielded only one clear biological interpreta- mates oftotal stopover. Despite the difference tion: capture histories with high minimum in stopover duration estimates obtained by es- stopover duration often had inestimable pa- timating total stopover and stopover-after, rameters (Fig. 2). Parameter estimability dur- stopover-after had a slightly wider range of ing GOF testing limited the numberofcapture CV values than total stopover. CV values for histories that could be analyzed; however, a stopover-after ranged from 13% to 365%. discriminant function analysis to predict pa- Most of the estimates involving CV values of rameterestimability during GOF testing ofthe <50% were capture histories from the fall mi- 49 capture histories that were estimable and gration season (18 ofthe 23 estimates fortotal had >10 observations per parameter was not stopover and 24 of 28 estimates for stopover- significant (Wilks' A = 0.83, F444 = 2.20, P after), approximately mirroring the distribu- = 0.085). tion of spring and fall capture histories (75% Optimal models for the capture histories that of estimable capture histories were collected passed GOF testing varied in the incorporation during the fall). These useful estimates were Morris et al. • UTILITY OF OPEN POPULATION MODELS 521 10 X xInestimable ing similar results. The capture histories were • Estimable also divided into different categories based only on total sample size (50 < n < 100, 100 ^ X < « < 150, and n > 150). The division by <>sD C>/3» x*x X* x sample size alone was not effective, because QO. -w0o3 sample size is a product of both sampling du- Ecn ' ' • ration and sampling intensity. E13 2 - mVV*- • Extrinsic parameterinestimability proved to c be the largest impediment to using open population models in our study, affecting both the 200 400 600 800 initial model fitting and GOF testing. The discriminant function analysis revealed that a Sample size long minimum stopover (>4 days) was a good FIG. 2. The relationshipbetweenparameterestim- indicator that the parameters would not be es- ability, minimum stopoverduration (days), andsample timable. Because most birds that are recap- msiizger.atAomroynsgtopcoavpetrurseitheissttohraiteshaodf1l0anodrbimrodresp(ebcyiesspea-t tured at stopover sites have minimum stop- cies) capture events per estimated parameter, those overs ofonly a few days, long minimum stop- with high minimum stopover duration often had ines- over statistics likely represent multiple birds timable parameters. with unusually long stopovers. Such a scenar- io would yield a large stopover estimate CV and indicate large biological differences obtained for a variety ofspecies including two among migrants at a given stopover site. Ex- vireos, Red-breasted Nuthatch (Sitta canaden- amining the 16 capture histories with >10 ob- sis), two kinglets, two thrushes. Gray Catbird servations per parameter but with inestimable (Dumetella carolinensis), many warbler spe- parameters revealed that 3 histories had no re- cies, and White-throated Sparrow {Zonotri- captures at all and 2 histories had only 2 re- chia leucophrys Table 2). captures. Ten of the capture histories were ; from three Nearctic-Nearctic migratory spe- DISCUSSION cies: five White-throated Sparrows (Zonotri- Our study provided some insights about the chia albicollis), four Yellow-rumped Warblers CMR conditions under which models can be (Dendroica coronata), and one Ruby-crowned effectively used to estimate migratory-stop- Kinglet Regulus calendula). Three ofthe oth- ( overduration. Dividing the data into three cat- er capture histories represented local breeding egories based on the number of observations species. All of these factors led us to believe per parameter revealed the importance of the that the inestimability in these cases might observation:parameter ratio in predicting the have been related to heterogeneous migration utility of CMR models. Models with >10 ob- behavior (either among individuals or subpop- servations per parameter were estimable and ulations). —62% satisfied GOF testing; most “failures” Unlike what we found for parameter estim- to satisfy GOF were due to the difficulty of ability, there was no clear single factor ex- estimating parameters during the GOF proce- plaining parameter inestimability in GOF test- dure when using simulations. If our banding ing. The discriminant function had low pre- data are representative, then the presence of dictive power, with only a 67% chance ofcor- >10 observations per parameter (roughly rectly predicting the outcome ofthe GOF test, three birds captured or recaptured per day) again indicating the lack of strong factors in- may connote a reasonable probability that fluencing estimability in GOF. Biological fac- CMR models will be useful for characterizing tors related to heterogeneity of the captured a given capture history. specimens (Pollock et al. 1990, Cooch and Although we present analyses based on to- White 2005) can easily lead to failures of tal number of observations (summed capture GOF testing. Additionally, there may be sta- and recapture events) per parameter, we also tistical reasons for some of the observed fail- conducted similar analyses using number of ures in GOF testing. 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UTILITY OF OPEN POPULATION MODELS: LIMITATIONS POSED BY PARAMETER ESTIMABILITY IN THE STUDY OF MIGRATORY STOPOVER

Similar UTILITY OF OPEN POPULATION MODELS: LIMITATIONS POSED BY PARAMETER ESTIMABILITY IN THE STUDY OF MIGRATORY STOPOVER