Supplemental Materials for the Analysis of Capture-Recapture 

Data for Polar Bears in Western Hudson Bay, Canada, 1984–2004 

Data Series 304 

U.S. Department of the Interior 

U.S. Geological Survey 
 

 

Cover: Photograph of a male polar bear on the sea ice north of Kaktovik, Alaska, USA, 2003. 

(Photograph taken by Eric V. Regehr, U.S. Geological Survey.) 
Supplemental Materials for the Analysis of 

Capture-Recapture Data for Polar Bears in 

Western Hudson Bay, Canada, 1984–2004 

By Eric V. Regehr, Nicholas J. Lunn, Steven C. Amstrup, and Ian Stirling 

Data Series 304 

U.S. Department of the Interior 

U.S. Geological Survey 
U.S. Department of the Interior 

DIRK KEMPTHORNE, Secretary 

U.S. Geological Survey 

Mark D. Myers, Director 

U.S. Geological Survey, Reston, Virginia: 2007 

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Suggested citation: 

Regehr, E.V., Lunn, N.J., Amstrup, S.C., and Stirling, Ian, Supplemental materials for the 

analysis of capture-recapture data for polar bears in western Hudson Bay, Canada, 1984–
 

2004: U.S. Geological Survey Data Series 304, 13 p. 

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iii 

Contents 

0BAbstract ........................................................................................................................................ 1
 

1BIntroduction................................................................................................................................... 1
 

2BNatural Survival ............................................................................................................................ 2
 

Sex and Age Composition ............................................................................................................ 3
 

4BGoodness of Fit ............................................................................................................................ 6
 

5BCapture-Recapture Analysis......................................................................................................... 7
 

6BSummary .................................................................................................................................... 12
 

References Cited........................................................................................................................ 12
 

Tables 

Table 1.	 Proportion of females among polar bears captured by the Canadian Wildlife Service 

(CWS) and Manitoba Department of Conservation (MDOC), and all polar bears 

harvested along the Nunavut coast of western Hudson Bay, 1984–2004. ................ 3
 

Table 2.	 Proportions of juveniles (0-1 years), subadults (2-4 years), prime-adults (5-19 years), 

and senescent-adults (≥20 years) among polar bears captured by the Canadian 

Wildlife Service (CWS) and Manitoba Department of Conservation (MDOC), and 

research-marked polar bears harvested along the Nunavut coast of western Hudson 

Bay, 1984–2004 ........................................................................................................ 4
 

Table 3.	 Mean ages of adult polar bears (age ≥ 5 years) captured by the Canadian Wildlife 

Service (CWS) and Manitoba Department of Conservation (MDOC), and researchmarked polar bears harvested along the Nunavut coast of western Hudson Bay, 

1984–2004 ................................................................................................................ 5
 

Table 4.	 Selected recapture probability (p) models fitted to capture data for polar bears 

collected by the Canadian Wildlife Service, 1984–2004 ............................................ 9
 

Table 5.	 Selected survival probability (φ) models fitted to capture data for polar bears 

collected by the Canadian Wildlife Service, 1984–2004 .......................................... 10
 

Table 6.	 Selected models including capture history dependence (CHD), fitted to capture data 

for polar bears collected by the Canadian Wildlife Service and the Manitoba 

Department of Conservation (MDOC), 1984–2004.................................................. 11
 
 

Supplemental Materials for the Analysis of 

Capture-Recapture Data for Polar Bears in 

Western Hudson Bay, Canada, 1984–2004 

By Eric V. Regehr, Nicholas J. Lunn, Steven C. Amstrup, and Ian Stirling 

0BAbstract 

Regehr and others (2007, Survival and population size of polar bears in western Hudson Bay in 

relation to earlier sea ice breakup: Journal of Wildlife Management, v. 71, no. 8, p. 2673-2683) 

evaluated survival in relation to climatic conditions and estimated population size for polar bears (Ursus 

maritimus) in western Hudson Bay, Canada. Here, we provide supplemental materials for the analyses in 

Regehr and others (2007). We demonstrate how tag-return data from harvested polar bears were used to 

adjust estimates of total survival for human-caused mortality. We describe the sex and age composition 

of the capture and harvest samples, and provide results for goodness-of-fit tests applied to capturerecapture models. We also describe the capture-recapture model selection procedure and the structure of 

the most supported model, which was used to estimate survival and population size. 

1BIntroduction 

Longer term and more consistent demographic data are available for the Western Hudson Bay 

(WH) polar bear population than for any other polar bear population in the world. Recently, Regehr and 

others (2007) analyzed capture-recapture data for the WH population collected from 1984 to 2004 under 

two sampling protocols. The Canadian Wildlife Service (CWS) captured free-ranging polar bears along 

the western coast of Hudson Bay. The Manitoba Department of Conservation (MDOC) captured 

problem bears in and around the community of Churchill, Manitoba, Canada. Regehr and others (2007) 

used Cormack-Jolly-Seber (CJS) type models to estimate survival, investigate the relationships between 

survival and sea ice conditions, and estimate the size of the WH population. This report provides 

supplemental materials that were not included directly in Regehr and others (2007) due to space 

limitations in the journal. 
 

 

2 Capture-Recapture Data for Polar Bears, Western Hudson Bay, Canada, 1984-2004 

2BNatural Survival 

Polar bears from the WH population were harvested as part of an annual, regulated hunt by Inuit 

along the Nunavut coast of western Hudson Bay. We estimated natural apparent survival (φ

N

) by 

adjusting the CJS estimates of total apparent survival (φ) using tag-return data from the harvest. For 

each sex and age class in the most supported CJS model for the combined CWS and MDOC data, we 

estimated natural apparent survival using the formula: 

N ˆ * φˆ j = φˆ j / 1( − H Mj / j) ; 

where 

Hj is the number of research-marked polar bears killed in the Nunavut harvest following 

sampling by the CWS and MDOC in year j, 

ˆ * 

j j / ˆ j − j + j M = m p m R is the estimated number of marked polar bears in the WH population

after sampling occasion j, 

mj is the number of marked polar bears captured at sampling occasion j, and 

Rj is the number of marked polar bears released at sampling occasion j. 

We could not estimate var (φˆ N ) using the methods of Taylor and others (2002) due to the complexity of 

the most supported model. We approximated average φ

N by constraining the annual estimates to [0, 1] 

and taking the arithmetic mean. Our approach was based on the assumption of additive mortality, 

because evidence is lacking for compensatory mortality among polar bears (Derocher and Taylor, 1994) 

and because the WH population appears to be declining despite a harvest that was thought to be 

sustainable. If density-dependent effects were compensating for harvest mortality, our estimates of 

φ

N could be biased high. 
 

 

 

 

 

 

Sex and Age Composition 3 

Sex and Age Composition 

We used standard regression techniques to describe trends in the sex and age composition of the 

CWS, MDOC, and Nunavut harvest samples (for example, Zar, 1996; Hosmer and Lemeshow, 2000). 

The proportion of females (table 1) was stable with time for the CWS capture sample (logistic regression 

5-year period model versus. time-constant model; G = 4.07, df = 3, P = 0.25) and for the Nunavut 

harvest sample (G = 4.72, df = 3, P = 0.19). The proportion of females in the MDOC sample varied 

among years (G = 14.30, df = 3, P < 0.001), with a trend toward a higher proportion of females (logistic 

regression on the proportion of females; linear trend model versus time-constant model, G = 9.24, 

df = 1, P = 0.002). 

Table 1. Proportion of females among polar bears 

captured by the Canadian Wildlife Service (CWS) 

and Manitoba Department of Conservation (MDOC), 

and all polar bears harvested along the Nunavut 

coast of western Hudson Bay, 1984–2004. 

Years CWS MDOC Harvest 

1984-1988 0.51 0.38 0.32 

1989-1993 0.54 0.35 0.27 

1994-1998 0.53 0.46 0.24 

1999-20041

 0.50 0.47 0.38 

1

Harvest data available through 2003 only. 
 

 

 

 

 

 

 

 

 

 

4 Capture-Recapture Data for Polar Bears, Western Hudson Bay, Canada, 1984-2004 

Age class composition (table 2) was relatively stable with time for the CWS capture sample 

(4 × 4 contingency table for 5-year periods; χ

2

 = 15.17, df = 9, P = 0.09). The age class composition of 

the MDOC sample varied among years (χ

2

 = 25.21, df = 9, P = 0.003), but there was no apparent trend. 

We described the age structure of the Nunavut harvest using research-marked polar bears only, because 

some ages were unknown for polar bears without a previous research capture. The age class composition 

of marked polar bears in the harvest varied among years (χ

2

 = 25.42, df = 9, P = 0.003), with a trend 

towards fewer subadults (logistic regression on the proportion of subadults; linear trend model versus 

time-constant model, G = 7.23, df = 1, P = 0.007). 

Table 2. Proportions of juveniles (0-1 years), subadults (2-4 years), 

prime-adults (5-19 years), and senescent-adults (≥20 years) among 

polar bears captured by the Canadian Wildlife Service (CWS) 

and Manitoba Department of Conservation (MDOC), and 

research-marked polar bears harvested along the Nunavut 

coast of western Hudson Bay, 1984–2004. 

[Juv, juvenile; Subad, subadult; Prime-ad, prime-adult; Senescent-ad, 

senescent-adult] 

Years Juv Subad Prime-ad Senescent-ad 

1984-1988 0.26 0.16 0.50 0.08 

1989-1993 0.27 0.14 0.49 0.09 

1994-1998 0.28 0.20 0.44 0.08 

1999-2004 0.27 0.15 0.49 0.09 

1984-1988 0.25 0.33 0.37 0.05 

1989-1993 0.18 0.48 0.27 0.07 

1994-1998 0.28 0.40 0.27 0.04 

1999-2004 0.29 0.36 0.29 0.06 

1984-1988 0.16 0.56 0.23 0.05 

1989-1993 0.06 0.46 0.43 0.06 

1994-1998 0.08 0.38 0.43 0.11 

1999-2003 0.13 0.38 0.34 0.15 

Harvest MDOC CWS 
 

 

 

 

Sex and Age Composition 5 

In the CWS capture sample, the age of adult females (table 3) increased from 1984 to 1993 

(linear regression; β

ˆ = 0.27, mSE ( ) β β = 0.06, ˆ = 0.09, P = 0.003) and was stable from 1994 to 2004 ( ˆ

ˆ SE β = 0.12, P = 0.64). The age of adult males varied among years, but there was not a significant 

ˆ 

m ( )

linear ( β

ˆ = -0.03, mSE ( ) β = 0.03, P = 0.35) nor quadratic (F = 2.39, df = 2, P = 0.09) trend. The age of 

ˆ adult females in the MDOC sample was stable from 1984 to 2004 ( β

ˆ = 0.05, mSE ( ) β = 0.07, P = 0.47), 

while the age of adult males decreased ( β

ˆ = -0.14, m ( ˆ SE β ) = 0.05, P = 0.007). The age of marked adult 

ˆ females in the Nunavut harvest increased from 1984 to 2003 ( β

ˆ = 0.54, mSE (β ) = 0.18, P = 0.004), 

while the age of marked adult males was stable ( β

ˆ ˆ = 0.13, mSE (β ) = 0.10, P = 0.22). 

 Table 3. Mean ages of adult polar bears (age ≥ 5 years) captured by the 

Canadian Wildlife Service (CWS) and Manitoba Department of Conservation 

(MDOC), and research-marked polar bears harvested along the Nunavut 

coast of western Hudson Bay, 1984–2004. 

[F, female; M, male] 

CWS MDOC Harvest 

 Years F M F M F M 

1984-1988 12.4 12.4 12.5 12.0 12.8 11.6 

1989-1993 13.5 12.6 14.1 11.6 12.1 11.3 

1994-1998 13.7 13.1 14.5 10.0 19.0 13.0 

1999-20041

 13.9 11.6 13.5 9.8 18.4 13.6 

1

Harvest data available through 2003 only. 
6 Capture-Recapture Data for Polar Bears, Western Hudson Bay, Canada, 1984-2004 

4BGoodness of Fit 

Assessment of goodness-of-fit is an important precursor to capture-recapture modeling (Cooch 

and White, 2005). Program RELEASE uses three contingency table components (TEST3.SR, 

TEST3.Sm, and TEST2.C) to compare the observed versus expected number of captures within 

categories that are designed to test the assumptions of the standard CJS model (Burnham and others, 

1987). The standard CJS model allows survival probability (φ) and recapture probability (p) to vary 

independently with time and is denoted φ (t pt ) ( ). At each sampling occasion (year), we evaluated the

significance and directionality of the program RELEASE components for sex- and age-based subsets of 

the WH capture-recapture data (Choquet and others, 2005). We also considered the interrelation of the 

components with regard to common assumption violations, such as temporary trap response (Pradel, 

1993) and transience (that is, permanent emigration following initial capture; Pradel and others, 1997). 

The CWS data for male polar bears did not fit model ( ) ( ) (overall χ2 φ t pt 118 = 166.08, P = 

0.002), due to lack of fit in TEST3.SR. The male data achieved an acceptable fit when partitioned into 

2 prime-adult versus other age classes (combined TEST3.SR χ38 = 37.18, P = 0.51). The female data fit 

model ( ) ( ) marginally well (overall χ125 φ t pt = 150.19, P 2 = 0.06). We started the CJS analysis with a 

model that allowed φ and p to vary with time for four independent strata: female prime-adults, male 

prime-adults, females of other ages, and males of other ages. This general model, denoted 

a3 a124 φ (t s × )φ {(t a a +[ 1, 24 ])×s p } (t a ×[ 124, 3a ]×s), also allowed for an additive effect in φ between 

juveniles and an aggregate subadult and senescent-adult age class. Model notation is described in 

section, ‘Capture-Recapture Analysis’. 

The combined CWS and MDOC data for male polar bears did not fit model φ ( )t pt( ) (overall

2 χ126 = 255.13, P < 0.001), due to significance in TEST3.SR and TEST2.C. The male data achieved an 

2 acceptable fit when partitioned into prime-adult vs. other age classes (combined TEST3.SR χ38 = 38.76, 

2 P = 0.44; combined TEST2.C χ76 = 66.84, P = 0.76). The lack of fit for males likely was due to the 

preponderance of young males in the harvest, as suggested by previous studies (Derocher and Stirling 

1995, Lunn and others, 1997). The combined CWS and MDOC data for females fit model φ (t pt ) ( )

2 reasonably well (overall χ134 = 153.34, overall P = 0.12). 
Capture-Recapture Analysis 7 

Violation of the CJS assumption of independent fates should not introduce bias into parameter 

estimates, but may affect model selection and lead to an overstatement of estimator precision (Burnham 

and Anderson, 2002). To investigate whether extended maternal care in polar bears (Derocher and 

Stirling, 1996) resulted in a significant lack of independence, we estimated an ancillary, biological value 

of the variance inflation factor ( cˆ) using the formula cˆ = n / (n – nc); where n is the total number of 

captures, and nc is the number of captures of dependent cubs (Taylor and others, 2002). The biological cˆ

of 1.3 suggested that potential overdispersion due to dependent fates was minor. We used cˆ = 1.0 in our 

modeling, based on the program RELEASE indication of cˆ < 1.0 for both the CWS-only and combined 

data. 

5BCapture-Recapture Analysis 

Model notation. ―We extended the CJS notation of Lebreton and others (1992) to allow for the 

specification of more complex models. We considered four age classes: juveniles (0-1 years; denoted 

a1), subadults (2-4 years; a2), prime-adults (5-19 years; a3), and senescent-adults (≥20 years; a4). We 

denoted combinations of age classes using the letter a followed by several integers. For example, an 

aggregate age class comprising juveniles (a1) and subadults (a2) is written a12. Overall age structure is 

written within square brackets. For example, model φ (t a ×[ 12, a3, a4]) estimates values of φj that vary 

independently with time for the age classes a12, a3, and a4. We partitioned survival notation into primeadult versus other age classes, which made it easier to interpret models with different types of 

a3 a124 interannual variation for the two groups. For example, in the model φ (s) φ (t + s) , prime-adult 

φa3 φa124 survival ( ) is time constant and varies by sex; survival of other age classes ( ) varies additively 

by time and sex. We denoted combinations of interactive (×) and additive (+) effects using nested 

brackets. For example, model φ {(t + [a12,a3,a4]) × s} has full time × sex and age class × sex 

interactions, but is additive in time and age class. We denoted effects that only apply to certain age 

classes using the symbol •. For example, model φ(trap' 24 ia ) includes the capture history dependence 

covariate trap’ for subadults and senescent-adults (a24), but not for juveniles or prime-adults. 

Sex effects in the most supported model required special notation. In the survival parameterization, the 

symbol s* denotes a common sex effect for prime-adults and other age classes. Therefore, female and 

male φ differ by the same constant for all polar bears, even though notation for the survival model was 
8 Capture-Recapture Data for Polar Bears, Western Hudson Bay, Canada, 1984-2004 

otherwise partitioned by age. In the recapture parameterization, the symbol s’ denotes an additive sex 

effect with two levels: female and male p differed by one constant from 1985 to 1995 and by another 

constant from 1996 to 2004. 

Model fitting.―We estimated survival and population size from the most supported model for 

the combined CWS and MDOC data, which contained 31 parameters and was denoted 

a3 a124 φ ( ) s* 1, 2, 4 φ {ice +[a a a s ]+ * +(trap ' •[a 24 ])} p t { + +s' (trap ' ×[a a 12, 34 ])} . In this model, survival was 

time constant for prime-adults, and varied as a function of the date of spring sea ice breakup in the WH 

management area (the environmental covariate ice) for polar bears of other ages. Survival of juvenile, 

subadult, and senescent-adult polar bears differed by additive age effects. Survival of females and males 

differed by an additive sex effect that was common to polar bears of all ages. Following capture in 

Churchill, survival of subadult and senescent-adult polar bears changed by a permanent, additive effect 

(i.e., the time-dependent individual covariate trap’). 

Recapture probabilities in the most supported model were time dependent for all polar bears. 

Recapture probabilities of females and males differed by an additive sex effect that differed in 

magnitude for 1985–95 versus 1996–2004. Following capture in Churchill, recapture probabilities of all 

polar bears changed by a permanent, additive effect (trap’) that differed for juveniles and subadults 

versus prime-adults and senescent-adults. Although permanent capture history dependence was 

universally denoted as trap’, the values of this covariate were estimated independently within the 

parameterizations for φ and p. 

We derived the most supported CJS model in several steps. Because the complexity of the 

analysis made it impractical to define an all-inclusive candidate model set (for example, Lebreton and 

others, 1992, Norman and others, 2004, Muths and others, 2006), we began by modeling the CWS-only 

data using a flexible approach that sequentially identified suitable parameterizations for separate parts of 

the model. We modeled p first (table 4) because we expected a simple parameterization that was 

primarily a function of study design and because the robust estimation of p is critical when estimating 

population size (Pollock and others, 1990). We also wanted to minimize the chance of confounding 

effects in φ and p. We then modeled φ sequentially for prime-adults, followed by other age classes 

(table 5). This limited the number of interactions between age, sex, and types of interannual variation, 

allowing us to focus on important biological hypotheses. Finally, we extended the CJS analysis to the 

combined CWS and MDOC data and evaluated all reasonable forms of capture history dependence 

associated with capture around Churchill (table 6). This approach assumed that the structure of the most 
 

 

 

 

 

 

 

Capture-Recapture Analysis 9 

supported model for the CWS-only data was sufficiently general to represent key biological (φ) and 

study design (p) effects for the entire WH population, and that potential differences in φ and p associated 

with CWS versus MDOC sampling protocols could be modeled as additive capture history dependence. 

These assumptions were corroborated by the goodness-of-fit analysis, which did not identify any 

unmanageable heterogeneity in the combined data, and by strong support within the data for the 

hypothesized forms of capture history dependence. We derived final parameter estimates from a single 

model because model-averaging was not appropriate without an all-inclusive candidate model set, and 

because model selection uncertainty was low with respect to the most important effects (that is, 

parameter estimates were similar for models with similar support in the data [Buckland and others, 

1997]). 

Table 4. Selected recapture probability (p) models fitted to capture data for polar bears 

collected by the Canadian Wildlife Service, 1984–2004. 

[We modeled p as a function of time (t), capture effort (effort), sex (s), an additive sex function with different effects for 

1985–95 versus 1996–2004 (s’), and the age classes a1 (juveniles, 0-1 years), a2 (subadults, 2-4 years), a3 (prime-adults, 

5-19 years), and a4 (senescent-adults, ≥20 years). All models had the general survival parameterization a3 a124 (t s) φ (t + a a ] ×s} . AICc = Akaike’s information criterion adjusted for small sample size; ΔAICc φ × { [ 1, 24 ) = difference 

in AICc relative to the most supported model in table 4; wi = normalized Akaike weight relative to models in table 4; np = 

number of estimated parameters] 

Model np AICc ΔAICc wi Description 

p (t a a s ×[ 124, 3]× ) 137 8936.31 40.79 0.00 

General p model. F and M p vary independently by time for two 

independent age classes. Age classes have different sex effects. 

p (t a a s ×[ 12, 34 ]× ) 136 8939.31 43.79 0.00 

F and M p vary independently by time for two independent age 

classes. The age classes have different sex effects. 

p {(t a a s +[ 12, 34 ])× } 102 8909.37 13.85 0.00 

F and M p vary independently by time for two additive age classes. 

Age classes have different sex effects. 

pt ( ×s ) 100 8905.52 10.00 0.01 F and M p vary independently by time. 

pt ( + s ) 87 8918.98 23.46 0.00 F and M p vary additively by time. 

p (t s + ') 85 8895.52 0 0.99 

Most supported p model. F and M p vary additively by time, with 

different sex effects for 1985–95 versus 1996–2004. 

p effort ( ×s ) 71 8957.97 62.45 0.00 F and M p are independent functions of effort. 

pt ( ) 84 8937.30 41.78 0.00 p varies by time. 

p s( ) 67 8985.99 90.47 0.00 F and M p are constant by time. 
 

 

 

 

 

 

 

10 Capture-Recapture Data for Polar Bears, Western Hudson Bay, Canada, 1984-2004 

Table 5. Selected survival probability (φ) models fitted to capture data for polar bears 

collected by the Canadian Wildlife Service, 1984–2004. 

[We modeled φ as a function of time (t), ice breakup date in the current year (ice), ice breakup date in the previous year 

(ice’), 3-year running mean of ice breakup date (ice’’), and a linear trend (linear); as well as sex and age effects. All models 

have the most supported recapture parameterization p t( + s '). AICc = Akaike’s information criterion adjusted for small 

sample size; ΔAICc = difference in AICc to the most supported model in the relevant section of table 5; wi = normalized 

Akaike weight relative to models in the relevant section of table 5; np = number of estimated parameters; Prime-ad, primeadult; Senescent-ad, senescent-adult; F, female; M, male] 

Model np AICc ΔAICc wi Description 

Model φ for prime-ad (descriptions apply to prime-ad only); retain the general φ parameterization for juv, subad, and senescent-ad. 

( ) {( [ ]) } a3 a124 φ t s t a a s × φ + 1, 24 × 85 8895.52 11.15 0.00 

General φ model. For prime-ad, F and M φ vary 

independently by time. For other ages, F and M φ

vary independently by time for two additive age 

classes. Age classes have different sex effects. 

( ) {( [ ]) } a3 a124 φ t s t a a s + φ + 1, 24 × 78 8904.55 20.18 0.00 F and M φ vary additively by time. 

( ) {( [ ]) } a3 a124 φ s φ t a a s + 1, 24 × 62 8884.37 0.00 0.78 F and M φ are constant by time. 

( ) {( [ ]) } a3 a124 φ ice s t a a s × φ + 1, 24 × 64 8887.01 2.64 0.21 F and M φ are independent functions of ice. 

( ) {( [ ]) } a3 a124 φ . φ t a a s + 1, 24 × 60 8892.46 8.09 0.01 φ is constant by time. 

Model φ for juv, subad, and senescent-ad (descriptions apply to these ages only); retain the most supported φ parameterization for prime-ad. 

( ) {( [ ]) } 3 124 1, 2, 4 a a φ s φ t a a a s + × 64 8878.04 11.48 0.00 

F and M φ vary independently by time for three 

additive age classes. Age classes have different sex 

effects. 

( ) ( [ ] ) 3 124 1, 2, 4 a a φ s φ t a a a s + + 47 8879.23 12.67 0.00 F and M φ vary additively by time for three additive 

age classes. Age classes have the same sex effect. 

( ) ([ ] ) 3 124 1, 2, 4 a a φ s φ a a a s + 28 8877.47 10.91 0.00 F and M φ are constant by time for three age 

classes. Age classes have the same sex effect. 

( ) ( [ ] ) a3 a124 φ s φ ice a a a s × 1, 2, 4 × 36 8873.28 6.72 0.02 

F and M φ are independent functions of ice for 

three independent age classes. Age classes have 

different sex effects. 

( ) ( [ ] ) 3 124 1, 2, 4 a a φ s φ ice a a a s + + 29 8868.17 1.61 0.30 

F and M φ are additive functions of ice for three 

additive age classes. Age classes have the same sex 

effect. 

( ) ( [ ] ) 3 124 ' 1, 2, 4 a a φ s φ ice a a a s + + 29 8877.12 10.56 0.00 

F and M φ are additive functions of ice’ for three 

additive age classes. Age classes have the same sex 

effect. 

( ) ( [ ] ) 3 124 '' 1, 2, 4 a a φ s φ ice a a a s + + 29 8875.06 8.50 0.01 

F and M φ are additive functions of ice’’ for three 

additive age classes. Age classes have the same sex 

effect. 

( ) ( [ ] ) 3 124 1, 2, 4 a a φ s φ linear a a a s + + 29 8879.74 13.18 0.00 

F and M φ are additive functions of linear for three 

additive age classes. Age classes have the same sex 

effects. 

( ) ( [ ]) 3 124 1, 2, 4 a a φ s φ ice a a a + 28 8877.10 10.54 0.00 φ is a function of ice for three additive age classes. 

( ) ( [ ] ) a3 a124 φ s * φ ice a a a s + 1, 2, 4 + * 288866.56 0 0.66 

Most supported φ model. For prime-ad, F and M φ

are constant by time. For other ages, F and M φ are 

additive functions of ice for three additive age 

classes. All age classes have the same sex effect. 

( ) ( [ ] ) a34 a12 φ s * φ ice a a s + 1, 2 + * 27 8945.77 79.21 0.00 

For prime-ad and senescent-ad combined, F and M 

φ are constant by time. For other ages, F and M φ

are additive functions of ice for two additive age 

classes. All age classes have the same sex effect. 
 

 

Capture-Recapture Analysis 11 

Table 6. Selected models including capture history dependence (CHD), fitted to capture 

data for polar bears collected by the Canadian Wildlife Service and the Manitoba Department 

of Conservation (MDOC), 1984–2004. 

[We modeled temporary (trap) and permanent (trap’) CHD following capture around Churchill by the MDOC, and all 

reasonable sex and age interactions. Parameterizations for CHD in recapture probabilities were evaluated retaining the a3 a124 general CHD survival parameterization φ s*+trap' φ ice + a a a 1, 2, 4 s* t a '× a a a . AICc ( ) { [ ] + +( r p [ 1, 2, 4 ])} = Akaike’s 

information criterion adjusted for small sample size; ΔAICc = difference in AICc to the most supported model in the relevant 

section of table 6; wi = normalized Akaike weight relative to models in the relevant section of table 6; np = number of 

estimated parameters. 

Model np AICc ΔAICc wi Description 

Model CHD in p; retain the general CHD φ parameterization. 

p t s trap a a a a s { + +' ' 1, 2, 3, 4 ( ×[ ]× )} 40 11954.06 9.57 0.01 

General model for 

CHD in p. Permanent 

CHD independent for F 

and M for four 

independent age classes. 

p t s trap a a a a { + +' ' 1, 2, 3, 4 ( ×[ ])} 36 11947.27 2.78 0.17 

Permanent CHD for 

four independent age 

classes. 

p t s trap a a a a { + +' 1, ( ×[ 2, 3, 4])} 36 11984.68 40.19 0.00 

Temporary CHD for 

four independent age 

classes. 

p t s trap a a { + +' ' 12, 34 ( ×[ ])} 34 11944.49 0.00 0.68 

Most supported model 

for CHD in p. 

Permanent CHD for two 

independent age classes. 

p t s trap a { + +' ' 1234 ( ×[ ])} 33 11948.80 4.31 0.08 Permanent CHD for one 

inclusive age class. 

p t s trap a { + +' ' 12 ( •[ ])} 33 12022.28 77.79 0.00 Permanent CHD for one 

non-inclusive age class. 

p t s trap a trap a { + +' 12 ( •[ ] & ' •[ 34 ])} 34 11949.21 4.72 0.00 

Temporary CHD for 

one non-inclusive age 

class; permanent CHD 

for one non-inclusive 

age class. 

p t s ( + ') 32 12071.46 126.97 0.00 No CHD in p. 

Model CHD in φ; retain the most supported CHD p parameterization. 

( ) { [ ] ( [ ])} 3 124 * ' 1, 2, 4 * ' 1, 2, 4 a a φ s trap ice a a a s trap a a a + φ + + + × 34 11944.49 4.64 0.05 

Permanent CHD for 

four independent age 

classes. 

( ) { [ ] ( [ ])} 3 124 * 1, 2, 4 * ' 1, 2, 4 a a φ s ice a a a s trap a a a φ + + + × 33 11942.62 2.77 0.14 

Permanent CHD for 

three independent, noninclusive age classes. 

( ) { [ ] ( [ ])} 3 124 * 1, 2, 4 * ' 2, 4 a a φ s ice φ + a a a + s + trap • a a 32 11941.40 1.55 0.25 

Permanent CHD for two 

independent, noninclusive age classes. 

( ) { [ ] ( [ ])} 3 124 * 1, 2, 4 * ' 24 a a φ s ice φ + a a a + s + trap • a 31 11939.85 0.00 0.55 

Most supported model 

for CHD in φ. 

Permanent CHD for one 

non-inclusive age class. 

( ) ( [ ] ) a3 a124 φ s * φ ice a a a s + 1, 2, 4 + * 30 11951.13 11.28 0.00 No CHD in φ. 
12 Capture-Recapture Data for Polar Bears, Western Hudson Bay, Canada, 1984-2004 

6BSummary 

This report provides supplementary materials for an analysis of capture-recapture data for polar 

bears in western Hudson Bay. We presented methods of estimating natural survival. We described the 

capture samples and provided results for goodness-of-fit tests. We also described the model selection 

procedure and the structure of the final model that was used for inference. 

References Cited 

Buckland, S.T., Burnham, K.P., and Augustin, N.H., 1997, Model selection: an integral art of inference: 

Biometrics, v. 53, p. 603-618. 

Burnham, K.P., and Anderson, D.R., 2002, Model selection and multimodel inference: 2nd edition, 

Springer-Verlag, New York, New York, USA. 

Burnham, K.P., Anderson, D.R., White, G.C., Brownie, C., and Pollock, K.H., 1987, Design and 

analysis methods for fish survival experiments based on release–recapture: American Fisheries 

Society Monograph 5, Bethesda, Maryland, USA. 

Choquet, R., Reboulet, A.M., Lebreton, J.D., Gimenez, O., and Pradel, R., 2005, U-CARE 2.2 User's 

Manual: CEFE, Montpellier, France. Available from ftp://ftp.cefe.cnrs-mop.fr/biom/Soft-CR/. 

Cooch, E., and White, G.C., 2005, Program MARK: a gentle introduction: 5th edition. 

http://www.phidot.org/software/mark/docs/book/. Accessed August 22, 2006. 

Derocher, A.E., and Stirling, I., 1995, Estimation of polar bear population size and survival in Western 

Hudson Bay: Journal of Wildlife Management, v. 59, no. 2, p. 215-221. 

Derocher, A.E., and Stirling, I., 1996, Aspects of survival in juvenile polar bears: Canadian Journal of 

Zoology, no. 74, p. 1246–1252. 

Derocher, A.E., and Taylor, M., 1994, Density-dependent population regulation of polar bears in Taylor, 

M. (ed.), Density-Dependent Population Regulation of Black, Brown, and Polar Bear: 9th 


International Conference on Bear Research and Management, Monograph 3, p. 25-37. 


Hosmer, D.W., and Lemeshow S., 2000, Applied logistic regression. John Wiley and Sons, New York, 

New York, USA. 

Lebreton, J.-D., Burnham, K.P. , Clobert, J., and Anderson, D.R., 1992, Modeling survival and testing 

biological hypotheses using marked animals: a unified approach with case studies: Ecological 

Monographs, v. 62, p. 67–118. 

Lunn, N.J., Stirling, I., Andriashek, D., and Kolenosky, G.B., 1997, Reestimating the size of the polar 

bear population in Western Hudson Bay: Arctic, v. 50, p. 234–240. 
References Cited 13 

Muths, E., Scherer, R.D., Corn, P.S., and Lambert, B.A., 2006, Estimation of temporary emigration in 

male toads: Ecology, v. 87, no. 4, p. 1048-1056. 

Norman, G.C., Conner, M.M., Pack, J.C., and White, G.C., 2004, Effects of fall hunting on survival of 

male wild turkeys in Virginia and West Virginia: Journal of Wildlife Management, v. 68, no. 2, 

p. 393-404. 

Pollock, K.H., Nichols, J.D., Hines, J.E., and Brownie C., 1990, Statistical inference for capture– 

recapture experiments: Wildlife Monographs 107. 

Pradel, R., 1993, Flexibility in survival analysis from recapture data: handling trap-dependence in

Lebreton, J.-D., and Nichols, J.D. (eds.), Marked individuals in the study of bird populations: 

Birkhauser Verlag, Basel, Switzerland, p. 29-37. 

Pradel, R., Hines, J.E., Lebreton, J.-D., and Nichols, J.D., 1997, Capture-recapture survival models 

taking account of transients: Biometrics, v. 53, p. 60-72. 

Regehr, E.V., Lunn, N.J., Amstrup, S.C., and Stirling, I., 2007, Survival and population size of polar 

bears in western Hudson Bay in relation to earlier sea ice breakup: Journal of Wildlife Management, 

v. 71, no. 8, p. 2673-2683. 

Taylor, M.K., Laake, J., Cluff, H.D., Ramsay, M., and Messier, F., 2002, Managing the risk from 

hunting for the Viscount Melville Sound polar bear population: Ursus, v. 13, p. 185–202. 

Zar, J.H., 1996, Biostatistical analysis: 3rd edition, Prentice Hall, Upper Saddle River, New Jersey, 

USA. 
For more information concerning the research in this report, contact 

Director, Alaska Science Center 

U.S. Geological Survey 

1011 East Tudor Road 

Anchorage, Alaska 99503 

http://alaska.usgs.gov/ 
Regehr and others— Supplemental Materials for the Analysis of Capture-Recapture Data for Polar Bears in Western Hudson Bay, Canada, 1984–2004—Data Series 304