Environ Ecol Stat (2008) 15:89–99 DOI 10.1007/s10651-007-0030-3
Closed population estimation models and their extensions in Program MARK Gary C. White
Received: 1 August 2005 / Revised: 7 July 2006 / Published online: 19 September 2007 © Springer Science+Business Media, LLC 2007
Abstract Program MARK provides >65 data types in a common configuration for the estimation of population parameters from mark-encounter data. Encounter information from live captures, live resightings, and dead recoveries can be incorporated to estimate demographic parameters. Available estimates include survival (S or φ), rate of population change (λ), transition rates between strata (), emigration and immigration rates, and population size (N ). Although N is the parameter most often desired by biologists, N is one of the most difficult parameters to estimate precisely without bias for a geographically and demographically closed population. The set of closed population estimation models available in Program MARK incorporate time (t) and behavioral (b) variation, and individual heterogeneity (h) in the estimation of capture and recapture probabilities in a likelihood framework. The full range of models from M0 (null model with all capture and recapture probabilities equal) to Mtbh are possible, including the ability to include temporal, group, and individual covariates to model capture and recapture probabilities. Both the full likelihood formulation of Otis et al. (1978) and the conditional model formulation of Huggins (1989, 1991) and Alho (1990) are provided in Program MARK, and all of these models are incorporated into the robust design (Kendall et al. 1995, 1997; Kendall and Nichols 1995) and robust-design multistrata (Hestbeck et al. 1991, Brownie et al. 1993) data types. Model selection is performed with AICc (Burnham and Anderson 2002) and model averaging (Burnham and Anderson 2002) is available in Program MARK to provide estimates of N with standard error that reflect model selection uncertainty. Keywords Closed capture models · Detection probabilities · Encounter histories · Huggins models · Maximum likelihood estimation · Pledger models
G. C. White (B) Department of Fishery and Wildlife Biology, Colorado State University, Fort Collins, CO 80523, USA e-mail:
[email protected]
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1 Introduction Program MARK (White and Burnham 1999) provides >65 different data types to estimate survival (S) and other population parameters, such as population size (N ) and population rate of change (λ), from the encounters of marked animals. This diversity of data types is needed to account for the various methods used to encounter marked animals (e.g., live recaptures, resightings of animals without actual capture, and recovery of dead animals possibly through harvest, but not exclusively so), as well as the method of marking animals. For example, recent innovations in marking include the use of DNA sampling, PIT tags, and camera “traps”. The objective of this article is to review the estimation of N of demographically and geographically closed populations (Otis et al. 1978, White et al. 1982) with procedures provided in Program MARK. Two types of estimators are provided. Otis et al. (1978) summarized the existing literature where N is included directly in the likelihood. Huggins (1989, 1991) and Alho (1990) suggested estimators of N where a conditional likelihood is used, and N is not included directly in the likelihood, but estimated as a derived parameter.
2 Estimation methods in MARK Typically, parameter estimates in Program MARK are obtained by the method of maximum likelihood estimation. Theory has been developed based on the log of the likelihood function to estimate the parameter value, its standard error, and profile likelihood confidence intervals. In Program MARK, maximum likelihood estimation is performed via numerical methods (default is Newton–Raphson with simulated annealing available) to accommodate the broad range of models possible. More recently, the MCMC algorithm has been added to Program MARK to provide estimates under the Bayesian framework. 2.1 Closed population estimators Otis et al. (1978) proposed a suite of eight models for the estimation of size of closed populations. They suggested three sources of variation in encounter probabilities. First, detection probabilities are not constant from occasion to occasion, giving time (t) variation. Second, animals respond behaviorally to initial capture, with recapture probabilities either greater (trap happy) or less than initial capture probabilities (trap shy). Behavioral response (b) may not be an attribute of the animals, but of the survey configuration. Third, encounter probabilities of individual animals may differ because of inherent differences of individuals, leading to individual heterogeneity (h). The combination of these three factors leads to a suite of eight models, where model M0 is the null model with constant detection probabilities across all three factors (Fig. 1). Otis et al. (1978) did not have estimators for Mtbh , Mth , or Mtb in their program CAPTURE (White et al. 1978, 1982), although estimators for Mth (Chao et al. 1992) and Mtb (Rexstad and Burnham 1991) were later added. Further, the estimators for Mth and Mh were not based on likelihood theory. Recently, however, maximum likelihood estimators for models incorporating individual heterogeneity have been developed based on mixture models (Norris and Pollock 1996, Pledger 2000). To illustrate these likelihood estimators of N , I will use a simple three-occasion example, where only eight possible encounter histories are possible. Considering only time and behavior variation, the expressions in Table 1 model the eight encounter histories, where pi is initial capture probability on occasion i, and ci is the recapture probability on occasion i
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Fig. 1 The suite of eight models proposed by Otis et al. (1978) to allow for time variation (t), behavioral response to initial capture (b), and individual heterogeneity (h) of encounter probabilities. Models pointed to by arrows are nested within the model above
Table 1 A three-occasion example, where only eight possible encounter histories are possible, is presented to illustrate the construction of encounter history probabilities considering only time and behavior variation, where pi is initial capture probability on occasion i, and ci is the recapture probability on occasion i (with no c1 ) Encounter history (i)
Animals observed (X i )
Probability (Pi )
100
X 100
p1 (1 − c2 )(1 − c3 )
010
X 010
(1 − p1 ) p2 (1 − c3 )
001
X 001
(1 − p1 )(1 − p2 ) p3
110
X 110
p1 c2 (1 − c3 )
101
X 101
p1 (1 − c2 )c3
011
X 011
111
X 111
(1 − p1 ) p2 c3 p1 c2 c3
000
X 000
(1 − p1 )(1 − p2 )(1 − p3 )
(with no c1 ). Note that the encounter history 000 is not observed, but is included in the likelihood. The number of animals assumed to have this encounter history is X 000 = N − Mt+1 , where Mt+1 is the number of animals actually observed, or just the sum of the observed X i values. The log likelihood can then be written as a multinomial distribution, but with one cell not observed: log L(p, c, N |X) ∝ log
N! (N − Mt+1 )!
+
X i log(Pi ).
i
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The constant term i X i ! for the observed X i is left out of the denominator in the above likelihood and in its computation in Program MARK, so that positive log likelihood values can result that produce negative AIC values. The likelihood models discussed above include N in the likelihood. Huggins (1989, 1991) and Alho (1990) proposed a conditional likelihood, where the 000 encounter history is conditioned out of the likelihood. Under this formulation, the observed cell probabilities are all divided by the term 1 − (1 − p1 )(1 − p2 ) (1 − p3 ) so that the sum of the cell probabilities in the resulting multinomial distribution sum to 1. The derived estimate of N is then Nˆ =
Mt+1 . 1 − (1 − pˆ 1 )(1 − pˆ 2 )(1 − pˆ 3 )
Pledger (2000) following Norris and Pollock (1996), proposed the use of a mixturedis pi A with Pr = π for tribution to model the pi and ci parameters. Thus, pi = pi B with Pr = 1 − π the case with two mixtures A and B, although the model can be extended to more than two mixtures. In the above equation, the parameter π is the probability the animal occurs in mixture A. For >2 mixtures, additional π parameters must be defined (i.e., π A , π B , . . .,), but constrained to sum to 1. A subtlety of the closed population models is that only the initial capture probabilities are used to estimate N , as demonstrated in the equation above for how N is estimated for the Huggins model. However, an even deeper subtlety occurs in that the last p parameter is not identifiable unless a constraint is imposed. So, for example, in model Mt , the constraint of pi = ci is imposed, providing an estimate of the last p from the last c. Likewise, under model Mb , the constraint of pi ≡ p. is imposed, so that the last p is assumed equal to all the other p values. A similar constraint is used for model Mbh , i.e., pi A ≡ p A , pi B ≡ p B , etc. Under model Mtb , the pi and ci are modeled as a constant offset (Obeh ) of one another, i.e., ci = pi + Obeh . This relationship will depend on the link function used, but the last pi is still obtained as ci minus the offset (where the offset is estimated from the data on the other pi and ci ). Under model Mtbh , the offset between the pi and ci is applied, with an additional offset(s) included to model the relationship among the mixtures, i.e., pi B = pi A + O B , piC = pi A + OC , with a different offset applied to each succeeding mixture. Similarly, ci B = pi B + Obeh = pi A + O B + Obeh , with the resulting relationship depending on the link function applied. With this model, the relationship between the mixtures of the pi is maintained, i.e., the ordering of the mixtures is maintained across occasions. Model Mth can also be modeled as an additive offset between the mixtures, although other relationships are possible because the last pi for each mixture is estimated from the corresponding last ci . Although other relationships than those of the preceding paragraph can be proposed to provide identifiability, the proposed models must provide identifiability of all the initial capture probabilities. When no constraint is imposed on the last pi , the likelihood is maximized with the last p = 1, giving the estimate Nˆ = Mt+1 . Thus, a diagnostic to determine whether the estimated model is of value is to check to see whether Nˆ = Mt+1 , and if so, to see if the last pi estimate equals 1. 2.2 Closed population estimation in Program MARK To simplify the mechanics, the above models have been implemented with three separate parameterizations (“Closed Captures”, “Closed Captures with Het.”, and “Full Closed Captures with Het.”, termed “data types” in Program MARK) for each of the full likelihood and Huggins parameterizations, giving six data types in Program MARK that provide closed
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population estimates, and combined with six additional models that incorporate the misidentification models of Lukacs and Burnham (2005) that are not discussed here. The first parameterization (“Closed Captures”) only includes a single mixture, and so only has the pi and ci parameters. This parameterization assumes no individual heterogeneity, so provides an easy way to generate estimates under models M0 , Mb , Mt , and Mtb . The second parameterization (“Closed Captures with Het.”) is a simple individual heterogeneity model, with parameters π, pi A ≡ p A , and pi B ≡ p B , and assumes no temporal or behavioral variation. Thus, estimates under model Mh are simple to obtain with this parameterization, with no other models possible (although π = 0 or π = 1 results in M0 ). Lastly, the full parameterization provides for all three effects of t, b, and h, so that estimates under models Mbh , Mth , and Mtbh are possible. Of course, any of the reduced models can be run from the full parameterization if the appropriate constraints are applied. Temporal and group covariates can be used to model p and c for any of the parameterizations. The primary advantage of the three Huggins data types is that individual covariates can be used to model p and c. Individual covariates cannot be used with the full likelihood approach because the term (1 − p1 ) (1 − p2 ). . .(1 − pt ) is included in the likelihood, and no covariate value is available for animals that were never captured. In contrast, the Huggins parameterization has conditioned this multinomial term out of the likelihood, and so an individual covariate can be measured for each of the animals included in the likelihood. When individual covariates are used, a Horvitz–Thompson estimator is used to estimate N :
Nˆ =
Mt+1 i=1
1 . 1 − [1 − pˆ 1 (xi )][1 − pˆ 2 (xi )] . . . [1 − pˆ t (xi )]
3 Example: Carothers (1973) Scheme A taxicab data To illustrate the use of these models in Program MARK, I will use the Scheme A taxicab data from Carothers (1973). The population of taxicabs in the city of Edinburgh, Scotland, was sampled on t = 10 occasions, with license plates recorded as the taxicab’s mark when first observed. The known N was 420 taxicabs. On each occasion, approximately 50 taxicabs were identified, so one would not expect any time (t) variation in these data. In Program MARK, the parameter space for a data type is defined in the parameter index matrices (PIMs). To illustrate, the PIMs for the p and c parameterization with N in the likelihood would consist of three screens (Fig. 2). The first would contain edit boxes for each of the p parameters. The second screen would contain one less edit box for each of the one fewer c parameters. The final screen would have just a single edit box for the single N parameter. To construct model M0 , all of the pand c parameters are set equal. An easy approach is to change the values in the PIM edit boxes for p and c shown in Fig. 2 to the same value, e.g., 1, and then give the parameter for N the value 2. However, this simple approach will not work for more complicated models, which require the use of a design matrix. Details of how the link functions that relate the design matrix to the real parameters (i.e., here the π, p, c, and N ) are provided in White and Burnham (1999). The number of rows in the design matrix is the number of unique values in the PIMs, and the user specifies the number of columns that are used to constrain the π, p, c, and N estimates. So, a simple design matrix consisting of only columns 1 and 12 from Table 2 will produce an estimate of N under model M0 . Similarly, using columns 1 and 3–12 from Table 2 produces model Mt , columns 1, 2, and 12 produces model Mb , and using all the columns in Table 2 provides model Mtb .
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Fig. 2 Example of the parameter index matrices (PIMs) from Program MARK for the Carothers (1973) Scheme A taxicab data with 10 occasions for the closed population data type (designated as “Closed Captures” in Program MARK)
The PIMs for the full heterogeneity data type (Fig. 3) require parameters for each of the mixtures, plus the mixture probability (or probabilities if > 2 mixtures are specified). As a result, considerably more parameters are necessary. However, the design matrix is used to impose reasonable, biologically justifiable constraints to reduce the number of parameters actually estimated. The design matrix in Table 3 will produce estimates under model Mtbh , where the mixture effects and the behavior effects are assumed to be additive effects before the link function is applied. In Table 3, model Mh is constructed by using just columns 1, 2, 13, and 14, model Mbh by using just columns 1, 2, 3, 13, and 14, and model Mth by using only columns 1, 2, 4–12, 13, and 14.
4 Additional features of Program MARK useful for closed population estimation Program MARK provides many additional features useful for the analysis of closed population estimators of N . Profile likelihood confidence intervals can be computed for π, p, c, and N . Previously mentioned is the capability to model the biological parameters as functions of time, group, and individual covariates. The modeling of individual heterogeneity with covariates often is a more biologically satisfying approach than approximating the heterogeneity with mixtures because individual covariates can help identify the causes of individual heterogeneity. In addition, parameters can be fixed to specified values, a useful feature for closed captures data sets where differing numbers of occasions are used for different trapping grids. That is, estimates of p and c can be fixed to zero for occasions where a grid was not sampled.
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Fig. 3 Example of the parameter index matrices (PIMs) from Program MARK for the Carothers (1973) Scheme A taxicab data with 10 occasions with the data type designated as “Full Closed Captures with Heterogeneity” in Program MARK. This data type implements the Pledger (2000) approach with two or more mixtures, with only two mixtures used in this example
Fig. 4 The Results Browser window from Program MARK showing the model selection results for the Carothers (1973) Scheme A example
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Table 2 Design matrix to produce estimates under model Mtb with Carothers (1973) Scheme A taxicab data Parameter
Columns 1
2
3
4
5
6
7
8
9
10
11
12
p1
1
0
1
0
0
0
0
0
0
0
0
0
p2
1
0
0
1
0
0
0
0
0
0
0
0
p3
1
0
0
0
1
0
0
0
0
0
0
0
p4
1
0
0
0
0
1
0
0
0
0
0
0
p5
1
0
0
0
0
0
1
0
0
0
0
0
p6
1
0
0
0
0
0
0
1
0
0
0
0
p7
1
0
0
0
0
0
0
0
1
0
0
0
p8
1
0
0
0
0
0
0
0
0
1
0
0
p9
1
0
0
0
0
0
0
0
0
0
1
0
p10
1
0
0
0
0
0
0
0
0
0
0
0
c2
1
1
0
1
0
0
0
0
0
0
0
0
c3
1
1
0
0
1
0
0
0
0
0
0
0
c4
1
1
0
0
0
1
0
0
0
0
0
0
c5
1
1
0
0
0
0
1
0
0
0
0
0
c6
1
1
0
0
0
0
0
1
0
0
0
0
c7
1
1
0
0
0
0
0
0
1
0
0
0
c8
1
1
0
0
0
0
0
0
0
1
0
0
c9
1
1
0
0
0
0
0
0
0
0
1
0
c10
1
1
0
0
0
0
0
0
0
0
0
0
N
0
0
0
0
0
0
0
0
0
0
0
1
Column 1 is the intercept for the p and c parameters, column 2 is the behavioral effect, columns 3–11 are time effects, and column 12 is the population estimate. Model M0 is constructed by just using columns 1 and 12, model Mb by just using columns 1, 2, and 12, and model Mt by using only columns 1 and 3–12
Model selection is performed with information-theoretic procedures (Burnham and Anderson 2002). As an example, model selection results based on AICc for the Carothers (1973) Scheme A example (Fig. 4) suggest model M0 as the best model for these data, while time variation is ranked low compared to behavior and individual heterogeneity effects. Most importantly, when considerable model selection uncertainty exists, parameter estimates can be model averaged to obtain unconditional standard errors, i.e., estimates and standard errors that take into account model selection uncertainty (Burnham and Anderson 2002). Model averaging results for the Carothers (1973) Scheme A data of Nˆ (Table 4) show a considerable increase in the standard error of the model-averaged estimate compared to estimates from most of the individual models. The 95% confidence interval on the model-average estimate includes the true N of 420. Program MARK also provides the capability to estimate variance components (White et al. 2001), e.g., the underlying process variance in a time series of survival or transition probability estimates. Estimates of process variances are needed to construct data-driven population viability analyses (White 2000). A feature recently added to Program MARK is Bayesian estimation using Markov chain Monte Carlo (MCMC) methodology. The main attraction of this feature is the ability to estimate process variances and covariances across series of parameters.
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Table 3 Design matrix to produce estimates under model Mtbh with Carothers (1973) Scheme A taxicab data Parameter
π p A1 p A2 p A3 p A4 p A5 p A6 p A7 p A8 p A9 p A10 p B1 p B2 p B3 p B4 p B5 p B6 p B7 p B8 p B9 p B10 c A2 c A3 c A4 c A5 c A6 c A7 c A8 c A9 c A10 c B2 c B3 c B4 c B5 c B6 c B7 c B8 c B9 c B10 N
Columns 1
2
3
4
5
6
7
8
9
10
11
12
13
14
1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 0
0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0
0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0
0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0
0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0
0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0
0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0
0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1
Column 1 is the π estimate, column 2 is the intercept for the p and c parameters, column 3 is the additive behavioral effect, columns 4–12 are time effects, column 13 is the additive mixture effect, and column 14 is the population estimate. Model Mh is constructed by just using columns 1, 2, 13, and 14, model Mbh by just using columns 1, 2, 3, 13, and 14, and model Mth by using only columns 1, 2, 4–12, 13, and 14
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Table 4 Model averaging results for estimates of population size (N ) for the eight models estimated for the Carothers (1973) Scheme A taxicab data Model
Weight
Estimate
Standard error
{M0 }
0.51778
368.1283
14.5926
{Mb }
0.29814
392.4802
{Mh2 }
0.12832
463.3255
{Mbh2 }
0.05551
396.7729
{Mtb }
0.00014
538.4863
35.14236 273.8799 42.04358 204.5754
{Mt }
0.00007
368.1178
{Mtbh2 additive mixtures}
0.00002
638.4588
506.1973
{Mth2 additve mixtures}
0.00002
463.4162
274.1122
Weighted average Unconditional standard error
389.226
14.59138
55.55765 62.63907
The 95% confidence interval for the weighted average estimate is 266–512, with 21% of the variation attributable to model selection variation. True N was 420. Note that individual heterogeneity models are designated with h2 to signify that two mixtures were modeled
Finally, Program MARK provides the ability to simulate data under the closed population models. This feature is useful for designing surveys and evaluating the robustness of estimators under various assumptions.
5 Availability of Program MARK Program MARK is available from http://www.cnr.colostate.edu/∼gwhite/mark/mark.htm as a downloadable ∼13 Mb setup file. The data used as an example here are distributed with Program MARK as part of this setup file. Other resources to learn Program MARK are linked to this same web page. Acknowledgements K. P. Burnham and D. R. Anderson continue to provide theoretical insights and the “big picture” concerning the development of Program MARK. Many of the ideas for improvements in Program MARK come from the users, particularly the Program MARK workshops and survivors of the Colorado State University graduate class FW663, where Program MARK is used extensively.
References Alho JM (1990) Logistic regression in capture–recapture models. Biometrics 46:623–635 Brownie C, Hines JE, Nichols JD, Pollock KH, Hestbeck JB (1993) Capture-recapture studies for multiple strata including non-Markovian transitions. Biometrics 49:1173–1187 Burnham KP, Anderson DR (2002) Model selection and multimodel inference: a practical information-theoretic approach. Springer-Verlag, New York, USA Carothers AD (1973) Capture-recapture methods applied to a population with known parameters. J Anim Ecol 42:125–146 Chao A, Lee S-M, Jeng S-L (1992) Estimating population size for capture-recapture data when capture probabilities vary by time and individual animal. Biometrics 48:201–216 Hestbeck JB, Nichols JD, Malecki RA (1991) Estimates of movement and site fidelity using mark-resight data of wintering Canada geese. Ecology 72:523–533 Huggins RM (1989) On the statistical analysis of capture-recapture experiments. Biometrika 76:133–140
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Huggins RM (1991) Some practical aspects of a conditional likelihood approach to capture experiments. Biometrics 47:725–732 Kendall WL, Nichols JD (1995) On the use of secondary capture-recapture samples to estimate temporary emigration and breeding proportions. J Appl Stat 22:751–762 Kendall WL, Pollock KH, Brownie C (1995) A likelihood-based approach to capture-recapture estimation of demographic parameters under the robust design. Biometrics 51:293–308 Kendall WL, Nichols JD, Hines JE (1997) Estimating temporary emigration using capture-recapture data with Pollock’s robust design. Ecology 78:563–578 Lukacs PM, Burnham KP (2005) Estimating population size from DNA-based closed capture-recapture data incorporating genotyping error. J Wildlife Manage 69:396–403 Norris JL, Pollock KH (1996) Nonparametric MLE under two closed capture–recapture models with heterogeneity. Biometrics 52:639–649 Otis DL, Burnham KP, White GC, Anderson DR (1978) Statistical inference from capture data on closed animal populations. Wildlife Monogr 62:1–135 Pledger S (2000) Unified maximum likelihood estimates for closed capture–recapture models using mixtures. Biometrics 56:434–442 Rexstad E, Burnham KP (1991) User’s guide for interactive program CAPTURE. Colorado Cooperative Fish and Wildlife Research Unit, Fort Collins, Colorado, USA, 29 pp White GC (2000) Population viability analysis: data requirements and essential analyses. In: Boitani L, Fuller TK (eds) Research techniques in animal ecology: controversies and consequences, Columbia University Press, New York, NY, USA, pp 288–331 White GC, Burnham KP (1999) Program MARK: survival estimation from populations of marked animals. Bird Study 46(Supplement):120–138 White GC, Burnham KP, Otis DL, Anderson DR (1978) User’s manual for program CAPTURE. Utah State University, Logan, Utah, USA, 40 pp White GC, Anderson DR, Burnham KP, Otis DL (1982) Capture-recapture and removal methods for sampling closed populations. Los Alamos National Laboratory Rep. LA-8787-NERP, Los Alamos, New Mexico, USA White GC, Burnham KP, Anderson DR (2001) Advanced features of Program Mark. In: Field R, Warren RJ, Okarma H, Sievert PR (eds) Wildlife, land, and people: priorities for the 21st century. Proceedings of the Second International Wildlife Management Congress, The Wildlife Society, Bethesda, Maryland, USA, pp 368–377
Author biography Gary C. White is a Professor in the Department of Fishery and Wildlife Biology at Colorado State University. His research interests are population dynamics and management of wildlife species, particularly the design and implementation of population monitoring methodologies. He has developed and maintained Program MARK for estimation of parameters from mark-encounter data of animal populations. Besides Program MARK, other software he has developed includes RELEASE, SURVIV, NOREMARK, and CAPTURE.
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