K-Sample Closed Capture-recapture Models UF 2015

K-Sample Capture-recapture Models for Closed Populations * 2

Modeling Capture Recapture Data: Model Parameters N = abundance or population size; number of animals in area exposed to sampling efforts p i = capture probability; probability that a member of N i is caught at time i Primary focus is on modeling p 3

Compare to L-P Operational difference L-PK-sample –Sampling periods2 K>2 Assumptions –Closed PopulationX X –No tag lossX X –Equal Capture ProbabilityX 4

Capture History Data Row vector of 1’s (indicating capture) and 0’s (indicating no capture) Examples: Caught in periods 1,3,4,7 of an 8-period study Caught in periods 2,3,5,7,8 of an 8-period study 5

Sources of Variation in Capture Probability Time (t) – Varies by time (permitted in LP method) Behavioral (Trap) Response (b) - marked and unmarked animals different. –trap-happy causes neg. bias in LP, –trap-shy causes pos. bias in LP. Heterogeneity (h) – each animal could be unique –Causes neg. bias in LP. 6

Closed Population Models (Otis et al. 1978) * ** * * MLE’s available various methods 7

Capture History Modeling: Some Example Model Structures Capture History M 0 M t M b 111 pppp 1 p 2 p 3 pcc 110 pp(1-p)p 1 p 2 (1-p 3 ) pc(1-c) 101 p(1-p)pp 1 (1-p 2 )p 3 p(1-c)c 100 p(1-p)(1-p)p 1 (1-p 2 )(1-p 3 ) p(1-c)(1-c) 011 (1-p)pp(1-p 1 )p 2 p 3 (1-p)pc 010 (1-p)p(1-p)(1-p 1 )p 2 (1-p 3 ) (1-p)p (1-c) 001 (1-p)(1-p)p(1-p 1 )(1-p 2 )p 3 (1-p)(1-p)p 000 (1-p)(1-p)(1-p)(1-p 1 )(1-p 2 )(1-p 3 ) (1-p)(1-p)(1-p) Note (for M b ): c = capture probability for marked animals p = capture probability for unmarked animals 8

Capture History Modeling: Some Example Model Structures Capture History M 0 M t M b 111 pppp 1 p 2 p 3 pcc 110 pp(1-p)p 1 p 2 (1-p 3 ) pc(1-c) 101 p(1-p)pp 1 (1-p 2 )p 3 p(1-c)c 100 p(1-p)(1-p)p 1 (1-p 2 )(1-p 3 ) p(1-c)(1-c) 011 (1-p)pp(1-p 1 )p 2 p 3 (1-p)pc 010 (1-p)p(1-p)(1-p 1 )p 2 (1-p 3 ) (1-p)p (1-c) 001 (1-p)(1-p)p(1-p 1 )(1-p 2 )p 3 (1-p)(1-p)p 000 (1-p)(1-p)(1-p)(1-p 1 )(1-p 2 )(1-p 3 ) (1-p)(1-p)(1-p) Note (for M b ): c = capture probability for marked animals p = capture probability for unmarked animals 9

Capture History Modeling: Some Example Model Structures Capture History M 0 M t M b 111 pppp 1 p 2 p 3 pcc 110 pp(1-p)p 1 p 2 (1-p 3 ) pc(1-c) 101 p(1-p)pp 1 (1-p 2 )p 3 p(1-c)c 100 p(1-p)(1-p)p 1 (1-p 2 )(1-p 3 ) p(1-c)(1-c) 011 (1-p)pp(1-p 1 )p 2 p 3 (1-p)pc 010 (1-p)p(1-p)(1-p 1 )p 2 (1-p 3 ) (1-p)p (1-c) 001 (1-p)(1-p)p(1-p 1 )(1-p 2 )p 3 (1-p)(1-p)p 000 (1-p)(1-p)(1-p)(1-p 1 )(1-p 2 )(1-p 3 ) (1-p)(1-p)(1-p) Note (for M b ): c = capture probability for marked animals p = capture probability for unmarked animals 10

M(0) t1t2t3t4 p p = c 11

M(t) t1t2t3t4 p p = c 12

M(b) t1t2t3t4 p p c 13

M(tb) t1t2t3t4 p p NOT identifiable!! c 14

Identifying the final p If K = 3 % of animals captured at t3 that were captured at t2 % of uncaptured animals at t2 that were captured at t3 15

Identifying the final p In practice, x 000 = 0 p 3 = 1 16

Identifying the final p M(t+1): 339 Real Function Parameters of {f0, p(t), c(t)} DM common intercept} 95% Confidence Interval Parameter Estimate Standard Error Lower Upper :p :p :p :p :p :p E :c :c :c :c :c :f E E E Estimates of Derived Parameters Population Estimates of {f0, p(t), c(t)} DM common intercept} 95% Confidence Interval Grp. Sess. N-hat Standard Error Lower Upper E

Model Mtb need to relate p t and c t in program CAPTURE c t = p t + β in program MARK 18

Model Mtb – Program MARK logit(p) = β 1 *(t 1) + β 2 *(t 2) + β 3 *(t 3) + β 4 *(t 4) + β 5 *recapture 19

Identifying the final c If K = 3 % of previously captured animals that were captured at t3 20

Capture Histories (l = 5 days) p 1 (1-p 1 )(1-p 1 )p 1 (1-p 1 ) (1-p 2 ) p 2 p 2 p 2 p 2 p 3 (1-p 3 ) p 3 (1-p 3 ) p 3 (1-p 4 )(1-p 4 )(1-p 4 )p 4 p 4 M(h): heterogeneity among animals NOT identifiable! 21

Model M h : Approaches to Estimation Jackknife estimators (Burnham and Overton 1978, 1979) Sample coverage estimators (Chao et al. 1992, Lee and Chao 1994) Conditional likelihood with covariates (Huggins 1989, Alho 1990) Finite mixture estimators (Norris and Pollock 1996, Pledger 2000) Parametric Bayesian approaches (Dorazio and Royle 2002) 22

M(h): Jackknife estimator Drop sampling occasions –If M jk is almost as big as M t+1, then M t+1 is close to N. –If M jk is much smaller, than M t+1 is much smaller than N. –Like species accumulation curve Not MLE –No AIC, no LRT 23

M(h): Sample coverage estimator Analyze capture frequencies, f i CV of f i varies with heterogeneity, so CV can be used to adjust estimates of N Not MLE –No AIC, no LRT 24

M(h): Covariates Detection may be a function of covariates –Sex –Size –Age Modeling these groups may account for heterogeneity BUT, can’t measure covariates for undetected animals 25

M(h): Huggins model Full Likelihood Conditional Likelihood (Huggins) Covariates allowed because f 0 removed from likelihood 26

M(h): Bayesian approaches Data augmentation –Add a large number of “all-zero” capture histories to data set –Estimate inclusion probability Ω for each individual 27

M(h): Bayesian approaches Data augmentation –If Ω is a parameter in a Bernoulli distribution, we can estimate it –If we model detection with an over-dispersed distribution, we can model heterogeneity i.e., replace binomial detection with beta-binomial or logit-normal –We can include covariates on p 28

Mixture Models Capture Histories (l = 5 days) p 1 (1-p 1 )(1-p 1 )p 1 (1-p 1 ) (1-p 1 ) p 1 p 1 p 1 p 1 p 2 (1-p 2 ) p 2 (1-p 2 ) p 2 (1-p 2 )(1-p 2 )(1-p 2 )p 2 p 2 M(h): Norris & Pollock, Pledger Mixture of groups  in group 1, (1-  ) in group 2 29

Mixture Model Let’s use example of 2-groups: p 1 =0.3 and p 2 =0.6 and π 1 = 0.4 and 4 samples + = 30

M(h): Pledger mixture Full Likelihood Pledger mixture model 31

M(h): Pledger mixture Pledger mixture model –MLE AIC for model selection 32

Heterogeneity is Bad News Bears Link’s Curse –Different heterogeneity models can yield equal fit (no way to pick the best one) and very different abundance estimates High p helps 33

Model Selection Non-MLE methods (jackknife, coverage) –Unresolved issue –Various ad-hoc approaches MLE methods –AIC, LRT Bayesian –Unresolved issue –BIC 34

Test Closure Otis et al 1978 –Examine time between first and last capture –Requires M(0) or M(h) CLOSETEST –Chi-square test –Requires M(0) or M(t) Open CR models –Estimate immigration or emigration –Sensitive to heterogeneity 35

Exceptions to Closure Assumption (k-sample) If movement in and out of study area completely random, method is unbiased for the super- population. If p varies only by session, estimators are unbiased for population when –Recruits enter between sessions (ingress-only), or –Marked and unmarked leave (or die) with equal probability (egress-only). –Note: data must be pooled. 36

Pooling for movement in or out If egress, pool all but first occasion and use LP estimator If ingress, pool all but last occasion and use LP estimator

Practical Advice Closure: short duration, K=5-10 –avoid migration, recruitment, deaths Tag loss: double tag animals Get p as high as possible –For 50 animals, p > 0.4 –For 200 animals, p > 0.2 (Otis et al. 1978) 38

Practical Advice Trapping grids –Use rectangles, not lines –Grid 10x10 or bigger –Space between traps = ¼ home-range diameter –Place more than one trap at each spot 39

Practical Advice Trap deaths –If low (5%), remove from analysis, add back to N ̂ –If higher, consider removal model Anomalous event (e.g., storm) –Remove affected data –Model effect Pilot study! 40

Program CAPTURE Otis et al. (1978), Rexstad and Burnham (1991) Estimates under 7 models (all but M tbh ) Variances and confidence intervals Closure test Goodness-of-fit and between-model tests Model selection algorithm 41

Program MARK Closed Captures (Mtb, Mt, Mb, M0) Huggins Closed Captures (N not in likelihood) Closed Captures with Het. (Mh mixture) Full Closed Captures with Het. (Mtbh mix.) Huggins Heterogeneity Huggins Full Heterogeneity 42