1. Mixture models for estimating population size with closed models Shirley Pledger Victoria University of Wellington New Zealand IWMC December 2003

2. 2 Acknowledgements Gary White Richard Barker Ken Pollock Murray Efford David Fletcher Bryan Manly

3. 3 Background Closed populations - no birth / death / migration Short time frame, K samples Estimate abundance, N Capture probability p – model? Otis et al. (1978) framework

4. 4 M(tbh) M(tb) M(th)M(bh) M(t)M(b)M(h) M(0)

5. 5 Models for p M(0), null model, p constant. M(t), Darroch model, p varies over time M(b), Zippin model, behavioural response to first capture, move from p to c M(h), heterogeneity, p varies by animal M(tb), M(th), M(bh) and M(tbh), combinations of these effects

6. 6 Likelihood-based models M(0), M(t) and M(b) in CAPTURE, MARK M(tb) – need to assume connection, e.g. c and p series additive on logit scale M(h) and M(bh), Norris and Pollock (1996) M(th) and M(tbh), Pledger (2000) Heterogeneous models use finite mixtures

7. 7 M(h) C animal classes, unknown membership. Animal i from class c with probability  c. Animal i Class 1 Class 2 Capture probability p 1 Capture probability p 2 11 22

8. 8 M(h 2 ) parameters N  1 and  2 p 1 and p 2 Only four independent, as  1 +  2 = 1 Can extend to M(h 3 ), M(h 4 ), etc.

9. 9 M(th) parameters N  1 and  2 (if C = 2) p matrix, C by K, p cj is capture probability for class c at sample j Two versions: 1.Interactive, M(txh), different profiles 2.Additive (on logit scale), M(t+h).

10. 10 M(t x h), interactive Different classes of animals have different profiles for p Species richness applications

11. 11 M(t+h), additive (on logit scale) For Class 1, log(p j /(1-p j )) =  j For Class 2, log(p j /(1-p j )) =  j   2 Parameter  2 adjust p up or down for class 2 Similar to Chao M(th) Example – Duvaucel’s geckos

12. 12 M(bh) parameters N  1   C (C classes,  ) p 1... p C for first capture c 1... c C for recapture Two versions: 1.Interactive, M(bxh), different profiles 2.Additive (on logit scale), M(b+h).

13. 13 M(b x h), interactive Different size of trap- shy response One class bold for first capture, large trap response Second class timid at first, slight trap response.

14. 14 M(b + h), additive (logit scale) Parallel lines on logit scale For Class 1, log(p/(1-p)) =  1 log(c/(1-c)) =  1   For Class 2, log(p/(1-p)) =  2 log(c/(1-c)) =  2   Common  adjusts for behaviour effect

15. 15 M(tbh) Parameters N and  1...  C (C classes) Interactive version – each class has a p series and a c series, all non-parallel. Fully additive version – on logit scale, have a basic sequence for p over time, use  to adjust for recapture and  to adjust for different classes. There are also other intermediate models, partially additive.

16. 16 M(t x b x h) For class c, sample j, Logit(p jc ) =   j +  +  c  +  j  +  jc  +  c  +  jc where  is a 0/1 dummy variable, value 1 for a recapture. (Constraints occur.)

17. 17 Other Models M(t+b+h) – omit interaction terms M(t x h) – omit terms with  M(t + h) – also omit (  ) interaction term M(b x h) – omit  terms M(0) has  only.

18. 18 M(t x b) Can’t do M(t x b) – too many parameters for the minimal sufficient statistics. Can do M(t+b) using logit. Similar to Burnham’s power series model in CAPTURE. Why can we do M(t x b x h) (which has more parameters), but not M(t x b)?

19. 19 M(txbxh) M(txb) M(txh)M(bxh) M(t)M(b)M(h) M(0) M(t+b)M(t+h)M(b+h) M(t+b+h) Now have these models:

20. 20 Example - skinks Polly Phillpot, unpublished M.Sc. thesis Spotted skink, Oligosoma lineoocellatum North Brother Island, Cook Strait, 1999 Pitfall traps April: 8 days, 171 adults, 285 captures Daily captures varied from 2 to 99 (av<40) November: 7 days, 168 adults, 517 captures (20 to 110 daily, av>70)

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23. 23 April: Rel(AIC c ) npar M(t + b + h) 0.00 12 M(t x h) 8.82 18 M(t x b x h) 9.79 26 M(t + h) 26.65 10 M(t) 63.43 9 M(t + b) 65.25 10 M(b x h) 200.56 6 M(b + h) 205.06 5 M(b) 267.15 3 M(h) 289.81 4 M(0) 328.53 2

24. 24 November: Rel(AIC c ) npar M(t x b x h) 0.00 22 M(t x h) 4.65 16 M(t + b + h) 7.82 11 M(t + h) 8.24 10 M(t + b) 145.08 9 M(t) 174.76 8 M(b + h) 190.50 5 M(h) 200.44 4 M(b x h) 219.41 6 M(0) 323.76 2 M(b) 325.60 3

25. 25 Abundance Estimates Used model averaging April, N estimate = 206 (s.e. = 33.0) 95% CI (141,270). November, N estimate = 227 (s.e. = 38.7) 95% CI (151,302).

26. 26 Using MARK Data entry – as usual, e.g. 00101 5; for 5 animals with encounter history 00101. Select “Full closed Captures with Het.” Select input data file, name data base, give number of occasions, choose number of classes, click OK. Starting model is M(t x b x h) Following example has 2 classes, 5 sampling occasions.

27. 27 Parameters for M(t x b x h) 11 1 p for class 1 2 3 4 5 6 p for class 2 7 8 9 10 11 c for class 1 12 13 14 15 c for class 2 16 17 18 19 N 20

28. 28 M(t x h): set p=c 11 1 p for class 1 2 3 4 5 6 p for class 2 7 8 9 10 11 c for class 1 3 4 5 6 c for class 2 8 9 10 11 N 12

29. 29 M(b x h): constant over time 11 1 p for class 1 2 2 2 2 2 p for class 2 3 3 3 3 3 c for class 1 4 4 4 4 c for class 2 5 5 5 5 N 6

30. 30 M(t) 11 1 (fix) p for class 1 2 3 4 5 6 p for class 2 2 3 4 5 6 c for class 1 3 4 5 6 c for class 2 3 4 5 6 N 7

31. 31 M(b) 11 1 (fix) p for class 1 2 2 2 2 2 p for class 2 2 2 2 2 2 c for class 1 3 3 3 3 c for class 2 3 3 3 3 N 4

32. 32 M(0) 11 1 (fix) p for class 1 2 2 2 2 2 p for class 2 2 2 2 2 2 c for class 1 2 2 2 2 c for class 2 2 2 2 2 N 3

33. 33 M(t + h): use M(t x h) parameters (as below), plus a design matrix 11 1 p for class 1 2 3 4 5 6 p for class 2 7 8 9 10 11 c for class 1 3 4 5 6 c for class 2 8 9 10 11 N 12

34. 34 Design matrix for M(t + h). Use logit link. B1B2B3B4B5B6B7B8 11 1 p class 1 1 1 1 1 1 p class 2 1 1 1 1 1 1 1 1 1 1 N 1  7 is   Adjusts for class 2

35. 35 M(b + h) Start with M(b x h) and use this design matrix, with logit link B1B2B3B4B5  1 1 p class 1 1 p class 2 1 c class 1 1 1 c class 2 1 1 N 1

36. 36 M(t + b + h) Start with M(t x b x h) Use one  to adjust for recapture For each class above 1 use another  for the class adjustment.

37. 37 Time Covariates Time effect could be weather, search effort Logistic regression: in logit(p), replace  j with linear response e.g.  x j +  w j where x j is search effort and wj is a weather variable (temperature, say) at sample j Logistic factors: use dummy variables to code for (say) different searchers, or low and high rainfall. Skinks: maximum daily temperature gave good models, but not as good as full time effect.

38. 38 Multiple Groups Compare – same capture probabilities? If equal-sized grids, different locations, N indexes density – compare densities in different habitats. Cielle Stephens, M.Sc. (in progress) – skinks. Good design - eight equal grids, two in each of four different habitat types. Between and within habitat density comparisons. Temporary marks.

39. 39 Discussion Advantages of maximum likelihood estimation – AIC c, LRTs, PLIs. Working well for model comparison. Two classes enough? Try three or more classes, look at estimates.

40. 40 If heterogeneity is detected, models including h have higher N and s.e.(N). If heterogeneity is not supported by AIC c, the heterogeneous models may fail to fit. See the parameter estimates. M(t x b x h) often fails to fit – see parameter estimates (watch for zero s.e., p or c at 0 or 1).

41. 41 Alternative M(h) – use Beta distribution for p (infinite mixture). Which performs better? - depends on region of parameter space chosen by the data. Often similar N estimates. Don’t believe in the classes or the Beta distribution. Just a trick to allow p to vary and hence reduce bias in N.

42. 42 All models poor if not enough recaptures. Warning signals needed. Finite mixtures, one class with very low p. Beta distribution, first parameter estimate < 1. Often with finite mixtures, estimates of  and p are imprecise, but N estimates are good.