1. Multistate models UF 2015

2. Outline  Description of the model  Data structure and types of analyses  Multistate with 2 and 3 states  Assumptions  GOF

3. Multistate Models  First developed by Arnason (1972, 1973) and almost completely ignored  Developed independently by Hestbeck et al. (1991)  Modern development: Brownie et al. (1993), Schwarz et al. (1993) Reviews by Lebreton et al. (2002, J. Appl. Stat), Kendall and Nichols (2004, Condor), Kendall (2004, Animal Biodiversity & Conservation)

4. Multistate Models: Data Structure  Open capture-recapture study  At each capture, animals are categorized by “state” Location (Hestbeck et al. 1991) Size class (Nichols et al. 1992) Breeding state (Nichols et al. 1994) Disease state (Jennelle et al., 2007)  State of an animal may change from 1 period to the next

5. Multistate Models: Capture History Data  Capture history: no longer adequate to use just 1’s and 0’s  0 still denotes no capture  Assign a positive number or letter to each state e.g., with 3 states: 1, 2, 3  Example: 3-site system Possible histories: 10310, 00201

6. Multistate Models: Notation   t rs = probability that animal in state r at sample period t is alive in state s at sample period t+1  p t r = probability that animal in state r at sample period t is captured

7. Multistate Models 3 states

8. Multistate Models: Capture History Expectations AAB: http://www.phidot.org/software/mark/docs/book/pdf/chap8.pdf

9. Multistate Models: Capture History Expectations http://www.phidot.org/software/mark/docs/book/pdf/chap8.pdf

10. Multistate Models: Decomposition of  t rs  Sometimes reasonable and desirable to decompose  t rs into survival and movement components:  S t r = probability that animal in state r at period t survives until period t+1   t rs = probability that animal in state r at period t and alive at period t+1, is in state s at t+1

11. Multistate Models 3 states

12. Parameter Estimation  Data: numbers of new releases each period and number of animals with each capture history  Model: probability structure for each capture history  Maximum likelihood (e.g., programs MARK, MSSURVIV, MSURGE)

13. Metapopulation model http://www.phidot.org/software/mark/docs/book/pdf/chap8.pdf

14. Evaluation of hypotheses

15. No diseaseDisease S i N  i ND S i D  i DN Disease States

16. Capture Histories (k = 3 years)  NNN  0BN  Survival depends on where you start.  Transition “ “ “ “ “  Capture depends on where you end up

17. B N History B0B (2 possibilities) 1 2 B N 1 2

18. Capture Histories (k = 3 years) NNN 0BN B0B

19. Multistate model assumptions  Transition from a given state must sum to 1

20. Multinomial logit transformation for 3 states  What about 2 states?

21. Multistate model assumptions  Within state, age, sex, etc., all animals equally likely to survive, move to given location, be detected  Marks do not affect survival or movement, are not lost, are recorded correctly  Each animal acts independently with respect to survival, movement, detection  Each state observable  State is correctly assigned each time

22. Multistate model assumptions   t rs = probability that animal in state r at period t and alive at period t+1, is in state s at t+1, given that alive at t+1. In other words if in state A at t, ind. survive at rate S A (e.g., not S B ). Then (move or stay) at t+1 to another state (e.g., B, or the same [A]): “survive and then move” “Move and then survive”

23. Multistate model assumptions  Transition probabilities do not depend on previous state S at t does not depend on S at t-1  Nichols et al. developed a “memory model” that relaxes this assumption S at t depends on S at t-1  MS-Surviv

24. Multistate models: why bother?  Transitions might be interesting biologically  Reduces heterogeneity in survival or capture probability by partitioning animals.

25. Multi-site model tools  MSSURVIV (J. Hines) Includes memory model Can define which movement probability by subtraction Can choose between estimating survival and transition separately or together  MARK (G. White) Can define which movement probability by subtraction Multinomial logit transformation Can include individual and group covariates (e.g., weight, age, climatological)  MSURGE (Choquet et al.) GOF test (UCARE) Powerful model specification feature

26. GOF testing  UCARE  Median c-hat

27. Local minima  Starting values  Simple models  Simulated annealing  MCMC: R-hat

28. MCMC

29. What you should know…  Parameters of interest  Assumptions  Example of applications  GOF