1. Wildlife Population Analysis
Lecture 08 - Time Symmetry and Robust Design

2. Detailed outlines of term project
Send to both and
Title
Questions
Description of the data
Hypotheses  models (more than one)
Examples
I will analyze mark recapture data from small mammals captured at Skyline Wildlife Management areas over 5 trap nights in each of 4 years to examine changes in abundance over time in two different habitat types using robust design models.
Hypothesis: White footed mice are more abundant in forest openings than in clearcuts each year after controlling for differences in trap response.
I will examine hypotheses related to survival of Alabama sturgeon using mark recapture data collected over 50 years with models for open populations
Hypothesis: Survival rates have declined as the number of dams has increased in the Alabama River
Bibliography:
A few citations to establish that you have reviewed the literature related to your hypotheses

3. Robust Design

4. Pollock’s Robust design
Readings
Kendall, W. L., J. D. Nichols, and J. E. Hines Estimating temporary emigration using capture-recapture data with Pollock's robust design. Ecology 78:
Resources
Williams, B.K., J.D. Nichols, and M.J. Conroy Analysis and Management of Animal Populations. Academic Press. San Diego, California.
Kendall, W. L Robustness of closed capture-recapture methods to violations of the closure assumption. Ecology 80:
Kendall, W. L., and J. D. Nichols On the use of secondary capture-recapture samples to estimate temporary emigration and breeding proportions. Journal of Applied Statistics 22:
Kendall, W. L., K. H. Pollock, and C. Brownie A likelihood-based approach to capture-recapture estimation of demographic parameters under the robust design. Biometrics 51:
Pollock, K. H., J. D. Nichols, C. Brownie, and J. E. Hines Statistical inference for capture-recapture experiments. Wildlife Monographs pp.

5. Apparent Mortality (1- Apparent Survival)
Population dynamics
Apparent Recruitment
Apparent Mortality (1- Apparent Survival)
Births
Emigration N
Changes in population size through time are a function of births, deaths, immigration, and emigration.
The open population CMR models we have considered until now have only estimated apparent survival rates.
That is, true survival, the portion of individuals surviving between sampling occasions is confounded with permanent emigration.
I spoke briefly on closed models that assume that size of the population remained unchanged between sampling events.
Robust design integrates the advantages of both types of models.
Although this model is complicated, it brings more biological reality to the analysis of population dynamics.
This method provides for the estimation of parameters that are not estimable under either open or closed models as well as more robust estimates of the familiar parameters of interest.
Immigration
Deaths

6. Robust Design
Most open population CMR models - captures occur instantaneously
Rarely the case.
Data are aggregated from capture sessions that may last several days or weeks.
Pollock’s Robust Design
break sessions into shorter sampling occasions
capture probabilities estimated among encounter occasions during these sessions.
Primary assumption over CJS
Close population during capture sessions (i.e., no births, deaths, emigration, or immigration).

7. Robust Design Closed models Open models Estimate N true survival
temporary emigration
immigration

8. Basic design Sample over two temporal scales. Only disadvantage - cost
more than one sampling occasion each session.

9. Motivation
Early work by Robson (1969) and Pollock (1975) w/CJS models:
Unable to incorporate:
Heterogeneity
Permanent trap response among individuals.
Survival estimates are robust
Abundance estimators ARE NOT

10. Data Structure K primary sampling occasions
Periods when population is assumed to be open.
Sampled at l secondary sampling occasions within each primary occasion
Population assumed closed.
Number can vary.

11. Robust design example
Robust design example, with 3 primary trapping sessions, each consisting of 4 secondary occasions.
Primary periods Ki 1 2 3
Secondary periods lij 4 5 6 7 8 9 10 11 12
Population status
Closed
Open

12. Robust design example Encounter history 12 live capture occasions
11 survival intervals
Unequal length
1s – open
0s - closed
Occasion 1 2 3 4 5 6 7 8 9 10 11 12
Interval 1 2 3 4 5 6 7 8 9 10 11
Length

13. Ad hoc approach Pollock’s (1981, 1982) original work
Combined open and closed models
Not based on a single likelihood.
Procedure
Select closed model(s) to estimate abundance during secondary periods
Select open model to estimate survival between the primary periods
Combined data from secondary periods (i.e., capture > 1 time = encounter)

14. Ad hoc approach Estimate recruitment using abundance and survival via:
Number of births during i i+1 ni
Animals removed at i
Population size at time t+1 i
Survival rate during the period i  i +1
Population size at time t Ri
Number released at i

15. Ad hoc approach Model selection independent
Single open model
K closed-models
Williams et al. (2002) recommend:
Single closed model for all of the secondary periods for consistency in assumptions and biases.

16. Robust Design 2 types of capture probabilities must be estimated:
pij - the capture probabilities associated with capture during the secondary sampling period j in the primary sampling period i , given that the animal was in the population at Ki and
pi* - the capture probabilities associated with capture at least once during the primary period i, given that the animal was in the population at Ki

17. Models Primary (open) capture probabilities (pi*) describe
temporal variation
‘permanent’ trap response.
Closed model capture probabilities (pij )
temporal variability
behavioral response
heterogeneity.

18. Models
Permanent trap response (different recapture probability after first capture)
splitting the data set into 2 groups
Fit single open model and
(2K-1) closed models for marked and unmarked animals in each of the primary periods except the first when all individuals are unmarked.
Kendall et al. (1995) describes 24 possible models based on all combinations of the primary and secondary capture probability models.

19. Assumptions - Kendall et al. (1995)
Closed-population models for secondary periods:
Population is closed to gains and losses during the period,
Marks are not lost nor incorrectly recorded,
Capture model structure is correct
Open models for the primary periods:
Survival probabilities equal for all individuals,
Capture probabilities equal for all individuals
Capture and survival probabilities of individuals are independent.

20. Additional parameters estimated
Not estimable under either CJS or closed-population models.
(K-1)pK - unidentifiable parameters in CJS
N1, and B1 not estimable in closed models
Under Robust Design:

21. Models
Primary and secondary periods are treated as independent ~ any closed or open models possible
Probability of temporary emigration - animal was not present on the study area during a given primary period
Estimated from the 2 types of capture probabilities.
Age specific models
Reverse time models
Multi-state models
Closed population models can incorporate individual capture heterogeneity

22. Likelihood-based approach
Kendall et al. (1995) described the combined likelihood of the data
products of the recapture components
mathematical relationships between capture probabilities.

23. Models
Any of the closed models for which MLE have been described can be used
Models incorporating heterogeneous capture probabilities.
Mixture models which incorporate random effects
Huggin’s models with covariates of capture probabilities (already in MARK)

24. Likelihood Example
Model with time variation in capture probabilities in both primary and secondary periods and only 2 samples from each secondary period:
P1 and P2 are unconditional components of the open population model. ui
number of unmarked caught on at least one secondary occasion within primary period i Ui
population of unmarked available for capture at i
mhi
number marked during h (prior to i) caught on at least one secondary occasion within primary period i. Thus, ui + mhi the number caught at least once during i. Ri
number of marked animals released at i ri
number from Ri ever recaptured
pi*
probability of capture at least once during the primary period i, given that the animal was in the population at Ki..
pij
probability of capture during the secondary sampling period j in the primary sampling period i , given that the animal was in the population at Ki
number previously unmarked exhibiting capture history  over the secondary occasions within primary period i,
number marked during primary period h unmarked animals exhibiting capture history  over the secondary occasions within primary period i

25. Components of the Likelihood
Capture of unmarked animals:

26. Components of the Likelihood
Conditional probability of the recaptures (mij): where  is the probability of not being recaptured.

27. Components of the Likelihood
Probability across the secondary periods in all of the different primary periods.

28. Capture probabilities linked
Probability of capture at least once during the primary period is probability of NOT going UNCAPTURED during all of the secondary periods.
Basis for the joint modeling of the data from both types of sampling periods.

29. Other Models
By constraining (reducing) the number of estimated parameters,
Use of covariates
time-specific
individual

30. Model assumptions Identical to those for the ad hoc approach and…
The relationship between the 2 types of capture probabilities.
Violated is when temporary emigration occurs among the primary sampling periods, thus some individuals are not available for capture during some secondary periods.

31. Estimation
In program MARK – Data types: Robust Design and Robust Design (Huggins est.).
The latter includes the ability to model heterogeneity as a function of recapture covariates.
Abundance estimation is via the equation:

32. Estimation
Thus the number of new recruits is also estimable as in the ad hoc approach via:

33. Model Selection, Estimator Robustness, and Assumptions
Models are based on the likelihood
model selection via AIC
LRTs
Robust GOF tests are not easily constructed,
Contingency tables and
Bootstrap are reasonable.

34. Effects of heterogeneity
Robust design models do not incorporate heterogeneity (but see Pledger 2000 and Huggins models),
With heterogeneity:
Survival estimates relatively unbiased,
Capture probabilities positively biased
Estimates of population size be negatively biased.

35. Special estimation problems
Temporary Emigration
violation of the assumed relationship between the two types of capture probabilities
biased parameter estimates.
Kendall and Nichols (1995) and Kendall et al. (1997)
estimators and models for use when temporary emigration occurs.

36. Examples Home ranges not completely sampled Migratory patterns
Example molt migrations
Periods of inactivity
Amphibians using vernal pools
Animals first appear as breeding adults.
Sea turtles

37. Two models of temporary emigration
Random temporary emigration - each individual has the same probability of becoming a temporary emigrant.
Capture probabilities are negatively biased
Abundance estimates show positive bias
“Markovian” temporary emigration - probability of temporary emigration at time i is affected by whether an individual was a temporary emigrant at i-1.
Direction and magnitude of bias dependent upon nature of process.

38. Concept
“Superpopulation,” number of animals associated with the sampling area.
Contrast with Ni, number of animals present at time i.
Requires the assumption that the population is essentially closed during the secondary sampling periods.

39. Additional variables and parameters
number of animals marked before primary period i and in the super population for the duration of the study. Mi
number marked animals in the area during the primary period i
number of animals entering the superpopulation between primary periods i and i+1 and remaining in the superpopulation for the duration of the study. Bi
number of animals entering the area during the primary period i
probability of capture for a member of the superpopulation during primary period i
probability of capture during i given present on the study area i
probability of temporary emigration (i.e., a member of but not available for capture).

40. Robust design with random emigration
Relationship between the superpopulation:
Relationship between capture probabilities for superpopulation and “available” population is:

41. Random emigration
Probability (rate) of random emigration can be estimated ad hoc from:
where is the probability of capture during the secondary sampling period i and is the capture probability after the open period
Only valid under models that do not include capture heterogeneity.

42. Likelihood – random emigration
only P2 changes
where ie is in the superpopulation at i (initial capture) and never seen again.
This model can be thought of as ptt.
For  = 0, this model becomes tpt.

43. Robust design with Markovian emigration
Temporary emigration is a first order Markov process, i.e., it depends upon the state of the individual during the prior time period.
Additional notation:
’i - probability of temporary emigration in primary period i given temporary emigration at i-1.
’’i probability of temporary emigration in primary period i given NOT a temporary emigration at i-1.
Thus if ’i = ’’i the random emigration model is obtained.
See Williams et al. (2001) or Kendall et al. (1997) for details of the likelihood.

44. Models
Require many constraints because they condition on probabilities pertaining to animals not observed in the previous period.
Particularly useful for animal that have low breeding propensity (i.e., they do not breed in some years).
Examples
Sea turtles
Marine mammals
Seaducks
Salamanders

45. Multiple Ages and Recruitment Components
Robust Design extended to multiple age groups
Components of recruitment can be separated
in situ reproduction
immigration from outside the study area (see Nichols and Pollock 1990 and Yoccoz et al. 1993).

46. Study design Critically important.
Time between primary sampling periods = time required for young to mature into adults
(or the next stage of interest).
Example
animals classified as young during i can be assumed to be adults at i+1, and
any other new adults on the area are assumed to be immigrants.
Extended to multiple age classes and multiple patches (Nichols and Coffman 1999).

47. Robust Design - study design
Number of secondary periods – trade off between precision and model complexity versus population closure.
If closure is false, open models can be used, but heterogeneity can not be examined.
Biological motivation for the study – temporary emigration rates may be of primary interest
Breeding rates
Recruitment – reproduction and immigration

48. Benefits
Robust Design minimizes covariation between estimated parameters thus producing more precise estimates of the parameters of interest.
Parameters that are otherwise inestimable – the final survival rate in open models, and in closed models the first and last recapture probabilities and abundances.
More precise estimates of demographic rates due to increased captures in primary periods.

49. Robust Design
Estimating recruitment components with multiple age classes
Williams et al. (2002, pp describe the use of reverse time robust design
Multi-age models of recruitment
Proportionate size of different age classes within populations.
Extension to multiple locations.