1. Time symmetrical (reverse-time) models

2. Resources:
Pollock, K. H., D. L. Solomon, and D. S. Robson Tests for mortality and recruitment in a k-sample tag- recapture experiment. Biometrics 30:77-87.
Pradel, R Utilization of capture-mark-recapture for the study of recruitment and population growth rate. Biometrics 52:

3. Motivation
Pollock et al. (1974) first suggested that capture history data were symmetrical with respect to time.
Survival models
Estimate probabilities of survival and recapture
given that an individual was known to be alive at some earlier time.
Time symmetrical or reverse-time models
Estimate the probabilities of membership and recapture in the population at earlier time periods
given capture at some later time.

4. Data structure
Live capture histories consisting of 1s and 0s – the same data that are required for CJS models.
m-array remains the same.
Redefine cell probabilities for the reverse-time models.

5. Model Structure - new parameters
gi – seniority probability, probability that an animal present just before time i was already present just after time i-1.
(i.e., the animal did not enter the population since the last sampling occasion i-1 to i.)
This animal is “old.”
Conversely, 1- gi is the probability than an animal was recruited to the sampled population since i -1.

6. Model Structure - new parameters
p’i – recapture probability at time i given present at (just after) i.
i – probability of not being seen prior to i given present just before i
(Analogous to i, and the final term in the multinomial)
Thus, this includes animals was not present or was in the population and not seen at or before i–1.

7. Losses or removal on capture
It is impossible for animals removed or killed on capture at i -1 to be recaptured at i.;
In open models, we deal with this situation implicitly by not including those individuals in the number of new releases just after i.
Furthermore, it has been shown that the estimates of  are relatively robust to heterogeneity in recapture probabilities in forward-time modeling.
Releases at i
Encounters at time j i
Ri – new releases 2 3 4 5 1 R1
m12
m13
m14
m15
R2+m12+m13+m14+m15
m23
m24
m25
R3+m23+m24+m25
m34
m35
R4+m34+m35
m45

8. Relationship of capture probabilities
The difference between the recapture probabilities is incorporated in the equation: where i is the rate of re-release after capture.
Thus, when i = 1, p’i = pi.

9. Capture history example – 011010
Capture at periods 2,3,5 in a 6-year study.
Estimate the probabilities of membership and recapture in the population at earlier time periods
P(011010|last capture at i =5) = 5(1-p’4) 4p’33p’22
Probability that, given capture at 5, this animal survived period 5, was not captured at 4, survived time 4, was captured at 3, survived time 3, was captured at 2, but was not seen before time 2.

10. Additional conditions
Conditions on the population size just prior to the first occasion and then calculates the expected population size at later times based on the expected change in population size over time (i ).
Thus, the expectation is that
Accounts for loss on capture

11. Alternative parameterizations
Pradel (1996) also described 3 other parameterizations
Accomplished this by simultaneous modeling of rates associated with forward-time and reverse-time models.
Allows modeling the effects of stationarity or covariates on parameters
Stationarity of  - useful for estimating elasticities comparable to population models
i , g i and f i as functions of environmental covariates
Time trend in i

12. Other parameterizations
Survival and fertility (recruitment):
Survival and lambda:

13. Assumptions Similar to CJS models:
Homogeneity is assumed with respect to seniority and capture.
The various ways of dealing with heterogeneity still apply – stratification, age-modeling, use of covariates.
Probability of seniority is the same for all animals in each period.
Capture probabilities are the same for each individual in the population at that time.
Capture and seniority probabilities of individuals are independent.

14. Estimation EH “forward order” and chose Pradel model
Reverse the order of the capture histories and running through software for forward- time analyses
(e.g., Recaptures only model in MARK).
MARK includes 6 different parameterizations for reverse time model
4 originally described
2 robust design

15. Estimation
Parameters estimable for a single-aged, time-specific K- period study
K, K-1, …, 3 and p’K-1, p’K-2, …, p’2.
2 and p’1 can not be estimated separately
w/no losses on capture the capture probabilities are identical to those obtained using the CJS model.

16. Interpretation - (After Nichols et al. 2000)
Consider an open population, where we define the growth rate, i, as the realized change in population size of the time period i. Then it is reasonable to think of this parameter as:

17. Contributions to population growth
Alternatively, population size at Ni+1 = Li +Bi
Li and Bi as binomial parameters given Ni+1
Bi = Ni+1 – Li
gi+1 can be viewed as the relative contributions to realized population growth. !

18. Example if gi+1 = 0.25 (and (1-gi+1) = 0.75),
Survivors have 3 times the impact on population growth in comparison to new recruits.
Contributions analogous to the elasticities we will examine with matrix models.

19. Alternative modeling Reduced parameter models
parsimony
increasing precision of estimates.
Multiple groups is also possible,
Covariates of gi and p’i.
Capture history dependence (e.g., behavioral response) are not presently possible

20. Model selection, robustness, and assumptions
Model selection via AIC is the same.
Goodness-of-fit tests based on contingency tables
not investigated in detail.

21. Additional applications of time symmetry
Multi-state & robust design models
Age-specific models (Nichols et al. 2000)
Primarily for contribution of age classes (gi+1)
Metapopulations
Relative contributions of patches

22. Cautionary notes Estimator robustness
Seniority estimator may not be robust to heterogeneity
Trap response has little influence on CJS survival estimates because they are conditioned upon prior captures.
Multiple trapping techniques
Trap response
Trap-happy (increase in capture probility) over- estimates of capture probability and under-estimates of seniority and vice-versa.
Similar situation when recapture probabilities of marked and unmarked individuals vary for other reasons.
Change in study area size leads to + bias in 

23. Robust Design

24. 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:

25. Kendall, W. L. , K. H. Pollock, and C. Brownie. 1995
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.

26. 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
1-’

27. 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).

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

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

30. 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

31. Data Structure K primary sampling occasions (sessions)
Periods when population is assumed to be open.
Sampled at l secondary sampling occasions within each primary occasion
Population assumed closed.
Number of occasions can vary (>2).

32. 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

33. Robust design example Encounter history 12 live capture occasions
11 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

34. 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)

35. 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

36. 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.

37. 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

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

39. 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.

40. 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.

41. 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:

42. 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

43. Models Reverse time models Multi-state models
Closed population models can incorporate individual capture heterogeneity
Estimate survival in the presence of capture heterogeneity.
Not possible using CJS open population models.

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

45. 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)

46. 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

47. Components of the Likelihood
Capture of unmarked animals:

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

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

50. 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.

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

52. 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.

53. 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:

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

55. 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.

56. 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.

57. 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.

58. 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

59. 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.
Probability of returning given emigration = (1- ’)
In MARK Markovian model by estimating ’’ and ’ separately

60. 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.

61. 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).

62. Robust design with random emigration
Relationship between the superpopulation:
Superpopulation less the proportion that emigrated
Relationship between capture probabilities for superpopulation and “available” population is:
Probability of capture given that didn’t emigrate

63. 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.

64. 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.

65. 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.
Thus if ’i and ’’i estimated separately then model is Markovian process
See Williams et al. (2001) or Kendall et al. (1997) for details of the likelihood.

66. 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

67. 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).

68. 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).

69. 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

70. 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.