1. Wildlife Population Analysis
Lecture 06 – Multinomials and CMR Models for closed populations

2. Multinomials
Lecture 06 – Part I

3. Multinomial Distribution and Likelihoods
Extension of the binomial coefficient with more than two possible mutually exclusive outcomes.
Nearly always introduced by way of die tossing.

4. Binomial Coefficient
The binomial coefficient was the number of ways y successes could be obtained from the n trials
Example 7 successes in 10 trials

5. Multinomial coefficient
The multinomial coefficient or the number of possible outcomes for die tossing (6 possibilities):
Example rolling each die face once in 6 trials:

6. Properties of multinomials
Dependency among the counts.
For example, if a die is thrown and it is not a 1, 2, 3, 4, or 5, then it must be a 6.
Face
Number
Variable 1 10 y1 2 11 y2 3 13 y3 4 9 y4 5 8 y5 6 y6
TOTAL 60 n

7. Multinomial pdf
Probability an outcome or series of outcomes:

8. Die example 1
The probability of rolling a fair die (pi = 1/6) six times (n) and turning up each face only once (ni = 1) is:

9. Die example 1
Dependency

10. Example 2
Another example, the probability of rolling 2 – 2s , 3 – 3s, and 1 – 4 is:

11. Likelihood
As you might have expected, the likelihood of the multinomial is of greater interest to us
We frequently have data (n, yi...m) and are seeking to determine the model (pi…m). The likelihood for our example with the die is:

12. Log-likelihood
This likelihood has all of the same properties we discussed for the binomial case.
Usually solve to maximize the ln(L)

13. Log-likelihood
Ignoring the multinomial coefficient (constant)

14. Capture histories & multinomials
Procedure in CMR studies
Capture a sample of animals,
Mark them so that they are uniquely identifiable,
Release them, and
Record
Recoveries of dead animals
Recaptures and re-observations (re-sightings) of live animals.

15. Data Tag No. Date 211 2/1/1991 3/2/1993 212 2/6/1991 213 3/1/1993 214
These data usually recorded in the form of a “vertical file” that may include ancillary information (i.e., covariates).
Tag No.
Date
211
2/1/1991
3/2/1993
212
2/6/1991
213
3/1/1993
214
3/1/1991
2/10/1993

16. Capture histories Occasion Tag No. 1 2 3 211 212 213 214 215 216 217
Typically converted to the form of a capture-history matrix
Each row represents an individual
Each column represents a sampling occasion.
On each occasion each individual
‘1’ if encountered (captured)
‘0’ if not encountered.
Ultimately for the closed population estimators discussed in this section these are converted to ‘LLLL’ data.
Occasion
Tag No. 1 2 3
211
212
213
214
215
216
217
218

17. Capture history - example
For individuals in a population marked and then on 3 occasions there are 8 (=2m = 23) possible capture histories
Data example:
1 - captured at 1, not seen again
2 - captured at 1, recaptured on 3
2 - captured at 1, seen on occasion 2
1 - captured at 1, recaptured on 2 & 3
1 – captured at 2 not recaptured
0 – captured at 2, recaptured at 3
3 - captured at 3
Rest of population – not captured yi
Capture History 1 2 3

18. Usefulness
Depending upon the study design and the assumptions can be used to estimate:
Population size,
Survival,
Re-sighting rate
Other demographic rates yi
Capture History 1 2 3

19. Capture histories and multinomials
Example of the roll of the die.
The likelihood for such a data set could be constructed as:
Face
Number (yi) 1 10 2 11 3 13 4 9 5 8 6
TOTAL 60

20. Capture histories and multinomials
Each capture history is a possible outcome,
Analogous to one face of the die (ni).
Data consist of the number of times each capture history appears (yi).
Each encounter history has an associated probability (pi)
Each pi may have an underlying sub-model yi
Capture History 1 2 3

21. Probabilities and capture histories
An example:
Capture history [1 0 1] could be interpreted as:
Pr{Given capture and live release at time 1, not recaptured on occasion 2, recaptured on occasion 3}
- Or -
Pr{R1|(p1)(1-p2)(p3)},
NOTE: pi here is the probability of capture on occasion i

22. Likelihood of capture histories
Define the model (probabilities) for each capture history
Construct the likelihood for our data of m capture histories of the form:

23. Dependency in the multinomial
Important because it allows us to infer the probability of the last capture history (i.e, the probability of never captured).

24. Derived from the dependency
How many individuals were never recaptured?
How many were never captured?
Probability of never captured

25. Closed Population CMR Models
Lecture 6 – part II

26. Resources
Reading
Chapter 14 in Williams, B.K., J.D. Nichols, and M.J. Conroy Analysis and Management of Animal Populations. Academic Press. San Diego, California. or
Paul Lukacs. Chapter 15: Closed population capture-recapture models in Cooch, E., and White (eds.), Program MARK: A gentle introduction September 2007.
Need to understand closed models before we move on to Robust Design which combined open and closed models

27. Resources Other resources
Otis, D. L., K. P. Burnham, G. C. White, and D. R. Anderson Statistical inference from capture data on closed animal populations. Wildlife Monograph pp.
Pollock, K. H., J. D. Nichols, C. Brownie, and J. E. Hines Statistical inference for capture-recapture experiments. Wildlife Monographs pp.
White, G. C., D. R. Anderson, K. P. Burnham, and D. L. Otis Capture-recapture and removal methods for sampling closed populations. Los Alamos National Laboratory, Rep. LA-8787-NERP. 235pp. – Chapter 1, Chapter 8

28. What’s a closed population?
Mortality (1 - S )
Natality
Fertility (F
Emigration
Immigration

29. Lincoln-Peterson Estimator
Forms the basis for virtually all CMR methods.
Requires > two sampling occasions.
Sample of animals (M) captured, marked, and released from the population of interest (N).
2nd sample of individuals captured (C) and the number of marked individuals in C are counted (R).
If the second sample was random, the proportion of marked individuals in the second sample reflects the proportion marked in the entire population.

30. Fundamentals Lincoln-Petersen estimator for N is:
tends to be biased and overestimates population size, especially with small samples.

31. Key assumptions: Population is closed Marks are not missed or lost.
Demographically – no births or deaths
Geographically – no movement in or out
Marks are not missed or lost.
Capture probabilities are equal among individuals.

32. Violations of assumptions
Closure
Trapping mortality positively biases the estimate of N
Emigration also creates positive bias
Immigration creates negative bias
Capture probability
Can differ between occasions
Should be equal among individuals
No differences due to age, sex, size, behavior, etc.
Trap happy – negative bias
Trap shy – positive bias
Tags lost or missed – positively biases estimate of N

33. Lincoln-Peterson Estimator
Probability distribution N
population size m2
number marked animals caught in second period (R) n1
number caught and marked in first period (M) p1
probability of capture in first period n2
number caught in second period (C) p2
probability of capture during second period

34. K-sample Capture-Recapture Models
More than two sampling occasions
Assumptions
Closure – demographic and geographic
Tags are not lost or missed
Capture probabilities are appropriately modeled (i.e., model is correct)

35. Data structure Capture histories
/* Closed capture data, number of encounter occasions = 7,
groups = 2 */ ; ; ; ; ; ; ; ; ; ; ; ; ; ; ;

36. Modeling approach
Like Lincoln-Peterson models these data follow a multinomial distribution:
ni – number of observations (encounters)
pi – probability of each capture history
N – population size

37. N, pcij, pcji (i = 1…K) (j = 1…N)
Models
Eight models possible in K-Sample Recapture Models.
Only the first 3 models (M0, Mt, and Mb) and Mtb are MLE
For the other 5 models, alternative assumptions and constraints are required.
Models Mh, Mbh, and Mth, estimated under assumption that capture probabilities are random samples of N distribution of probabilities
Model Name
Sources of capture probability variation
Estimated parameters M0
Constant capture probability.
N,p Mt
Temporal variation
N, pi (i = 1…K) Mb
Behavioral response (trap-happy or trap shy)
N, pc, pr Mh
Individual heterogeneity
N, pi
(i = 1…N)
Mtb
Temporal and behavioral
N, pci, pri (i = 1…K)
Mbh
Behavioral response and individual heterogeneity
N, pci, pri (i = 1…N)
Mth
Temporal variation and individual heterogeneity
N, pij (i = 1…K) (j = 1…N)
Mtbh
Temporal variation, behavioral response and individual heterogeneity
N, pcij, pcji (i = 1…K) (j = 1…N)

38. Table 14.1 from Williams et al. 2002
Capture history
Probability models (pi) M0 Mt Mb
111 p3
p1 p2 p3
pc pr2
110
p2(1-p)
p1 p2(1-p3)
pc pr(1-pr)
101
p1(1-p2)p3
100
p(1-p)2
p1(1-p2 )(1-p3)
pc (1-pr)2
011
(1-p1)p2 p3
(1-pc)pc pr
010
(1-p1)p2(1-p3)
(1-pc)pc(1-pr)
001
(1-p1)(1-p2)p3
(1-pc)2 pc
000
(1-p)3
(1-p1)(1-p2)(1-p3)
(1-pc)3
pi represent encounter probability on occasion i

39. Constant capture probability – M0
All capture probabilities are equal
Probability distribution:
xw – number capture history w observed
n. – number of captures
MK+1 – number of unmarked animals captured
K – number of occasions

40. Temporal variation in capture probability – Mt
Extension of the Lincoln-Peterson index
Additional sampling occasions.
Number of estimated parameters (N, p1…pK)
K+1 - one more than the number of sampling occasions.
The probability distribution of this model is:
When K = 2 this is the Lincoln-Peterson estimator.

41. Behavioral trap response – Model Mb
Describes a change in capture probability after the first encounter with an animal.
Notation - probability of capture
unmarked individuals - pc
marked animals - pr
Behavioral response
positive (trap-happy)
negative (trap-shy).
Model estimates only three parameters N, pc, pr.

42. Behavioral trap response – Model Mb
The probability function for this model is:
mj - marked animals caught on occasion j
Mj - total marked animals in the population at j
MK+1 - total number of animals marked

43. Behavioral trap response – Model Mb
The probability function for this model is:
pc – probability of initial capture
MK+1 – total number of animals marked in the study

44. Behavioral trap response – Model Mb
The probability function for this model is:
(1-pc) – probability of not captured
KN - MK+1 - M. – estimated number not captured

45. Behavioral trap response – Model Mb
The probability function for this model is:
pr – probability of recapture
m. – total number of recaptures
(1-pr) – probability of not recaptured
M.-m. – number marked, not recaptured

46. Heterogeneity among individuals – Model Mh
Truly different conceptually
Capture probabilities
do not vary temporally
Do not vary based on behavior
each individual is assumed to have a unique capture probability
Conceptually
random sample of capture probabilities (p1…pN)
underlying distribution F(p).
Given a cell probability, j, - average probability that an individual is captured j times:

47. Heterogeneity among individuals – Model Mh
The model is described based on fj, the number of animals caught on j occasions:
Not MLE
not possible to use AIC or LRTs for model selection.

48. Heterogeneity among individuals – Model Mh
Nonparametric MLEs for this model have been proposed by Norris and Pollock (1995, 1996).
Estimate the number of pi
Pledger’s (2000) finite mixture models are MLEs based on a fixed (small) number of groups with unique capture probabilities.
MLE models proposed by Huggins (1989, 1991) and Alho (1990) estimate capture probabilities using the familiar logit functions of individual covariates:

49. Heterogeneity among individuals – Model Mh
Do not include N in the likelihood
Huggins
Estimates of abundance are based on the Horvitz and Thompson (1952) estimator:
Probability of capture across all occasions
Sum of the inverse of the probability of capture across all occasions and all individuals

50. Closure All models assume no change in N during the study.
Two general approaches have been proposed for testing
The null hypothesis test in program CAPTURE (Otis et al )
compares the pij (probability of capture for individual i at time j) for all individuals captured more than twice.
alternative hypothesis suggests that prior to first and subsequent to last capture some individuals had pij = 0,
suggesting that they were recruited to the population sometime after the first or before the last capture period.

51. Model selection
Balance the precision of estimates in parsimonious models
Bias of more general (informative) models.
Seeking to select the simplest model that fits our data.

52. What no AIC? 4 of 8 models are not based on likelihood
(but see Pledger 2000 and Huggins 1991).
Otis et al. (1978) and Rexstad and Burnham (1991)
Approach based on goodness of fit and between-model tests.

53. Discriminant Analysis
CAPTURE uses discriminant function analysis to compare models based on test statistics and probabilities.
Stanley and Burnham (1998) improved approach
incorporates linear and multinomial discriminant functions.
model averaging.
Mixture models of Pledger (2000)
MLEs emulating all 8 of the closed-populations models
Allows use of AIC for model selection and model averaging
Huggins models
MLEs that allow heterogeneity based on covariates
allows use of AIC for model selection and model averaging

54. Between model tests
Where MLEs can be computed - likelihood ratio tests
compare nested models (e.g., Mb and Mtb).
conditional on the more general (i.e., parameterized) model fitting the data
ask whether the less general model adequately represents the data.
CAPTURE compares:
M0 versus Mb and Mt
Mh and Mh versus Mbh.

55. Goodness of fit Just like open models:
Use multinomial distributions to calculate expected values
Compare to observed values
Similar to bootstrap GOF and RELEASE GOF
Program CAPTURE, which will run under MARK (or alone)
Computes GOF for 4 of the models
Mb, Mt, Mh, and Mtb.

56. Minimizing violations of assumptions
Closure
Keep it short - maintain closure and minimize gains and losses.
Timing is everything – e.g., avoid periods of seasonal movements or dispersal
Trap mortality bias estimates of N
If substantial, use removal models (behavioral response)
Tag loss
Some models can use individuals “known to be marked”
Individual marks – methods really don’t work with “batch marks.”

57. Precision of estimates
High capture probabilities increase precision and reduce bias.
Increasing occasions (> 5) increases precision
AIC aims to maximize precision and minimize bias

58. Precision of estimates
Behavioral response – problematic to “closure” (GOF) and precision
Pre-baiting – can minimize effects
Minimize trapping deaths – extreme behavioral response
Minimize handling time to reduce trap shyness
Heterogeneity – problematic to fit and precision
Stratify areas
Covariates – use Huggins model
Distribution of trapping effort
Trap density v. home range
Spatial arrangement

59. Lab this week Multinomials Models for closed populations
Closed models in MARK
Huggins’ model
Pledger’s mixture models