How to assess individual heterogeneity in demographic parameters and detectability using capture-recapture models.

1. Heterogeneity & capture-recapture

2. Accounting for individual heterogeneity in
mark-recapture models
– Standard mark-recapture models assume
parameter homogeneity
– From a statistical point of view, heterogeneity
can induce bias in parameter estimates
– From a biological point of view, heterogeneity
is of interest – individual quality

3. Accounting for individual heterogeneity in
mark-recapture models
– If the variability is observed and measured
in some way, use this information
• individual covariates
• group effects, …
– If not, use mixture/random-effect models

4. Prob of an encounter history
• Under homogeneity, the capture history ‘101’
has probability
• φ is survival
• p is detection for all individuals
( ) ( ) pp ⋅⋅−⋅= φφ 1101Pr

5. p Under heterogeneity:
n π is the probability that the individual belongs
to state L
n φL is survival for low quality individuals
n φH is survival for high quality individuals
( ) ( ) ( ) ( ) pppp HHLL
⋅⋅−⋅⋅−+⋅⋅−⋅⋅= φφπφφπ 111101Pr
Pledger et al. (2003) model for
heterogeneity

6. Allowing movements among
classes (2 classes e.g.)
p Need to rewrite Pledger model as a
hidden Markov model à la Roger
(multievent)
p Relates to dynamic heterogeneity!
p The big D matrix in Hal’s model (?)

7. Matrix models and finite mixtures.
CR Workshop 2008 7

8. Example of zones of unequal accessibility
Resightings of Black-headed Gulls Chroicocephalus ridibundus,
La Ronze pond, France

9. Example of zones of unequal accessibility
Guillaume Péron’s PhD, Roger’s work
Resightings of Black-headed Gulls Chroicocephalus ridibundus,
La Ronze pond, France
The detection strongly depends on the bird’s position

10. zone 1: nests inside the
vegetation
La Ronze pond, central France
due to high fidelity, movements
between zones should be
relatively rare
zone 2: nests on the edge of
vegetation clusters

11. Example: results
zone 1
(inside vegetation?)
Estimates:
p1= 0.089 (0.018)
π1= 0.948 (0.056)
Estimated survival : φ= 0.827 (0.018)
zone 2
(vegetation edge?)
Estimates:
p2= 0.481 (0.099)
π2= 0.052
ψ21= 0.094 (0.108)
ψ12= 0.022 (0.012)

12. Impact of ignoring heterogeneity in
detection – wolfs in French Alps
64 [29 ; 111]
33 [17 ; 54]
Time (years)
Strong bias in population size estimates
Cubaynes et al. 2010 in Cons. Biol.
Homogeneity vs.
heterogeneity in
detection
Populationsize

13. Impact of ignoring heterogeneity in
detection – wolfs in French Alps
• Marie-Caroline Prima is currently
working on modelling transitions
between heterogeneity classes (social
status)

14. • « Over time, the observed hazard rate will
approach the hazard rate of the
more robust subcohort »
Vaupel & Yashin (1985, Amer.Statistician)
• See Péron et al. (2010, Oïkos) for a case study
on Black-headed gulls
• Using simulations here
Dealing with heterogeneity in
survival – senescence

15. 0 2 4 6 8 10 12 14
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
sub-cohort 2
senescence
sub-cohort 1
constant survival
Age
Survival

16. 0 2 4 6 8 10 12 14
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
« population » - level fit
Age
Survival
sub-cohort 2
senescence
sub-cohort 1
constant survival

17. 0 2 4 6 8 10 12 14
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
« individual » - level fit
2-class survival
Age
Survival
sub-cohort 2
senescence
sub-cohort 1
constant survival
« population » - level fit

18. Con$nuous
mixture
of
individuals
p What if I have a continuous mixture of
individuals?
p Use individual random-effect models
p CR mixed models (Royle 2008 Biometrics; Gimenez &
Choquet 2010 Ecology, Sarah Cubaynes’ PhD)

19. p Explain individual variation in survival
p No variation – homogeneity
p Individual random effect – in-between (frailty)
p Saturated – full heterogeneity
iφ
( )2
,~ σµφ Ni
φ
Individual
random-­‐effect
models

20. Con$nuous
mixture
of
individuals
p What if I have a continuous mixture of
individuals?
p Use individual random-effect models (Royle
2008 Biometrics, Gimenez & Choquet 2010 Ecology)
p Mimic examples in Vaupel and Yashin (1985)
with p < 1 using simulated data

21. 0 2 4 6 8 10 12 14
0.4
0.5
0.6
0.7
0.8
0.9
1 300 individuals
logit(φi(a)) = 1.5 - 0.05 a + ui
ui ~ N(0,σ=0.5)
Survival
Age

22. 0 2 4 6 8 10 12 14
0.4
0.5
0.6
0.7
0.8
0.9
1 Expected pattern
E(logit(φi(a))) = 1.5 - 0.05 a
Age
Survival

23. 0 2 4 6 8 10 12 14
0.4
0.5
0.6
0.7
0.8
0.9
1 Fit at the population level
Age
Survival

24. 0 2 4 6 8 10 12 14
0.4
0.5
0.6
0.7
0.8
0.9
1 Fit at the individual level
with an individual random effect
Age
Survival

25. Senescence
in
European
dippers

26. with IH: onset = 1.94
Senescence
in
European
dippers
Marzolin et al. (2011) Ecology

27. without IH: onset = 2.28
with IH: onset = 1.94
Marzolin et al. (2011) Ecology
Senescence
in
European
dippers

28. Conclusions
• Ignoring heterogeneity in detection or
survival can cause bias in parameter
estimation (survival, abundance)
• Ignoring heterogeneity in detection or
survival can cause bias in biological
inference
• Heterogeneity in itself is fascinating
• Multievent models provide a flexible
framework to incorporate heterogeneity in
capture-recapture models (E-SURGE)

29. Conclusions
• Caution: big issues of parameter
redundancy and local minima
• Mixture models: choice of the number of
classes based on prior biological
assumptions – model selection using AIC
(Cubaynes et al. 2012 MEE)
• Random-effect models: significance via
LRT (halve the p-value of the standard test;
Gimenez & Choquet 2010 Ecology)

30. Current work
p Validity of normal random effect assumption?
p Parametric approach assumes a distribution
function on the random effect
p Non-parametric (Bayes) approach
p Main idea: Any distribution well approximated
by a mixture of normal distributions
p More to come…