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Heterogeneity & capture-recapture
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
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
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
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
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 (?)
Matrix models and finite mixtures.
CR Workshop 2008 7
Example of zones of unequal accessibility
Resightings of Black-headed Gulls Chroicocephalus ridibundus,
La Ronze pond, France
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
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
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)
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
Impact of ignoring heterogeneity in
detection – wolfs in French Alps
•  Marie-Caroline Prima is currently
working on modelling transitions
between heterogeneity classes (social
status)
•  « 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
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
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
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
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)
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	
  	
  
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
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
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
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
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
Senescence	
  in	
  European	
  dippers	
  
with IH: onset = 1.94
Senescence	
  in	
  European	
  dippers	
  
Marzolin et al. (2011) Ecology
without IH: onset = 2.28
with IH: onset = 1.94
Marzolin et al. (2011) Ecology
Senescence	
  in	
  European	
  dippers	
  
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)
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)
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…

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Individual Heterogeneity in Capture-Recapture Models

  • 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
  • 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…