1. All equal, really? Individual variability in
capture-recapture models from biological and methodological
perspectives Olivier Gimenez (Montpellier) Emmanuelle Cam
(Toulouse) Jean-Michel Gaillard (Lyon)
2. Process in the wild Investigating process in natural
populations Long-term individual monitoring datasets Methodological
issues when moving from lab to natural conditions
3. Process in the wild Investigating process in natural
populations Long-term individual monitoring datasets Methodological
issues when moving from lab to natural conditions Issue 1:
detectability < 1 Issue 2: individual heterogeneity (IH)
4. Issue 2: individual heterogeneity Simple capture-recapture
models assume homogeneity From a statistical point of view, IH can
cause bias in parameter estimates See also L. Cordes talk: Band
reporting rates of waterfowl: Does individual heterogeneity bias
estimated survival rates?
5. Issue of individual heterogeneity Simple CR models assume
homogeneity From a statistical point of view, IH can cause bias in
parameter estimates From a biological point of view, IH is of
interest individual quality 2010
6. Accounting for individual heterogeneity Biologists rely on
empirical measures (mass, gender, age, experience, etc.)
Statistician attempt to filter out the signal from noisy
observations? Focus shifting from mean to variance? How to account
for IH?
7. How to account for IH Case study 1: detecting trade-offs
Case study 2: describing senescence Does IH have a genetic basis?
Case study 3: quantifying heritability How to determine the amount
of IH? Case study 4: non parametric Bayesian approach Perspectives
Outline of the talk
8. Outline of the talk How to account for variation in IH Case
study 1: detecting trade-offs Case study 2: describing senescence
Does IH have a genetic basis? Case study 3: quantifying
heritability How to determine the amount of IH? Case study 4: non
parametric Bayesian approach Perspectives
9. Natural selection favors individuals that maximize their
fitness Limited energy budget: strategy of resource allocation
Trade-off between traits related to fitness IH may mask trade-offs
(Van Noordwijk & de Jong 1986 Am Nat) Assessing trade-offs in
the wild
10. IH as covariates If IH is measurable, then use it! Often,
continuous individual covariate changing over time: issue of
missing data Work by S. Bonner and R. King on how to handle with
continuous covariate
11. IH as covariates If IH is measurable, then use it! Often,
continuous individual covariate changing over time: issue of
missing data Work by S. Bonner and R. King on how to handle with
continuous covariate Use states instead of sites in multisite
models (categorical covariate)
12. Use breeders / non-breeders states (Nichols et al. 1994
Ecology) State-dependent survival Sstate : reproduction vs future
survival State-dependent transitions ij : present vs. future
reproduction Numerous applications Trade-offs and multistate
models
13. Kittiwakes (Cam et al. 1998 Ecology) B NB S B 0.79 NB 0.65
0.90 0.10 0.67 0.33
14. How to account for IH Case study 1: detecting trade-offs
Case study 2: describing senescence Does IH have a genetic basis?
Case study 3: quantifying heritability How to determine the amount
of IH? Case study 4: non parametric Bayesian approach Perspectives
Outline of the talk
15. Outline of the talk How to account for IH Case study 1:
detecting trade-offs Case study 2: describing senescence Does IH
have a genetic basis? Case study 3: quantifying heritability How to
determine the amount of IH? Case study 4: non parametric Bayesian
approach Perspectives
16. Over time, the observed hazard rate will approach the
hazard rate of the more robust subcohort Vaupel and Yashin 1985 Am
Stat Suggest that analyses conducted at the population vs.
individual level should differ (Cam et al. 2002 Am Nat) What if
detection p < 1 ? Impact of IH on age-varying survival
17. Finite mixture of individuals Use mixture models (Pledger
et al. 2003 Biometrics) Latent variable for the class to which an
individual belongs (Pradel 2009 EES) 2 classes of individuals (low
vs. high quality)
18. Probabilities in a mixture model Under homogeneity is
survival p is detection pp 1101Pr
19. Under heterogeneity is the probability that the individual
belongs to state L L is survival for low quality individuals H is
survival for high quality individuals Probabilities in a mixture
model
20. Under heterogeneity is the probability that the individual
belongs to state L L is survival for low quality individuals H is
survival for high quality individuals pppp HHLL 111101Pr
Probabilities in a mixture model
21. Finite mixture of individuals Use mixture models (Pledger
et al. 2003) A model with a hidden structure, with a latent
variable for the class to which an individual belong to (HMM;
Pradel 2009) Mimic examples in Vaupel and Yashin (1985 Am Stat)
with p < 1 using simulated data
22. 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 1 400 individuals (the most fragile) Sub-cohort 2 100
individuals (the most robust) Survival Age
23. 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 Fit at
the population level Sub-cohort 2 100 individuals (the most robust)
Sub-cohort 1 400 individuals (the most fragile) Survival Age
24. 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
Fit at the individual level using a 2-class mixture Fit at the
population level Sub-cohort 1 400 individuals (the most fragile)
Sub-cohort 2 100 individuals (the most robust) Survival Age
25. Real case study on Black-headed Gulls Not so simple in real
life Case study on (famous) Black-headed gulls (J.-D. Lebreton)
Suspicion of IH
26. Zones of unequal accessibility Detection strongly depends
on birds position Detection heterogeneity
27. Detection heterogeneity (1) zone 1: nests inside the
vegetation La Ronze pond
28. Detection heterogeneity (1) zone 1: nests inside the
vegetation zone 2: nests on the edge of vegetation clusters La
Ronze pond
29. Results - Pron et al. (2010) Okos Absence of survival
IH
30. 00.20.40.60.81 0 10 20 Age Survivalprobabilities Absence of
survival IH Estimation of survival senescence Results - Pron et al.
(2010) Okos
31. 00.20.40.60.81 0 10 20 Age Survivalprobabilities Absence of
survival IH Presence of detection and emigration IH Estimation of
survival senescence Results - Pron et al. (2010) Okos
32. Absence of survival IH Presence of detection and emigration
IH If IH ignored on temporary emigration, then senescence
undetected Results - Pron et al. (2010) Okos
33. Results - Pron et al. (2010) Okos Absence of survival IH
Presence of detection and emigration IH If IH ignored on temporary
emigration, then senescence undetected See M. Lindbergs talk:
Individual heterogeneity in black brant survival and recruitment
with implications for harvest dynamics
34. Continuous mixture of individuals What if I have a
continuous mixture of individuals? Use individual random-effect
models CR mixed models (Royle 2008 Biometrics; Gimenez &
Choquet 2010 Ecology)
35. Explain individual variation in survival No variation
homogeneity Random effect in-between Saturated full heterogeneity i
Individual random-effect models 2 ,~ Ni
36. Explain individual variation in survival No variation
homogeneity Random effect in-between Saturated full heterogeneity i
2 ,~ Ni Individual random-effect models
37. Explain individual variation in survival No variation
homogeneity Individual random effect in-between Saturated full
heterogeneity i 2 ,~ Ni Individual random-effect models
38. Continuous mixture of individuals What if I have a
continuous mixture of individuals? Use individual random-effect
models (Royle 2008 Biometrics, Gimenez & Choquet 2010 Ecology)
Mimic examples in Vaupel and Yashin (1985) with p < 1 using
simulated data
41. 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 Survival Age
42. 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 effectSurvival Age
43. Senescence in European dippers
44. with IH: onset = 1.94 Marzolin et al. (2011) Ecology
Senescence in European dippers
45. without IH: onset = 2.28 with IH: onset = 1.94 Marzolin et
al. (2011) Ecology Senescence in European dippers
46. How to account for IH Case study 1: detecting trade-offs
Case study 2: describing senescence Does IH have a genetic basis?
Case study 3: quantifying heritability How to determine the amount
of IH? Case study 4: non parametric Bayesian approach Perspectives
Outline of the talk
47. Outline of the talk How to account for IH Case study 1:
detecting trade-offs Case study 2: describing senescence Does IH
have a genetic basis? Case study 3: quantifying heritability How to
determine the amount of IH? Case study 4: non parametric Bayesian
approach Perspectives
48. Heritability in the wild Quantitative genetics: joint
analysis of a trait and genealogical relationships Increasing used
in animal and plant pops
49. Heritability in the wild Quantitative genetics: joint
analysis of a trait and genealogical relationships Increasing used
in animal and plant pops Animal models: mixed models incorporating
genetic, environmental and other factors. Heritability: proportion
of the phenotypic var. attributed to additive genetic var.
50. Heritability in the wild Quantitative genetics: joint
analysis of a trait and genealogical relationships Increasing used
in animal and plant pops Animal models: mixed models incorporating
genetic, environmental and other factors. Heritability: proportion
of the phenotypic var. attributed to additive genetic var.
Combination of animal and capture- recapture models ?
51. The idea is the air (Cam 2009 EES) " [The animal model has]
been applied to estimation of heritability in life history traits,
either in the rare study populations where detection probability is
close to 1, or without considering the probability of detecting
animals (...) "
52. The idea is the air (Cam 2009 EES) " [The animal model has]
been applied to estimation of heritability in life history traits,
either in the rare study populations where detection probability is
close to 1, or without considering the probability of detecting
animals (...) " I think its Emmanuelle
53. Introducing the threshold model Main issue: survival is a
discrete process, but theory well developed for continuous
distributions
54. Main issue: survival is a discrete process, but theory well
developed for continuous distributions Trick/idea: Survival is
related to an underlying latent variable that is continuous
Introducing the threshold model
55. Liability ind. i dies on (t,t+1) li,t N(i,t ,2) ind. i
survives on (t,t+1)
56. It can be shown that survival and mean liability are linked
For some function G, we have: Plug in the animal model iittii,t
aebG ,
57. It can be shown that survival and mean liability are linked
For some function G, we have: mean survival iittii,t aebG , Plug in
the animal model
58. It can be shown that survival and mean liability are linked
For some function G, we have: yearly effect mean survival 2 ,0~ tt
Nb iittii,t aebG , Plug in the animal model
59. It can be shown that survival and mean liability are linked
For some function G, we have: yearly effect mean survival
non-genetic effect 2 ,0~ tt Nb 2 ,0~ ei Ne iittii,t aebG , Plug in
the animal model
60. It can be shown that survival and mean liability are linked
For some function G, we have: additive genetic effect yearly effect
mean survival non-genetic effect 2 ,0~ tt Nb 2 ,0~ ei Ne AMNaa aN 2
1 ,0~,, iittii,t aebG , Plug in the animal model
61. Case study on blue tits in Corsica Blue tits Corsica 1979
2007 654 individuals, 218 fathers (sires), 215 mothers (dams), 12
generations. Mark-recapture data Social pedigree
62. median = 0.110 95% cred. int. = [0.006; 0.308] Additive
genetic variance Papax et al. 2010 J of Evolutionary Biol.
63. Is IH significant? General question (Bolker et al. 2009
TREE) median = 0.110 95% cred. int. = [0.006; 0.308] Additive
genetic variance Papax et al. 2010 J of Evolutionary Biol.
64. Is IH significant? General question (Bolker et al. 2009
TREE) median = 0.110 95% cred. int. = [0.006; 0.308] Additive
genetic variance Papax et al. 2010 J of Evolutionary Biol. See T.
Chamberts talk: Use of posterior predictive checks for choosing
whether or not to include individual random effects in
mark-recapture models.
65. How to account for IH Case study 1: detecting trade-offs
Case study 2: describing senescence Does IH have a genetic basis?
Case study 3: quantifying heritability How to determine the amount
of IH? Case study 4: non parametric Bayesian approach Perspectives
Outline of the talk
66. Short musical interlude (ACDC) Wake up!
67. Outline of the talk How to account for IH Case study 1:
detecting trade-offs Case study 2: describing senescence Does IH
have a genetic basis? Case study 3: quantifying heritability How to
determine the amount of IH? Case study 4: non parametric Bayesian
approach Perspectives
68. Fit models with 1, 2, 3, classes of mixture, and use AIC
(Pledger et al. 2003 Biometrics) This strategy does the job in
simulations (Cubaynes et al. 2012 MEE) Number of classes for finite
mixtures?
69. Fit models with 1, 2, 3, classes of mixture, and use AIC
(Pledger et al. 2003 Biometrics) This strategy does the job in
simulations (Cubaynes et al. 2012 MEE) CR encounter histories are
short in time, which ensures low number of classes Problem solved!
Number of classes for finite mixtures?
70. Number of classes for finite mixtures? Fit models with 1,
2, 3, classes of mixture, and use AIC (Pledger et al. 2003
Biometrics) This strategy does the job in simulations (Cubaynes et
al. 2012 MEE) CR encounter histories are short in time, which
ensures low number of classes Problem solved! See Arnold et al.
(2010 Biometrics) for an automatic method (RJMCMC)
71. Parametric approach assumes a distribution function F on
the e Validity of normal random effect assumption? What if
random-effect models? Non parametric Bayesian approach
72. Parametric approach assumes a distribution function F on
the e Validity of normal random effect assumption? Main idea: Any
distribution well approximated by a mixture of normal distributions
where is a discrete mixing distribution What if random-effect
models? Non parametric Bayesian approach F x( )= N x q,s 2 ( )Q dq(
) Q dq( )
73. Parametric approach assumes a distribution function F on
the e Validity of normal random effect assumption? Main idea: Any
distribution well approximated by a mixture of normal distributions
where is a discrete mixing distribution Dirichlet process: What if
random-effect models? Non parametric Bayesian approach F x( )= N x
q,s 2 ( )Q dq( ) Q dq( ) F x( ) ph h=1 N N x qh,s 2 ( )
74. Case study on wolves (95-03) Wolf is recolonizing France
Problematic interactions with human activities Heterogeneity
suspected in the detection process Wide area Social species
75. Results on wolves 1 2 3 4 5 nb of clusters050150250
77. Wolf survival 0.6 0.7 0.8 0.9 02468 SURVIVAL
homogeneity
78. 0.80 0.85 0.90 0.95 1.00 0246810 SURVIVAL Wolf survival
mixture of normals 0.6 0.7 0.8 0.9 02468 SURVIVAL homogeneity
79. How to account for IH Case study 1: detecting trade-offs
Case study 2: describing senescence Does IH have a genetic basis?
Case study 3: quantifying heritability How to determine the amount
of IH? Case study 4: non parametric Bayesian approach Perspectives
Outline of the talk
80. Outline of the talk How to account for IH Case study 1:
detecting trade-offs Case study 2: describing senescence Does IH
have a genetic basis? Case study 3: quantifying heritability How to
determine the amount of IH? Case study 4: non parametric Bayesian
approach Perspectives
81. Conclusions CR methodology is catching up with p=1 world IH
needs to be accounted for Whenever possible, adopt a biological
view and measure quality in the field If not, well, mixture or
random-effect models
82. Tribute to MARK Rmi ChoquetGary White E-SURGE
83. Conclusions CR methodology is catching up with p=1 world IH
needs to be accounted for Whenever possible, adopt a biological
view and measure quality in the field Mixture of random-effect
models Interpretation difficult / hazardous though How to choose
between the two approaches? See T. Arnolds talk: Modeling
individual heterogeneity in survival rates: mixtures or
distributions?
84. Perspectives 1. More biology in heterogeneity 2. Fixed or
dynamic heterogeneity? Only suggestions for future research
85. Perspectives 1. More biology in heterogeneity Detection is
often considered nuisance Understanding the biology of IH in
detection? Link with literature on personality See C. Senars talk:
Selection on the size of a sexual ornament may be reverse in urban
habitats: a story on variation in the black tie of the great
Tit
86. Heterogeneity in detection
87. Daily detection probability for cliff swallows at two sites
when flushing was (black) and was not done (grey)
88. Perspectives 2. Fixed or dynamic heterogeneity?
89. Diversity in life histories: traits (size, age at
maturity), physiology, appearance Understanding diversity of life
histories
90. Fixed heterogeneity: fixed differences in fitness
components among individuals determined before or at the onset of
reproductive life (Cam et al. 2002). This diversity is explained
by?
91. Fixed heterogeneity: fixed differences in fitness
components among individuals determined before or at the onset of
reproductive life (Cam et al. 2002). Dynamic heterogeneity:
diversity of state sequences due to stochasticity (Tuljapurkar et
al. 2009 Ecol. Letters) Current debate on dynamic vs fixed
heterogeneity This diversity is explained by?
92. Fixed or dynamic heterogeneity?
93. Fixed or dynamic heterogeneity? Multistate models with
individual random effects and first- order Markovian transitions
between states
94. Fixed or dynamic heterogeneity? Multistate models with
individual random effects and first- order Markovian transitions
between states Diversity better explained by models incorporating
unobserved heterogeneity than by models including first- order
Markov processes alone, or a combination of both
95. Fixed or dynamic heterogeneity? Multistate models with
individual random effects and first- order Markovian transitions
between states Diversity better explained by models incorporating
unobserved heterogeneity than by models including first- order
Markov processes alone, or a combination of both To be reproduced
on other populations / species