My talk at EURING 2013 on individual variability in capture-recapture models

All equal, really? Individual variability in
capture-recapture models from
biological and methodological
perspectives
Olivier Gimenez (Montpellier)
Emmanuelle Cam (Toulouse)
Jean-Michel Gaillard (Lyon)

Process in the wild
 Investigating process in natural populations
 Long-term individual monitoring datasets
 Methodological issues when moving from lab
to natural conditions

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)

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?

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

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?

 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

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

 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

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

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)

 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

Kittiwakes (Cam et al. 1998 Ecology)
B NB S
B 0.79
NB 0.65
0.90 0.10
0.67 0.33

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

 « 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

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)

Probabilities in a mixture model
 Under homogeneity
  is survival
 p is detection
    pp   1101Pr

 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

 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

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

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

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

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

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

 Zones of unequal accessibility
 Detection strongly depends on birds position
Detection heterogeneity

Detection heterogeneity (1)
zone 1: nests inside
the vegetation
La Ronze pond

Detection heterogeneity (1)
zone 1: nests inside the
vegetation
zone 2: nests on the
edge of vegetation
clusters
La Ronze pond

Results - Péron et al. (2010) Oïkos
• Absence of survival IH

00.20.40.60.81
0 10 20
Age
Survivalprobabilities
Estimation of survival senescence

00.20.40.60.81
0 10 20
Age
Survivalprobabilities
• Presence of detection and
emigration IH
Estimation of survival senescence

emigration IH
• If IH ignored on temporary
emigration, then senescence
undetected

emigration IH
• If IH ignored on temporary
emigration, then senescence
undetected
See M. Lindberg’s talk: Individual heterogeneity in black brant survival
and recruitment with implications for harvest dynamics

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)

 Explain individual variation in survival
 No variation – homogeneity
 Random effect – in-between
 Saturated – full heterogeneity
i
Individual random-effect models
 2
,~  Ni


 Random effect – in-between
i
 2
,~  Ni


 Individual random effect – in-between
i
 2
,~  Ni


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

0 2 4 6 8 10 12 14
0.4
0.5
0.6
0.7
0.8
0.9
1
300 individuals
logit(i(a)) = 1.5 - 0.05 a + ui
ui ~ N(0,=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(i(a))) = 1.5 - 0.05 aSurvival
Age

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

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

Senescence in European dippers

with IH: onset = 1.94
Marzolin et al. (2011) Ecology

without IH: onset = 2.28
with IH: onset = 1.94
Marzolin et al. (2011) Ecology

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.

 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 ?

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 (...) "

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 it’s
Emmanuelle

Introducing the threshold model
 Main issue: survival is a discrete process,
but theory well developed for continuous
distributions

 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

Liability
ind. i dies on (t,t+1)
li,t  N(µi,t ,σ2)
ind. i survives on (t,t+1)

 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   ,

mean survival
  iittii,t aebG   ,

yearly effect
mean survival
 2
,0~ tt Nb 
  iittii,t aebG   ,

yearly effect
mean survival
non-genetic effect
 2
,0~ tt Nb 
 2
,0~ ei Ne 
  iittii,t aebG   ,

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   ,

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

median = 0.110
95% cred. int. = [0.006; 0.308]
Additive genetic variance
Papaïx et al. 2010 J of Evolutionary Biol.

 Is IH significant? General question (Bolker et al. 2009
TREE)
median = 0.110
95% cred. int. = [0.006; 0.308]

 Is IH significant? General question (Bolker et al. 2009
TREE)
median = 0.110
95% cred. int. = [0.006; 0.308]
See T. Chambert’s talk: Use of posterior predictive checks for
choosing whether or not to include individual random effects in
mark-recapture models.

Short musical interlude…
(ACDC)
Wake up!

 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?

Biometrics)
 CR encounter histories are short in time,
which ensures low number of classes
 Problem solved!

Biometrics)
 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)

 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

function F on the e
 Main idea: Any distribution well approximated by a
mixture of normal distributions
 where is a discrete
mixing distribution
F x( )= N x q,s 2
( )Q dq( )ò Q dq( )

function F on the e
 Main idea: Any distribution well approximated by a
mixture of normal distributions
 where is a discrete
mixing distribution
 Dirichlet process:
F x( )= N x q,s 2
( )Q dq( )ò Q dq( )
F x( ) » ph
h=1
N
å N x qh,s 2
( )

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

Results on wolves
1 2 3 4 5
nb of clusters050150250

-0.2 0.2 0.6 1.0
0.01.5
detectability cluster 1
-0.2 0.0 0.2 0.4
024
0.00 0.10 0.20
02040
0.00 0.10 0.20
0150
Results on wolves

Wolf survival
0.6 0.7 0.8 0.9
02468
SURVIVAL
homogeneity

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

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

Tribute to…
MARK
Rémi ChoquetGary White
E-SURGE

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. Arnold’s talk: Modeling individual heterogeneity
in survival rates: mixtures or distributions?

Perspectives
1. More biology in heterogeneity
2. Fixed or dynamic heterogeneity?
Only suggestions for future research…

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. Senar’s 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

Daily detection probability for cliff swallows at two sites
when flushing was (black) and was not done (grey)

Perspectives
2. Fixed or dynamic heterogeneity?

Diversity in life histories: traits (size, age at maturity), physiology,
appearance…
Understanding diversity of life histories

• 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?

• 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?

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

• 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

My talk at EURING 2013 on individual variability in capture-recapture models

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My talk at EURING 2013 on individual variability in capture-recapture models