Post on 30-Jul-2015
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
Pop
ulat
ion
size
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
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sub-cohort 2 senescence
sub-cohort 1 constant survival
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« population » - level fit
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« individual » - level fit 2-class survival
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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
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1 300 individuals logit(φi(a)) = 1.5 - 0.05 a + ui
ui ~ N(0,σ=0.5)
Survival
Age
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1 Expected pattern E(logit(φi(a))) = 1.5 - 0.05 a
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1 Fit at the population level
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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…