individual heterogeneity in capture-recapture models

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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« population » - level fit

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« 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 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…