A Unified Capture-Recapture Model - University of …rbarker/rjb_pdf/Schofield and Barker.pdf · A...
Transcript of A Unified Capture-Recapture Model - University of …rbarker/rjb_pdf/Schofield and Barker.pdf · A...
A Unified Capture-Recapture Model
Matthew R. Schofield∗†and Richard J. Barker‡
November 22, 2006
∗The author would like to thank the Tertiary Education Commission
for the Bright Futures Ph.D. scholarship that funded this research.
†Department of Mathematics and Statistics, University of Otago, P.O.
Box 56, Dunedin, New Zealand
‡Department of Mathematics and Statistics, University of Otago, P.O.
Box 56, Dunedin, New Zealand
1
1 Summary
A hierarchical framework is developed for capture-recapture data
that separates the capture process from the demographic pro-
cesses of interest, such as birth and survival. This allows users to
parameterize in terms of meaningful demographic parameters.
The model is very flexible with many of the current capture-
recapture models shown to be special cases. The hierarchical
nature of the model allows natural expression of relationships,
both between parameters and between parameters and the re-
alization of random variables, such as population size. Previ-
ously, many of these relationships, such as density dependence
have been unable to be explored using capture-recapture data.
We fit a density dependent model to male Gonodontis bidentata
data and report evidence of negative density dependence in per-
capita birth rates and weak evidence of negative density depen-
dence in survival. Demographic analysis; Density dependence;
Hierarchical analysis; Missing data; Open population estimation
2
2 Introduction
The uses and extensions of open population capture-recapture
modeling have been many and varied since the foundational pa-
pers of Darroch (1959), Cormack (1964), Jolly (1965), and Seber
(1965). The last 20 years in particular have seen a proliferation
of mark-recapture models with more than 100 distinct models
now included in the mark-recapture software MARK (White
and Burnham 1999). The choice of model is governed by fea-
tures of the sampling process and the parameters that are of
interest. Some models are based on more informative study
designs than others, for example the robust design (Pollock
1982). Other models include covariates that allow missing and
possibly uncertain components, for example multi-state models
(Brownie et al. 1993; Schwarz et al. 1993b), multi-event mod-
els (Pradel 2005) and time-varying continuous-covariate models
(Bonner and Schwarz 2006). Further models incorporate addi-
tional information about the parameters in the model, for ex-
ample re-sighting models (Burnham 1993; Barker 1997). Other
models include reparameterizations that are more meaningful
for biological application, for example Pradel (1996) and Link
3
and Barker (2005).
Computer software packages such as MARK, or M-SURGE
(Choquet et al. 2004) for multi-state models, are very good at
allowing classical analyses based on fitting fixed-effects mod-
els using maximum likelihood. These packages also use GLiM-
type structures to allow constraints on parameters. Where these
packages are weak is in allowing users to fit hierarchical mod-
els including those that express stochastic relationships among
parameters in the model.
Mark-recapture models are naturally hierarchical in the sense
that biologists commonly model demographic parameters such
as survival rate, birth rate, population size as population vari-
ables regulated by probability distributions. Moreover, it is nat-
ural to expect relationships among parameters. For example,
the important concept of density dependence implies that pop-
ulation vital rates depend on population abundance (or density).
As noted by Armstrong et al. (2005): “There is evidence for den-
sity dependence in a wide range of species... but most studies
can be challenged on statistical grounds”. Typically methods
are used where the data is used twice; once to estimate abun-
dance and then again to use the abundance estimate in a density
4
dependent relationship. As pointed out by Seber and Schwarz
(2002): “Tools to investigate the whole issue of density depen-
dence and dependence upon the actions of other individuals are
not yet readily available [for capture-recapture data]. Models
that estimate abundance (e.g., Jolly-Seber models) are avail-
able, but the feedback loop between abundance and subsequent
parameters has not yet been complete”.
There has been relatively little development of hierarchical
models, especially those that allow flexibility in the way hier-
archical relationships are expressed. Link and Barker (2005)
proposed a modification to the likelihood of Pradel (1996) that
allowed stochastic dependence between survival and per capita
birth rates. While a step toward flexible hierarchical model-
ing, their likelihood does not allow for relationships to be ex-
pressed in terms of abundance, thus limiting its use for exploring
density-dependent relationships. The lack of a suitable choice
of likelihood has been an impediment to flexible hierarchical
modeling (Barker and White 2004). Parameterizations that are
convenient for likelihood-based estimation are not necessarily
the best to adopt for exploring biological relationships. Also,
some constraints of interest, such as density-dependence, can-
5
not be written in terms of deterministic functions of parameters
that are explicitly expressed in the likelihood.
Mark-recapture models can be naturally thought of as mod-
els for missing data subject to informative censoring. In open
population studies, the time that an animal first entered the
population is often of interest but all that is known is that it
entered the population sometime before the first capture. Sim-
ilarly, the time of death of an individual may be of interest but
all we know is that if it died during the study, it was sometime
after the last time it was caught. The data are also usually
interval-censored as in most designs the population is sampled
at discrete times.
The standard approach to mark-recapture modeling is to
find an observed data likelihood (ODL) expressed in terms of
parameters of interest after first writing a complete data likeli-
hood (CDL). Unobserved terms are then either integrated out
of the model or left in as parameters to be estimated. This step
is needed so that a likelihood function is obtained for parameter
estimation. However, Bayesian inference methods, in particular
McMC allow easy imputing of the unknowns to give the CDL.
Working with the CDL allows modeling to be focused on ob-
6
taining meaningful biological models instead of concentrating
on the intricacies of capture-recapture study design. An addi-
tional advantage of models based on direct use of the CDL is
that it increases the range of parameter constraints that may be
considered, including stochastic constraints.
In this paper we describe a missing data model that exploits
Bayesian multiple imputation to allow any demographic param-
eter to be explicitly incorporated in the model. Our model is
based on an individual-specific factorization that allows rela-
tionships among parameters and also between parameters and
the outcome of random variables, such as population size. A
further advantage of our approach is that it provides a single
unified modeling framework that includes virtually all of the
standard models as special cases.
Our factorization is similar to that of Dupuis (1995), who
imputed the missing data values for a multi-state model con-
ditional on first capture, where death was considered a state.
However, we have separated death from the multi-state model
and extended the model to include birth.
7
3 Model
The observed data from the t-sample capture-recapture study
is the matrix Xobs, where Xobsij = 1 if individual i was caught
in sample j and Xobsij = 0 otherwise. We define the number
of individual ever available for capture as N , like Crosbie and
Manly (1985) and Schwarz and Arnason (1996). Given N , the
capture histories for the unseen individuals, denoted Xmis are
known. This gives the complete matrix of capture-recapture
histories for all individuals in N , denoted X.
To model the demographic changes in the population due to
birth and death we require the interval censored times of birth
and death for each individual. These are expressed through
partially observed birth and death matrices, b (for births) and
d (for deaths). The value bij = 1 means that individual i was
born between sample j and j + 1, with bij = 0 otherwise (note
that bi0 = 1 means that i was born before the study started).
The value dij = 1 means that individual i died between sample
j and j + 1, with dij = 0 otherwise (note that dit = 1 means
that i was still alive at the end of the study). Individuals must
be born before they can die and can only be born and die once,
8
leading to the constraints,
k∑
j=0
bij−
k∑
j=1
dij ≥ 0,∀i, k,∑
j
bij = 1,∀i,∑
j
dij = 1,∀i.
As we are uncertain whether values of Xij = 0 prior to first
capture and after last capture are because individual i was not
able to be caught in sample j, or because i was not alive at
time of sample j the b and d matrices comprise observed values,
denoted bobs or dobs and missing values, denoted bmis or dmis.
We assume no error in the capture histories, so all values of b
after first capture are observed as bij = 0 and all values of d
before the final capture are observed as dij = 0. Consider the
capture history 0110 for a t = 4 period study. As the individual
had to be born before sample 2, the birth matrix b comprises
an observed component, bobs = (bi2 = 0, bi3 = 0) and a missing
component bmis = (bi0, bi1). As the individual could not have
died before sample 3, the death matrix d comprises dobs = (di1 =
0, di2 = 0) and dmis = (di3, di4). The complete b and d matrices
allow us to model the demographic processes of interest directly.
The missing data mechanisms for b and d are modeled through
X.
The complete b and d matrices allow us to obtain demo-
9
graphic summaries of interest, such as the number of individuals
in the population at time of sample j, denoted Nj,
Nj =N
∑
i=1
(
j−1∑
k=0
bik −
j−1∑
k=1
dik
)
.
Note the difference between N and Nj. The parameter N is the
total number of individuals ever available for capture during the
study. It is a nuisance parameter used to specify the model fully
and to include realizations of random variables, such as Nj in
the model.
Other demographic summaries, such as the number of births
between sample j and j + 1, denoted Bj, and the number of
deaths between sample j and j + 1, denoted Dj, can also be
found directly from the b and d matrices.
We introduce the notation bj, dj and Xj to denote the jth
column of the matrices b, d and X respectively, and the notation
b0:j and d1:j to denote the columns 0 through j for the b matrix
and columns 1 through j in the d matrix respectively.
3.1 Modeling the Capture Process
For a multiple recapture study, the complete capture matrix
is assumed to be the outcome from a series of independent
10
Bernoulli trials. While alive, each individual is assumed to be
caught in sample j with probability pj,
[X|b, d, s, p,N, z] ∝N !
u.!(N − u.)!
N∏
i=1
ti2∏
j=ti1
pXij
j (1 − pj)(1−Xij) .
where u. =∑
j uj is the total number of observed individuals,
ti1 is the sample individual i was first available for capture and
ti2 is the last sample that individual i was available for capture.
For example, if the individual was born between sample j and
j + 1 and died between sample k and k + 1 then ti1 = j + 1 and
ti2 = k. The notation [y|φ] is used to denote the probability
distribution or probability density function of y conditional on
φ.
3.2 Modeling the Deaths
Conditional on individual i being alive at time of sample j, death
between samples j and j + 1 is assumed to be the outcome of a
Bernoulli trial. The probability of death is 1 − Sj, where Sj is
defined as the survival probability from period j to j + 1. The
full conditional distribution used in the model is,
[dij|b0:j, d1:j−1, Sj, N ] ∝ S(1−dij)j (1 − Sj)
dij .
11
Note that we assume that an individual cannot die in the
same period that it was born. Incorporating different types of
data or assumptions can relax this assumption, for example,
Crosbie and Manly (1985); Schwarz et al. (1993a); Schwarz and
Arnason (1996) introduce assumptions to allow individuals to
die before they are available for capture.
3.3 Modeling the Births
Many of the current birth parameterizations are “hybrid” in na-
ture, combining aspects of the study with the birth process. For
example, the Jolly-Seber model parameterizes in terms of {Uj},
the total number of unmarked individuals in the population in
sample j, which reflects the intensity with which sampling is
carried out as well as the birth process. The models of Crosbie
and Manly (1985) and Schwarz and Arnason (1996) parameter-
ize birth in terms of {βj}, the probability of being born between
sample j and j + 1 conditional on ever being available for cap-
ture. The βj parameters reflect aspects of the study design as
well as the birth process. Consider a t = 3 study with N = 75
and parameters β0 = 1/3, β1 = 1/3 and β2 = 1/3. Extending
12
the study to a t = 4 periods with N = 100 gives different pa-
rameters β0 = 1/4, β1 = 1/4 and β2 = 1/4. A more natural
parameterization is in terms of per capita birth rates,
ηj = E[Bj|N ]/Nj.
This approach is adopted by Pradel (1996) and Link and Barker
(2005). However, as with the Jolly-Seber model and the formu-
lations of Burnham (1991) and Schwarz and Arnason (1996)
quantities such as Bj and Nj are not explicitly included in the
model. To overcome this, Pradel (1996) and Link and Barker
(2005) replace Nj with E(Nj|N). However, as Nj is explicit in
our model, we can use this to obtain a per-capita birth rate in
terms of Nj.
We assume that the observed birth matrix is the outcome of
a series of individual multinomial trials. Writing this in terms
of a series of binomial trials gives
[bij|b0:j−1, d1:j−1, β,N ] ∝ β′
j
bij(1−β′
j)(1−
∑jk=0
bik), j = 0, . . . , t−2
where β′
j = βj/∏j−1
k=0(1 − β′
k), β′
0 = β0 and βj is the multino-
mial probability of birth used by Crosbie and Manly (1985) and
Schwarz and Arnason (1996). There is no combinatoric term
13
because the arbitrary ordering of the data has already been ac-
counted for in the modeling of the capture histories.
We propose re-parameterizing and modeling in terms of the
parameter β0 and the per-capita birth rate
ηj = E(Bj|N)/Nj, j = 1, . . . , t − 2,
using the transformation ηj = βjN/Nj, where Nj = f(b0:j−1, d1:j−1).
Note that the parameter ηj−1 (or βj−1) is obtained through the
constraint∑
k βk = 1.
3.4 Posterior
For Bayesian inference we require the posterior distribution,
[p, S, η, bmis, dmis, N |Xobs].
This is proportional to the complete data likelihood
[X, b, d|p, S, η,N ],
which can be factored using the rules of conditional probability
to obtain the series of conditional distributions shown in sections
3.1, 3.2 and 3.3. The model can also be represented as a directed
acyclic graph (figure 1).
14
For the model where we assume probability of capture, sur-
vival and per-capita birth rates are all period specific fixed ef-
fects, denoted p(t)S(t)η(t), we are able to choose prior distri-
butions and re-parameterize so that we obtain full conditional
distributions of known form for all parameters except N , for
details see appendix A.
4 Covariates
Fully observed covariates, z, with associated parameters, θz,
that provide information about parameter(s) of interest can be
included in the usual way. To include partially observed covari-
ates, we need to model the covariate z and impute any missing
values in z every iteration, that is, the missing data gets treated
as another unknown updated with the parameters. Examples
of partially observed covariates that can be included in this way
are given below.
15
4.1 Categorical Individual-Specific Time-Varying
Covariates
Categorical individual-specific time-varying covariates are com-
monly collected in capture-recapture studies. For example, one
could assume that the breeding status of an individual affects
its survival probability, however, these covariates can only be
known when the individual is observed and are usually missing
when the individual is not observed. Such data motivated the
multi-state model (Schwarz et al. 1993b) which assumes that the
“state” occupied in sample j only depends on the state occupied
in sample j − 1, that is,
[zij = k|zij−1 = h] = ψhk, j = 1, . . . , t
with the constraint that∑
k ψhk = 1,∀k. We also model the
initial allocation to “state” as
[zij = h|bij−1 = 1] = πjh, j = 1, . . . , t
with the constraint∑
h πjh = 1, where h denotes the “state”.
This model can be extended to allow the “state” occupied
in sample j to depend on the “states” occupied in both sample
j − 1 and j − 2 (Brownie et al. 1993).
16
4.1.1 Multi-Event
A useful recent development is the multi-event framework that
allows for categorical covariates to be uncertain as well as par-
tially observed (Pradel 2005). The framework has a “state”
covariate of interest that we denote z1 and an “event” covariate
z2 that provides information about z1. This can be included
into our framework by having z1 modeled in terms of z2 and θz2.
The covariate z1 together with the parameters θz1then provide
information about the parameter(s) of interest.
4.1.2 Movement
A commonly used categorical covariate is availability for cap-
ture, where individual i in sample j is either available for cap-
ture (zij = 1) or unavailable for capture (zij = 2). This is an
example where one value of the covariate is never observed be-
cause no individual can be caught while unavailable for capture.
In the first sample after birth, we model the value of the co-
variate for individual i as the outcome of a Bernoulli trial with
probability πj = [zij = 1|bij−1 = 1]. For the complementary
allocation zij = 2, the probability is 1 − πj. Three common
17
assumptions about subsequent movement are first order Marko-
vian emigration, random emigration and permanent emigration
(Barker 1997). First order Markovian emigration is when move-
ment between the time of sample j and j + 1 depends only on
the covariate for individual i at time of sample j. The transition
matrix Ψj for Markovian emigration is,
Ψj =
Fj 1 − Fj
F ′
j 1 − F ′
j
,
where
Fj = probability that individual with zij = 1 has zij+1 = 1.
F ′
j = probability that individual with zij = 2 has zij+1 = 1.
Under random emigration the movement probability does
not depend on the previous value of the covariate, that is, F ′
j =
Fj. Under permanent emigration, once an individual becomes
unavailable for capture, it can never be available again, that is
F ′
j = 0.
In the presence of movement, we can model as if there were
no movement under two assumptions: (i) There is permanent
emigration with the times of birth and immigration combined to
give additions to the population and times of death and emigra-
tion combined to give deletions to the population. This results
18
in survival probabilities becoming deletion rates and birth rates
becoming addition rates. (ii) There is random emigration where
the initial allocation rate are the same as subsequent movement
rates, that is, πj = Fj. This results in the probability of cap-
ture becoming joint probabilities of capture and availability for
capture. For more information on movement assumptions under
permanent, Markovian and random emigration see appendix B.
4.2 Continuous Individual-Specific Time-Varying
Covariates
Continuous individual-specific time-varying covariates, for ex-
ample, individual length or weight can be included in the same
way as the partially observed categorical covariate, except the
model for z is continuous (Bonner and Schwarz 2006; Schofield
and Barker 2006). Note that survival and probability of capture
become individual specific due to the effect of the continuous co-
variate.
19
5 Density Dependence
An important feature of the hierarchical framework is the abil-
ity to model relationships between parameters. For example,
one could believe that parameters are drawn from a common
distribution, that is, a random effect. Specifying multivariate
distributions allow parameters to be related to each other, as
in Link and Barker (2005) where survival and per-capita birth
rates are correlated. An important feature of our model is that
parameters can also depend not only on other parameters, but
on the realization of the random variables b and d prior to the
current period. For example, the survival and birth rates for
the next period could be related to the current population size,
that is, density dependence.
5.1 Example: Gonodontis bidentata with Den-
sity Dependence
The data used are of male Gonodontis bidentata, a dataset pre-
viously used by Bishop et al. (1978), Crosbie (1979), Crosbie
and Manly (1985) and Link and Barker (2005). The data is
available from Bishop et al. (1978) and consists of u. = 689
20
unique individuals tagged over 17 periods.
To incorporate density dependence, we assume that
logit(Sj) ∼ N(γ0 + γ1Nj, τS), j = 1, . . . , t − 1
log(ηj) ∼ N(α0 + α1Nj, τη), j = 1, . . . , t − 2
where
Nj ≡ log(Nj) − 5.5
which is centered to reduce the sampling correlation between
parameters. We assume that that the probability of capture is
sample dependent and that either (i) there is no movement, or
(ii) there is permanent emigration, or (iii) there is random em-
igration. All three assumptions require no movement covariate
in the model and have the same algebraic structure. However,
each assumption gives a different interpretation of parameters,
see appendix B.
We use a Gibbs sampler to update all of the unknowns in
the model. We are able to sample all parameters from their full
conditional distributions directly except Sj, β0, ηj and N , which
we do with the Metropolis-Hasting algorithm or extensions of
it, see appendix C.
After an adaptive phase of 20,000 iterations and a burn in of
21
100,000 iterations a posterior sample of size 400,000 was drawn.
We used flat Beta(1, 1) priors for pj, vague N(0, 0.0001) priors
for α0, α1, γ0 and γ1, vague G(0.001, 0.001) priors for τη and τS
and a flat discrete uniform prior for N with a lower bound of u.
and an upper bound of 200, 000. To confirm that the model had
mixed suitably, multiple chains were fitted with over-dispersed
starting values and checked with the Gelman-Rubin converge
diagnostic (Gelman and Rubin 1992). The posteriors of par-
ticular interest are those on α1 and γ1, the density dependent
parameters (figure 2). The 95% central credible interval for α1
is (−1.49,−0.11) which excludes 0 suggesting that per-capita
birth rates are negatively associated with population size. The
95% central credible interval for γ1 is (−1.87, 0.37) which in-
cludes 0, with approximately 82% of the posterior mass below
0. This suggests a negative relationship between survival and
population size, but is far from convincing. These results in-
dicate that the population size is stable, at least in regards to
birth; when the population size becomes large or small, birth
rates adjust so that the population returns to somewhere near
equilibrium.
The analysis of Link and Barker (2005) suggested a positive
22
correlation between the logit of survival and the log of the per-
capita birth rates. Our evidence of density dependence raises
the possibility that the correlation identified by Link and Barker
(2005) was one induced by survival and birth rates both being
negatively density dependent.
6 Discussion
The hierarchical framework we describe offers a unified approach
to modeling capture-recapture data. It allows the investigation
of biologically interesting relationships among parameters, as
well as between parameters and external covariates. The re-
liance of most studies on classical inference, particularly maxi-
mum likelihood estimation, has meant that the machinery avail-
able to allow this sort of analysis has been limited. The devel-
opment of posterior simulation methods for Bayesian inference,
and McMC in particular, has been an important advance in this
regard.
We have shown that it is conceptually easy to incorporate
into our framework standard capture-recapture models as well
as models that incorporate partially observed covariates. How-
23
ever, many different models are easily included in this frame-
work, including models with different study designs and addi-
tional data.
An example of a model with a different design is the robust
design (Pollock 1982), that consists of a series of closed popula-
tion samples within each open population sample. This allows
the probability of capture and sample size to be primarily es-
timated from the closed population samples, with survival and
birth parameters estimated from the open population samples.
To fit the robust design, Sj must be constrained to 1 and ηj
(or βj) must be constrained to 0 between the closed population
samples. Assumptions about the probability of capture within
the closed population periods are expressed through the capture
model specified for the basic case in section 3.1.
Other study designs can introduce errors in the capture his-
tory, for example, genetic tagging using non-invasive methods
(Lukacs and Burnham 2005). The observed capture matrix, de-
noted X2, together with corruption parameters θX provide in-
formation about the true capture matrix X1. At each iteration
values for X1 are imputed and used in the sampling model.
Models that have additional data can also be incorporated.
24
For example, if a census is undertaken that counts the total
population size at time of sample j, denoted Cj, we can use this
to get a better estimate of population size Nj. A reasonable
model could be that the counts are distributed as a negative
binomial with mean Nj and over-dispersion parameter θC , Cj ∼
NB(Nj, θC), j = 1, . . . , t.
Another source of data could be re-sightings of the tags out-
side the study periods (Burnham 1993; Barker 1997). This pro-
vides additional information on survival and probability of cap-
ture, see Schofield and Barker (2006) for a description of fitting
this model in the framework.
Even though including these models is conceptually easy,
the computational implementation of these models is far more
difficult. An attractive feature of the hierarchical framework is
that the models can be thought of as products of conditional
distributions. For example, the p(t)S(t)η(t) model was sepa-
rated into the conditional distributions for capture, birth and
death as shown in sections 3.1, 3.2 and 3.3. More complex mod-
els, for example, density dependence, multi-event or a model
incorporating census data have the p(t)S(t)η(t) core with ad-
ditional distributions that account for the extension, whether
25
covariates or additional data sources, etc. This allows condi-
tional distributions to be specified and pieced together to form
a user-defined model, an important step toward the concept of
the “mother-of-all-models” (Barker and White 2004). However,
many of these conditional distributions are complex and allow-
ing full flexibility in model specification comes as a trade-off
against computational speed. With the ever increasing advances
in computational power and the prospect of biologists being able
to fit models with meaningful parameters and relationships of
interest, we believe that this area is one of great promise.
The use of density dependent modeling illustrates the po-
tential of our analysis and also reminds us of the need to collect
high quality data to achieve reasonable mixing. The quality of
data can be improved either by more intense sampling on each
occasion, or by increasing the number of sampling occasions, or
by using a better design, such as the robust design of Pollock
(1982). Data collected during periods when the population is
closed provide high quality information on pj and Nj, and data
collected between times when the population is open provide in-
formation on the survival and birth parameters, Sj and ηj. The
two components of the density dependence relationship, for ex-
26
ample ηj and Nj, are informed primarily by different subsets of
data. In contrast, with simple multiple recapture studies, all
parameters are estimated from the same information.
Link and Barker (2005) showed that including N is unnec-
essary for the estimation of identifiable parameters that appear
explicitly in the likelihood. However, we include the parame-
ter N in the model so that we can use demographic summaries
such as Nj in the presence of individual specific parameters, even
though there is very little, if any information about N in the
data. This extends Huggins (1989) to open population models,
giving posterior predictions of Nj when there are individual-
specific covariates affecting the parameters. It should be noted
however, that it is possible to include demographic summaries
such as Nj in the model when there are no individual specific
parameters.
References
Armstrong, D. P., Davidson, R. S., Perrott, J. K., Royguard,
J., and Buchanan, L. (2005), “Density-dependent population
growth in a reintroduced population of North Island saddle-
27
backs,” Journal of Animal Ecology, 74, 160 – 170.
Barker, R. J. (1997), “Joint Modeling of Live-Recapture, Tag-
Resight, and Tag-Recovery Data,” Biometrics, 53, 666 – 677.
Barker, R. J., Burnham, K. P., and White, G. C. (2004),
“Encounter history modeling of joint mark-recapture, tag-
resighting and tag-recovery data under temporary emigra-
tion.” Statistica Sinica, 14, 1037 – 1055.
Barker, R. J. and White, G. C. (2004), “Towards the mother-
of-all-models: customised construction of the mark-recapture
likelihood function,” Animal Biodiversity and Conservation,
27, 177 – 185.
Bishop, J. A., Cook, L. M., and Muggleton, J. (1978), “The re-
sponse of two species of moths to industrialization in north-
west England. II Relative fitness of morphs and population
size.” Philosophical Transaction of the Royal Society of Lon-
don, 281, 517 – 540.
Bonner, S. J. and Schwarz, C. J. (2006), “An Extension of the
Cormack-Jolly-Seber Model for Continuous Covariates with
28
Application to Microtus pennsylvanicus,” Biometrics, 62, 142
– 149.
Brownie, C., Hines, J. E., Nichols, J. D., Pollock, K. H., and
Hestbeck, J. B. (1993), “Capture-recapture studies for multi-
ple strata including non-Markovian transition probabilities,”
Biometrics, 49, 1173 – 1187.
Burnham, K. P. (1991), “On a unified theory for release-
resampling of animal populations.” in Proceedings 1990 Taipei
Simposium in Statistics., eds. Chao, M. T. and Cheng, P. E.,
Institute of Statistical Science, Academia Sinica, pp. 11 – 35.
— (1993), “A theory for combined analysis of ring recovery and
recapture data,” in Marked Individuals in Bird Population
Studies., eds. Lebreton, J.-D. and North, P., Birkhauser Ver-
lag, Basel., pp. 199 – 213.
Choquet, R., Reboulet, A. M., Pradel, R., Gimenez, O., and
Lebreton, J. D. (2004), “M-SURGE: new software specifically
designed for multistate capture-recapture models,” Animal
Biodiversity and Conservation, 27, 1 – 9.
29
Cormack, R. M. (1964), “Estimates of Survival from the Sight-
ing of Marked Animals,” Biometrika, 51, 429 – 438.
Crosbie, S. F. (1979), “The mathematical modelling of capture-
mark-recapture experiments on animal populations.” Ph.D.
thesis, University of Otago, Dunedin, New Zealand.
Crosbie, S. F. and Manly, B. F. J. (1985), “Parsimonious Mod-
elling of Capture-Mark-Recapture Studies,” Biometrics, 41,
385 – 398.
Darroch, J. N. (1959), “The Multiple-Recapture Census: II. Es-
timation When There is Immigration or Death,” Biometrika,
46, 336 – 351.
Dupuis, J. A. (1995), “Bayesian estimation of movement
and survival probabilities from capture-recapture data,”
Biometrika, 82, 761 – 772.
Gelman, A. and Rubin, D. (1992), “Inference from Iterative
Simulation using Multiple Sequences,” Statistical Science, 7,
457 – 511.
Green, P. (1995), “Reversible jump Markov chain Monte
30
Carlo computation and Bayesian model determination,”
Biometrika, 82, 711 – 732.
Huggins, R. (1989), “On the Statistical Analysis of Capture
Experiments,” Biometrika, 76, 133 – 140.
Jolly, G. M. (1965), “Explicit Estimates from Capture-
Recapture Data with Both Death and Immigration-Stochastic
Model,” Biometrika, 52, 225 – 247.
Link, W. A. and Barker, R. J. (2005), “Modeling Association
among Demographic Parameters in Analysis of Open Popu-
lation Capture-Recapture Data,” Biometrics, 61, 46 – 54.
Lukacs, P. M. and Burnham, K. P. (2005), “Review of capture-
recapture methods applicable to noninvasive genetic sam-
pling,” Molecular Ecology, 14, 3909 – 3919.
Pollock, K. H. (1982), “A Capture-Recapture Design Robust to
Unequal Probability of Capture,” Journal of Wildlife Man-
agement, 46, 752 – 757.
Pradel, R. (1996), “Utilization of Capture-Mark-Recapture for
the Study of Recruitment and Population Growth Rate,” Bio-
metrics, 52, 703 – 709.
31
— (2005), “Multievent: An Extension of Multistate Capture-
Recapture Models to Uncertain States,” Biometrics, 61, 442
– 447.
Schofield, M. R. and Barker, R. J. (2006), “A Joint Recap-
ture, Recovery and Resighting Model with Individual Covari-
ates: growth of Rainbow Trout in the Rangitikei River, New
Zealand,” In Preparation.
Schwarz, C. J. and Arnason, A. N. (1996), “A General Method-
ology for the Analysis of Capture-Recapture Experiments in
Open Populations,” Biometrics, 52, 860 – 873.
Schwarz, C. J., Bailey, R. E., Irvine, J. R., and Dalziel, F. C.
(1993a), “Estimating Salmon Spawning Escapement Using
Capture-Recapture Methods,” Canadian Journal of Fisheries
and Aquatic Science, 50, 1181 – 1197.
Schwarz, C. J., Schweigert, J. F., and Arnason, A. N. (1993b),
“Estimating migration rates using tag-recovery data,” Bio-
metrics, 49, 177 – 193.
Seber, G. A. F. (1965), “A Note on the Multiple-Recapture
Census,” Biometrika, 52, 249 – 259.
32
Seber, G. A. F. and Schwarz, C. J. (2002), “Capture-recapture:
before and after EURING 2000,” Journal of Applied Statis-
tics, 29, 5 – 18.
White, G. C. and Burnham, K. P. (1999), “Program MARK:
survival estimation from populations of marked animals.”
Bird Study, 46S, 120 139.
33
7 Figures
Figure 1: Directed acyclic graph of the jth period for the
p(t)S(t)η(t) model. The bold arrows represent deterministic
links and the plain arrows represent stochastic links.
34
−3 −2 −1 0 1 20
0.2
0.4
0.6
0.8
1
γ1
−3 −2 −1 0 1 20
0.5
1
1.5
α1
Figure 2: Posterior density estimates for the density depen-
dent effects on survival (γ1) and per-capita birth rates (α1) for
Gonodontis bidentata.
35
A Appendix: Computation of p(t)S(t)η(t)
Model
The set of parameters {pj} have Beta(αp, γp) priors that yield
Beta(nj +αp, Nj −nj + γp) full conditional distributions, where
nj is the total number of individuals caught at time of sample
j. The set of parameters {Sj} have Beta(αS, γS) priors that
yield Beta(Nj −Dj +αS, Dj +γS) full conditional distributions.
To obtain updates from β0 and {ηj}t−2j=1, we re-parameterize in
terms of β′
j, where
β′
j = βj/
j−1∏
k=0
(1 − β′
k), j = 1, . . . , t − 2
and β′
0 = β0. The set of parameters {β′
j} have Beta(αβ′ , γβ′) pri-
ors that yield Beta(Bj +αβ′ , N−∑j
k=0 Bk +γβ′) full conditional
distributions which are then transformed to {ηj} by taking
β′
j
j−1∏
k=0
(1 − β′
k)N/Nj.
The times of birth/death are obtained by calculating the full
conditional probability of each plausible period of birth/death
for each individual at every iteration. A period of birth/death is
then sampled using these probabilities. The parameter N has a
discrete uniform distribution and is updated using a reversible
36
jump algorithm (Green 1995). A new candidate value for N
is proposed along with associated values of b, d and Xmis and
the group are accepted or rejected together. The chain is run
for a fixed number of iterations to obtain an optimal jumping
distribution.
B Appendix: Discussion on Movement
Assumptions
B.1 Permanent Emigration
Permanent emigration is assumed in most models that include
first captures, (Jolly 1965; Seber 1965; Schwarz and Arnason
1996; Link and Barker 2005). However, when averaging across
the various combinations for b, d and z for each capture his-
tory, the parameters Sj and Fj are confounded and there is not
enough information to separately estimate birth, movement and
survival parameters prior to the first capture.
The standard approach is to consider additions and deletions
instead of births and deaths. Additions combine individuals be-
ing born available for capture with immigrants becoming avail-
37
able for capture. Deletions combine deaths of those available
for capture with emigrants leaving to become unavailable for
capture. The random variables b and d are used to express time
of addition and deletion, instead of birth of death respectively.
This means the covariate z is no longer required because every
individual is available for capture from the sample of addition
until the sample of deletion, when it either leaves or dies before
the next sample. The meaning of the parameters changes so that
ηj becomes the per capita addition rate and 1−Sj becomes the
deletion rate.
B.2 First order Markovian Emigration
Without strong assumptions, first order Markovian emigration
is not identifiable unless more complex study designs are used,
such as the robust design, or models incorporating different
types of re-encounter data. Even with these designs, additional
constraints about the time-specific covariate parameters, Fj and
F ′
j are required. A common constraint is that movement param-
eters are fixed through time, Fj = F and F ′
j = F ′ (Barker et al.
2004). A further problem is that there is not enough informa-
38
tion to separate the per capita birth rates, ηj, and the allocation
probabilities πj. One possible solution is to gather additional
information that can be used to separate ηj and πj. Another is
to assume that all individuals are born unavailable for capture,
that is, specify a distribution for πj with all mass on 0. This as-
sumption can be relaxed by having some mass on πj > 0, allow-
ing some individuals to be born available for capture. Another
potential solution combines individuals being born available for
capture with immigrants becoming available for capture for the
first time, with b parameterizing addition. The advantage of this
is that assumptions are no longer required for the initial alloca-
tions, πj, however, addition rates are being estimated instead of
birth rates, which are of biological interest.
B.3 Random Emigration
Under random emigration Fj is confounded with pj+1. The stan-
dard approach is to consider the identifiable parameter p′j+1 =
Fjpj+1, the joint probability of being available for capture and
caught in sample j + 1. Including the first captures means that
πj is also confounded with pj+1. As with Markovian emigra-
39
tion, one possible solution is to specify a distribution for πj with
all/most mass on 0. Another possible solution is to assume that
initial allocations are the same as subsequent movement proba-
bilities, that is, πj = Fj, (Barker 1997). Under this assumption
the algebraic structure for the model is identical to that of per-
manent emigration with additions and deletions.
C Appendix: Computation of Den-
sity Dependent Model
All parameters updated using Metropolis-Hastings or reversible
jump McMC have an adaptive phase of 20,000 iterations to find
an optimal jumping distribution. We update Sj and β0 using
a Metropolis-Hastings algorithm with a flat beta distribution
for β0 (note that Sj has a hierarchical “prior” distribution). To
update ηj, we change variable for computational reasons and
update βj and use a Metropolis-Hastings algorithm with the
hierarchical “prior” distribution. The parameter N is updated
using a reversible jump algorithm in the same way as in section
A. The parameters {pj} have a Beta(αp, γp) prior that yields a
40
Beta(nj + αp, Nj − nj + γp) full conditional distribution. The
parameter α0 has a N(0, τα0) prior distribution that yields a
N(τη
∑t−2k=1(log(ηk) − α1Nk)/τ, τ) posterior distribution, where
τ = τη(t−2)+τα0. The parameter α1 has a N(0, τα1
) prior distri-
bution that yields a N(τη
∑t−2k=1((N)k(log(ηk)−α0))/τ, τ) poste-
rior distribution, where τ = τη
∑t−2k=1 N2
k +τα1. The parameter γ0
has a N(0, τγ0) prior distribution that yields a N(τS
∑t−1k=1(logit(Sk)−
γ1Nk)/τ, τ) posterior distribution, where τ = τS(t − 1) + τγ0.
The parameter γ1 has a N(0, τγ1) prior distribution that yields
a N(τS
∑t−1k=1((N)k(logit(Sk) − γ0))/τ, τ) posterior distribution,
where τ = τS
∑t−1k=1 N2
k +τγ1. The parameter τS has a G(aτS
, bτS)
prior distribution that yields a G(aτS+(t−1)/2, bτS
+∑t−1
k=1(logit(Sk)−
(γ0 +γ1Nk))2/2) posterior distribution. The parameter τη has a
G(aτη, bτη
) prior distribution that yields a G(aτη+(t−2)/2, bτη
+
∑t−2k=1(log(ηk) − (α0 + α1Nk))
2/2) posterior distribution. The
times of birth/death are obtained by calculating the full con-
ditional probability of each plausible period of birth/death for
each individual at every iteration. A period of birth/death is
then sampled using these probabilities.
41