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
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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