Extending the Model

An extension to the Jolly-Seber model developed by Crosbie and Manly (1985) and Schwarz and Arnason (1996) was presented in Section 2.4 in which the birth process was explicitly incorporated into the likelihood formulation. A similar extension is investi- gated by Link and Barker (2005) who develop a hierarchical extension to the CJS model to model relationships amongst demographic parameters. They develop an analysis of

capture-recapture data that allows inference to be made about the relationships between survival and birth processes rather than focussing on survival or abundance alone. The Schwarz-Arnason model (Schwarz and Arnason, 1996) is analysed and an alternative pa- rameterisation is presented. They introduce the parameterfi which is defined as a birth

rate parameter and is given by fi = β_di_i where

di+1 =diφi+βi (2.5.1)

for i = 1,2, . . . , K−1 and d1 =β0. The fi are then used to replace the entry probabil-

ities βi in the Schwarz-Arnason model. The di can be thought of as approximating the

proportion of the superpopulation N that constitute the population in the ith _{period N} i

under the assumption of no losses on capture. Link and Barker (2005) note that fi ≈ _NBi

i represents an index to per capita birth rates as opposed to the index of total births rep- resented by the Schwarz-Arnason parameter βi ≈ B_Ni where both these approximations

are based on assuming no losses on capture. They also demonstrate that all of the in- formation on the parameters β0 and N is contained in the Schwarz-Arnason likelihood component P₁0_a({u·} |N) and note that, without making some untestable assumptions,

the capture histories contain no information about β0 and N. Consequently, they deter- mine the statistic u· to be approximately ancillary to the parameters in the model that

require estimation. Hence, their extensions to the CJS model are based on the P₁0_b (see Eq. (2.4.4)) likelihood component of the Schwarz-Arnason model (Schwarz and Arnason, 1996).

The formulation of their likelihood is analogous to that of the Schwarz-Arnason ex- tension (see Eq. 2.4.5) to the three component product for the Jolly-Seber model (see Eq. 2.3.10). Their reparameterisation is chosen so that the model is expressed entirely in terms of identifiable parameters.

They define

λi =φipi+1+φi(1−pi+1)λi+1, i= 1,2, . . . , K−1 (2.5.2)

where λK−1 = φK−1pK. Thus, λi represents the probability of an animal remaining in

the population and being recaptured after sample iconditional on it being released after capture in samplei. Another new parameterisation is established with the definition

τi =

φi−1pi

λi−1

, i= 2,3, . . . , K−1 (2.5.3)

which represents the probability of being recaptured in sampleirelative to the probability of being recaptured after sample i−1. These two definitions are used in an alternative parameterisation of the CJS component of the likelihood formulation. Link and Barker (2005) decompose this component (which is equivalent to P3 - see Eq. (2.3.14)) into conditionally independent binomial distributions. The first component

L2a= K−1 Y i=1 Ri ri (λi)ri(1−λi)Ri−ri (2.5.4)

then models the future recaptures of each cohort of animals that are marked and released following the ith _{sampling occasion and takes the product of these terms. The second}

component L2b = K−1 Y i=2 Ti mi (τi)mi(1−τi)Ti−mi (2.5.5)

includes the new termTi which is formally defined as the number of animals marked and

released prior to sampling period i that are then recaptured at some sampling occasion

j, i ≤ j ≤ K. It is defined recursively with T2 = r1, and Ti+1 = Ti −mi+ri, for i =

released before sampling period i that are known to be recaptured on some future sam- pling occasion but have not yet been recaptured (hence the subtraction of themi terms).

Therefore, the component P₃0_b (Eq. (2.5.5)) models the split between animals that are recaptured in theith _{sample and those that are recaptured after thei}th _{sample and again}

takes the product of these terms.

Then the likelihood, as a function of all model parameters θ, specified by (Link and Barker, 2005) is expressed as the product of three components:

L(θ)∝L1({νi})L2({pi},{φi})L3({pi},{φi},{fi}), (2.5.6)

where the first component L1 contains the νi parameters which are the release proba-

bilities for animals following capture at the ith _{sample. It is analogous to the losses on}

capture component P2 (Eq. (2.3.12)) of the Jolly-Seber model, although the losses on capture are not modelled separately for marked and unmarked animals in the Link and Barker (2005) formulation. As noted previously, the second component L2 = L2a×L2b

represents the CJS model and is equivalent toP3 (Eq. (2.3.14)) in the Jolly-Seber model. The third component L3 is equivalent to P10b (see Eq. (2.4.4)) and models the temporal

distribution of the animals’ initial captures. The likelihoods are then expressed solely in terms of identifiable parameters and MLEs can be obtained in closed form. Link and Barker (2005) note that the MLEs for the parametersφi, pi and νi are the classical Jolly-

Seber solutions (Jolly, 1965; Seber, 1965; Pollock et al., 1990). Details of the MLE form of the other parameters are provided by Link and Barker (2005).

Link and Barker (2005) also specify a bivariate distribution for transformations of the fecundity and survival parameters fi and φi respectively. They choose a bivariate

normal distribution with mean vector µ and variance-covariance matrix Σ and spec- ify a log transform for the fi and a logit transform for the φi. Therefore, their full

likelihood consists of the statistics D = [{ui},{mi},{Ri},{ri}] and the parameters

θ= [{φi},{pi},{νi},{fi},{µi},{Σi}].The full model expression also requires the spec-

ification of a prior due to the use of Bayesian model fitting methods. Link and Barker (2005) briefly detail the assumptions they make and the choice of prior this results in.

As discussed in his review paper (Schwarz, 2001) notes the lack, at the time, of in- vestigations into the utility of the JS model in estimating population abundance. As noted by Link and Barker (2005), one possible reason for this, as suggested by Schwarz (2001), is the lack of appropriate parameterisation of the model to obtain direct inference about birth rates. Specifically, Schwarz (2001) notes the lack of comparable procedures for the birth parameters when compared to those developed by Lebreton et al. (1992) for survival and capture parameters. By focussing on modelling the interactions of de- mographic parameters, Link and Barker (2005) chose to express the likelihood in terms of Eq. (2.5.6) which enables them to express the model in terms of parameters relating directly to fecundity fi instead of the less biologically relevant βi used by Schwarz and

Arnason (1996). Link and Barker (2005) also claim that the factorisation of the likelihood simplifies the incorporation of hierarchical modelling and is necessary for the generation of suitable candidates as part of the MCMC fitting-algorithm. If losses on capture do occur then they can be modelled using theL1 component (see Eq. (2.5.6)) and, although

fi can still be considered a birth parameter, its interpretation is modified to being the

number of births per animals that would have existed in the population during period i