Multiple Age Models

The focus of the preceding sections has been on the single-age Jolly-Seber model (Jolly, 1965; Seber, 1965) which defines animals as belonging to a single state. Fully age-specified models were considered in Section 2.2.2 where the sampling design allowed the age of each

captured animal to be accurately determined. Consequently, it is possible to obtain age- specific abundance estimates for each of the central K −2 sampling periods and for all but the first age class. Pollock’s robust design (Pollock et al., 1990; Kendall and Nichols, 1995; Williamset al., 2002, pp.523-544) can be used to obtain estimates of abundance for the age class 0.

The other form of age-cohort models discussed in Section 2.2.2 in which age is known only for marked animals typically do not allow abundance estimation. These cohort mod- els (Buckland, 1982; Pollock et al., 1990) are typically used for animals marked as young or age 0. These models are appropriate when it is not possible to easily age animals but there is reason to believe that the survival rate may be age-dependent. Due to the fact that the unmarked animals caught in periodi,ui, cannot be assigned an age-class no age-

specific capture rates can be determined and, consequently, no abundance estimation is possible for these forms of cohort-models. It is possible to obtain cohort and time-specific survival estimates (Buckland, 1982; Pollock et al., 1990). Pollock et al. (1990) obtains survival estimates for a study in which the newly marked animals in each year are defined as a cohort and the time since marking is recorded. For this form of study Buckland (1982) notes that survival rates are functions of both time since marking and temporal effects and describes analyses which emphasise one or other of these effects.

2.3.2.2 Multistate Models

Once again, the discussion on extending the Cormack-Jolly-Seber model to a multistate modelling framework (see Section 2.2.3) covered many of the issues that are relevant to performing a similar expansion for the Jolly-Seber model. When populations were classified by states for both the Markovian and memory models discussed in Section 2.2.3,

the capture probability was modelled as being both state and time-specific. Hence, if the focus of inference was on the animals belonging to state g during sampling period i, a Horvitz-Thompson like estimate (Borchers et al., 2002) of the abundance for that state would be given as ˆ N_ig = n g i ˆ pg_i

where ˆng_i is the number of animals in state g at time i that are caught during period

i, and ˆpg_i is the estimated capture probability for that specific group of animals. If the population was classified into a total of G states then the total abundance for animals across all states is obtained by summing the state-specific abundance estimates:

ˆ N P i = G X g=1 ˆ N_ig.

Once again, a state specific estimate of capture, ˆpg_i, is required to obtain an abundance estimate and this implicitly assumes that the marked and unmarked animals in state g

during sample periodi possess the same capture probabilities.

This is still an active area of research and no efficient parameterisation of the multistate Jolly-Seber model has yet been published. A framework respecting proper conditioning has yet to be developed.

2.3.3

Auxiliary Data

Many of the ideas for incorporating auxiliary information into Jolly-Seber models for analysing capture-recapture data have been covered in the discussion on Cormack-Jolly- Seber models in Section 2.2.5. Many of the techniques outlined in Lebretonet al. (1992) and reviewed by Pollock (2002) are applicable to the Jolly-Seber models. By extend- ing the analysis of capture-recapture data to estimate population abundance Jolly-Seber

models incorporate unmarked animals which can result in auxiliary information being un- recorded for animals that are not observed. The problem of modelling missing covariate values is well suited to Bayesian modelling techniques and recent developments of exten- sions to capture-recapture models (Link and Barker, 2004, 2005; Bonner and Schwarz, 2006; Schofield and Barker, 2008; Dupuis and Schwarz, 2007) use a Bayesian framework to impute missing covariate values when modelling demographic parameters as functions of covariates (see Section 2.5).

2.3.4

Model Selection

An alternative approach to Eq. (2.3.1) for expressing the distribution function for the observed data in a Jolly-Seber model is now presented. This approach is described in detail in Williams et al. (2002, 495-522) and is based on the approach of Seber (1982). This approach models the data using the mij-array statistics introduced in Section 2.2.

The significant property of this approach is that it decomposes the distribution function into three separate likelihood components. Using the original notation in Table 2.6 and the extra notation in Table 2.7 the model can be expressed as

P r({ui},{di, d0i},{mij}) = P1({ui} | {Ui},{pi})

×P2(di, d0i|mi, ui), ηi, ηi0)

×P3(mij|Ri, φi, pi) (2.3.10)

The first component of this decomposition models the capture of unmarked animals and takes the same binomial form as the first term in Eq 2.3.1:

P1({ui} | {Ui},{pi}) = K Y i=1 Ui! ui! (Ui−ui)! pui i (1−pi)Ui−ui (2.3.11)

The second component of Eq. 2.3.10 models the marked and unmarked animals that are not released back into the population after capture and can be expressed as:

P2({di, d0i} | {mi, ui}),{ηi, ηi0}) = K Y k=1 mi di (ηi)di(1−ηi)mi−di × K Y k=1 ui d0_i (η_i0)d0i₍₁−_η0 i) ui−d0i (2.3.12)

The third component of Eq. 2.3.10 models the conditional probability distribution of the animals recaptured at each period and is given by:

P3({mij} | {Ri},{φi, pi}) = K−1

Y

i=1

Ri!

(mi,i+1)!(mi,i+2)!. . . mi,K!(Ri−ri)!

(φipi+1)mi,i+1 × {[φi(1−pi+1)φi+1pi+2]mi,i+2. . . (2.3.13) . . .[φi(1−pi+1)φK−1pK]mi,Kχ Ri−_ri i (2.3.14)

This component is simply the Cormack-Jolly-Seber conditional model for the marked animals that are released back into the populationRi. It could also be written using the

expression in Eq. (2.2.4) if the conditioning was on the number of unmarked animals ui

caught in each sampling period.

Assessing the goodness-of-fit for the JS models and comparing nested models is based on component P3 of the likelihood representation in Eq.(2.3.10). This component con- ditions on the number of releases in each time period and incorporates the subsequent capture-history data on these marked animals. Therefore, since the classical goodness-of- fit and model comparison tests are calculated using the capture histories of the animals, they are typically based on the P3 component of the likelihood. The second component

P2 is included for completeness but typically does not form a part of classical inference on Jolly-Seber models, although tests could be performed if issues such as the temporal

Parameters

ηi = The probability that a marked animal captured during sampling

periodi is released back into the population.

η_i0 = The probability that an unmarked animal captured during sampling periodi is released back into the population.

Statistics

mij = The number of marked animals captured and released on the ith

sample that are next caught on sampling periodj (i = 1, . . . , K − 1 andj =i+ 1, . . . , K).

di = The numbers ofmi that are not released back into the population

ati.

d0_i = The numbers of ui that are not released back into the population

ati.

Table 2.7: Extra Parameters and Statistics for an alternative representation of

the single-age Jolly-Seber model.

variation of release rates or equality between the release rates for marked and unmarked animals were of interest. The first component, P1, is useful in the estimation of popula- tion abundance but is not used to assess model goodness of fit. However in alternative parameterisations (Crosbie and Manly, 1985; Schwarz and Arnason, 1996) the entry of previously unmarked animals, ui, is explicitly modelled using entry probabilities. These

parameterisations then allow the ui to be incorporated into the model selection and fit

assessment inference.

The mechanics of model selection and goodness-of-fit testing for the Jolly-Seber model follow much the same procedures as discussed in Section 2.2.1.4 for the Cormack-Jolly- Seber models. The use of an information-theoretic approach to model selection such as AIC and its small-sample analogues are preferred to the more traditional approaches involving likelihood ratio tests. Incorporating model uncertainty into inference by aver- aging across multiple models and weighting each parameter estimate by the model AIC

can provide more robust parameter estimates and is also recommended.