Model Structure

The Jolly-Seber (Jolly, 1965; Seber, 1965) approach assumes that the unmarked animals in the population, the Ui are fixed parameters and that the recruits to the population

can be expressed in terms of theUi’s as ˆBi = ˆUi−1φˆi( ˆUi−ui). This then allowed the first

component, P1, to be expressed in the products of binomials form given in Eq. (2.3.11). From P1, an estimate of Ui can be obtained: ˆUi =

ui

ˆ

pi

. The relationship between Bi and

Ui can then be used to estimate the net recruits ˆBi. However, Schwarz and Arnason

(1996) noted several problems with this approach. With the births not appearing in the likelihood it is difficult to impose any restrictions on the estimates of Bi. For example,

there is no simple way to ensure that known periods experiencing no recruitment can be implemented as a model constraint. The numerical evaluation of the likelihood can result in negative estimates for the Bi; there is no clear way to maximize the likelihood

ensuring that all estimated Bi are non-negative. The reduced-parameter models of the

or both, cannot be expressed using the reduced form of P1.

One approach to ensuring the numerical optimisation of the likelihood results in non- negative birth estimates centered on log-linear models (Cormack, 1989). These models included a parameter Ψi defined such that the number of unmarked individuals in the

population at the time of the (i+ 1)st_{sample is Ψ}

i times the number of unmarked animals

surviving from theith _{sample. From this formulation, the relationship between uncaught} animals and births is given by

Ui+1 =Ui(1−pi)φiΨi =Ui(1−pi)φi+Bi.

The parameterisation of the models formulated by Cormack (1989) has a direct corre- spondence with the rates for capture, survival and recruitment in the usual JS models. The advantage of this approach is that negative estimates of recruits can be avoided by constraining the Ψi to be non-negative. However, there are some disadvantages to

this approach. The Ψi are only an indirect estimate of the Bi and it can be difficult to

obtain the equivalent restriction on the Ψi to impose a constraint on the Bi. Deriving

standard errors of the back-transformed parameters of biological interest is a non-trivial process and no method is presented to do so. This issue is later resolved (Cormack, 1993).

The approach taken by Schwarz and Arnason (1996) builds on the work of Crosbie and Manly (1985) who reparameterised the Jolly-Seber model by defining a new “super- population” parameter. This parameter, Ni, is defined as the total number of animals

that exist in the population at i and survive until the next sample time, thus it gives the number of animals available to be sampled in the population at any point during the study. They define the parameter,Bi, as the number of animals that enter the population

relationship between these two parameters is then N = K−1 X i=0 Bi (2.4.1)

whereB0 =N1since all the animals in the population initially,N1, are new entries relative to the first sampling occasion. Schwarz and Arnason (1996) then model these random variables Bi as realisations from a multinomial distribution. The Bi’s are referred to

as births and in this context the term birth is used to define any mechanism by which new animals can enter the population and can encompass birth, immigration, etc. The multinomial model for the Bi includes entry parameters βi which determine the point

at which the new births enter the sampling population. Hence the births model can be expressed as {B1, . . . , BK−1} ∼Multinomial(N;β1, . . . , βK−1) and B0 =N − K−1 X i=1 Bi. (2.4.2)

They then define another set of parameters which are given, recursively, as Ψ1 =β0 and Ψi+1 = Ψi(1−pi)ψi+βi. These parameters are used to model the number of unmarked

animals that, as a proportion of the superpopulation N, are in the population and cap- tured on each sampling occasion. The sum for Ψi+1 consists of the two components, firstly the probability that an animal was in the previous population, avoided capture ini and survived until i+ 1 and secondly the probability that the animal was a new recruit in periodi. From this definition it can be seen that Ψipi corresponds to the capture of un-

marked animals at sampling periodi. This results in a multinomial model for the capture of previously unmarked animals at each sampling occasion

This corresponds to the first componentP1 of the standard Jolly-Seber distribution func- tion (Eq. 2.3.11) and can be written as:

P₁0({ui} |N,{βi},{pi},{φi}) = N! u1!u2!. . . uK!(N −u·)! " 1− K X i=1 Ψ1pi #N−u· _K Y i=1 (Ψipi)ui (2.4.3) whereP₁0 corresponds to the Schwarz-Arnason form ofP1, andu·denotes the total number

of unmarked animals captured during the study,

u· =

K

X

i=1

ui.

TheN−u·term represents all the animals that exist in the superpopulation but are never

caught. The expression P₁0 is a function of the parameters N,{βi},{pi} and {φi}, which

are subject to the constraint:

K−1

X

i=0

βi = 1.

Then, by conditioning on the total number of unmarked animals observed during the study (u·) a further factorisation ofP10 can be performed:

P₁0({ui} |N,{βi},{pi},{φi}) = P₁0_a({u·} |N)P₁0_b({ui} |u·,{βi},{pi},{φi}) =    N! u·!(N −u·)! " _K Y i=1 Ψipi #u·" 1− K Y i=1 Ψipi #N−u·   × ( u· u1!u2!. . . uK! K Y i=1 Ψipi PK i=1Ψipi !ui) (2.4.4)

Of these two components the first, P₁0_a, models the split between the animals in the su- perpopulation that are caught at some point during the study and those that are not. The second component,P₁0_b, is restricted to the animals that are caught at least once and models the temporal distribution of their initial captures.

Combining the standard forms for P2 and P3 (see Eq. 2.3.10) with the new Schwarz- Arnason (Schwarz and Arnason, 1996) representation ofP1 produces the entire distribu- tion function:

P r({ui},{di, d0i},{mij}) = P10a(u·|N)

×P₁0_b({ui} |u·,{βi},{pi}, φi)

×P2({di, d0i} | {mi, ui}),{ηi, η0i})

×P3({mij} | {Ri},{φi, pi}) (2.4.5)

The likelihood model formulations in Eq. (2.3.11) and Eq. (2.4.4)) are different in the pa- rameterisations but they are both based on exactly the same observed capture-recapture data. They also have common componentsP2 and P3. Therefore, the alternative expres- sions in Eq. (2.3.11) and Eq. (2.4.4) are statistically equivalent.

2.4.2

Model Assumptions

The alternative expression for the model componentP1 required the specification of some new parameters, the entry probabilitiesβi. These parameters model the probabilities that

a member of the superpopulation is not present on the study area (i.e. available to be sampled) until sampling period i, but then enters the population and can be caught on sampling occasion i+ 1. The key assumption relating to these entry probabilities is that of homogeneity. It is assumed that all animals in the superpopulation that have yet to enter the population have an equal chance of belonging to the next cohort of recruits. For example, animals of different gender are not assumed to have different probabilities of making the transition to the population during the same period.

The use of the multinomial distribution implicitly requires the assumption that the fates of the animals are independent. By including the entry probabilities, the “fate” of an animal incorporates not only the usual capture and survival parameters but also the entry of an animal into the sampling population. The independence assumption should still hold unless there are strong reasons why the entry of one animal to the population may influence the entry of another, for example, multiple births per family.

2.4.3

Estimation

The estimation of the model parameters is performed chiefly through numerical maximi- sation of the likelihood equations (Schwarz and Arnason, 1996). The full likelihood can be expressed as the product of a sequence of multinomial and binomial terms (Schwarz and Arnason, 1996) and it can be demonstrated that the conditional MLEs derived in this way are asymptotically equivalent to the unconditional MLEs5_.

Schwarz and Arnason (1996) chose to estimate the model parameters using a logit link function as a means of ensuring that all estimates are kept within the parameter space. The estimates are then derived from the following components of the data: the estimates of survival and capture rates are based on the recaptures of previously marked animals, the estimates of new recruits to the sampling population are based on the pro- portions of unmarked animals that are captured on each occasion and the estimate of the superpopulation size is based on the total number of unmarked animals captured during the sampling study. Hence, the capture {ˆpi}, survival

n ˆ φi o and entry n ˆ βi o probability estimates can be obtained from numerical maximisation of the product P₁0_bP2P3. These

5_{The conditional likelihood estimates will differ from the MLEs if constraints are applied to the model}

estimates ˆpi, ˆφi and ˆβi can then be used withP10ato estimate the superpopulation size N. ˆ N = u· PK i=1Ψipˆi (2.4.6)

As with both the Cormack-Jolly-Seber (Section 2.2.1.2) and the Jolly-Seber (Section 2.3.1.3) models not all of the parameters are identifiable. The capture probability on the first sam- ple,p1, cannot be estimated andφK−1pK can only be estimated as a product. Along with

the constraint that PK−1

i=0 βi = 1 there are 3K −3 parameters that can be estimated

under the full time-specific model (Schwarz and Arnason, 1996). The lack of identifiable estimates for p1 and pK makes estimation of the superpopulation total ˆN difficult for

the fully time-specific model. Imposing the constraints p1 = pK = 1 is recommended

by Schwarz and Arnason (1996). Estimates of time-specific abundance and recruitment are also given and it is demonstrated that the usual estimates (Crosbie and Manly, 1985; Pollocket al., 1990) are obtained.

The variances are derived asymptotically and those obtained for φi and pi are the

same as in Pollock et al. (1990). The recruits Bi are assumed to be random variables

which adds a component of variance to the asymptotic variances of ˆBi and ˆNi. If there

are no constraints on the βi, or only simple constraints such as βi = constant then this

extra source of variance is removed. However, for more general constraints Schwarz and Arnason (1996) note that there is no general method for deriving the variance.