Link-Barker Jolly-Seber Model

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Link-Barker Jolly-Seber Model

Link and Barker (2005) reparameterized the recruitment likelihood of Pradel (1996), ending up at a starting point of the likeliehood from Schwarz and Arnason (1996), used in the POPAN data type.  Basically, the probability of entry parameters of Schwarz and Arnason (1996) are translated to a recruitment (births and immigration) parameter (f).  The Link-Barker model conditions on the total number of animals ever caught and so the super-population size (N) and the probability that an animal has entered the population prior to the experiment [beta(0)] are no longer required in the likelihood.  This obviates the need for the identifiability constraint on p(1) and p(t) needed for the Schwarz and Arnason (1996) POPAN model.  If there are no losses on capture the Link-Barker likelihood and the Pradel (1996) likelihood are equivalent.

The Link-Barker data type has 3 PIMs for each group: phi, p, and f.  For t occasions, the phi PIM has t – 1 parameters, the p PIM has t parameters, and the f PIM has t – 1 parameters.  The f parameters need not be constrained to the [0, 1] interval, so you may want to use a log or identity link function for these parameters if estimates exceed 1.  The default link for f is the log link, regardless of what the link type is for phi and p.  That is, unless a parameter-specific link is used, f will be computed with the log link.

Two main advantages of this reparameterized model are that the f parameters are biologically interpretable as the birth rates (actually recruitment to the population from either immigration or births) to predict the number of new animals in the population, and the super-population size (N) is no longer required in the likelihood.  Historically, the weakest feature of the Jolly-Seber model has been the estimates of the population sizes and births for each occasion, because these parameters are based on untestable assumptions, i.e., that the unmarked animals have exactly the same capture probabilities as the marked animals.  Any behavioral response to initial capture will violate this assumption.

Use of individual covariates in the Link-Barker model is allowed, but the meaning of the estimates can be difficult to interpret biologically.  That is, lambda = phi + f.  If phi and/or f are modeled as functions of individual covariates, then the population parameter lambda is also a function of these individual covariates.

Competing models in MARK for this data type are the Pradel recruitment (f) model, Burnham’s Jolly-Seber model (which often suffers from optimization problems) and the POPAN data type.  Burnham’s Jolly-Seber model provides an estimate of the population size on the first occasion, and the rate of population change (lambda) for each succeeding interval.  The POPAN data type does provide estimates of population sizes and births (B) by occasion as derived parameters.  None of these models have the same likelihood, so AIC values are not comparable.

Both the Pradel recruitment and the Link-Barker model have the f parameter.  However, these parameters are not exactly equivalent unless there are no losses on capture.  The main advantage of the Link-Barker model over the Pradel parameterization is that the distribution for losses on capture factors out of the likelihood.  Unless you have many losses on capture the parameter estimates under the two parameterisations will not differ greatly.  If you do have a large number of losses on capture then the experiment is probably of questionable value as the removal of animals may cause a change in the demographics of the study population.  Note that the Pradel recruitment (f) model as parameterized in MARK, does not handle the losses on capture correctly, and does not incorporate the losses on capture parameter described in Pradel’s (1996) paper.

The Link-Barker parametrization has been extended by incorporating the Pledger mixture model on p, allowing for heterogeneity of capture probabilities.