Bayesian Multistate Models

A further extension to the general multistate Jolly-Seber capture-recapture model (Schwarz and Arnason, 1996) is developed by Dupuis and Schwarz (2007). To address potential problems with estimator bias incurred by heterogeneity in the capture and survival rates methodology has been developed in which animals are classified into states. This clas- sification can be based on static variables (e.g. gender) (Schwarz and Arnason, 1996) or on stochastic variables (e.g. movement between geographical regions) (Schwarz et al., 1993). When there is movement between states these models are unable to produce esti- mates of abundance. A Bayesian treatment for the movement case was presented by King and Brooks (2002b) although they did not address the issue of abundance. The super- population approach described in Schwarz and Arnason (1996) is extended to model the appearance of new entrants into the population between sampling occasions. The missing value problem is given an alternative approach owing to the model structure requiring that both unrecorded states prior to an animal’s first capture as well as the full set of unobserved states for animals that are never marked are incorporated into the modelling framework.

The observed data are decomposed into two interlinked vectors for the ith _{animal: x} i

simply records the standard capture history pattern of 1’s and 0’s and zi records the

covariate value observed on each occasion corresponding to a 1 (the animal is observed) in xi. No covariate is recorded on occasions corresponding to 0’s in xi. Independence

is assumed with respect to the processes of capture, survival and movement and move- ment is assumed to be a first-order Markov process. The model probabilities are now defined to be a function of state as well as time which would increase the complexity of the likelihood-based models defined in Schwarz and Arnason (1996). Dupuis and Schwarz (2007) define zi,t as the state of animal i at time t such that if zi,t = r then animal i

is present in the sampling population and is alive in state r at time t. Also, if zi,t = ∗,

animalihas not yet entered the sampling population at timet. Using these variables, the number of animals from the superpopulation that enter into the sampling population in stater between times tand t+ 1 is given by: Et(r) =

PN

i=1Izi,t=∗,zi,t+1=r, where I(c) is an

indicator function taking the value 1 if condition cis true and 0 otherwise. Similarly, the number of animals in the population in staterat timetis given byNt(r) =PN_i=1Izi,t+1=r.

Hence, the number of animals per state and the number of new entrants are not param- eters in their own right but simply functions of other variables. The priors are assumed to be independent of each other with specific emphasis placed on the independence of the priors on the survival parameters and the superpopulation.

Dupuis and Schwarz (2007) introduce a blocking design to simulate missing covariate data. The blocks are determined by capture history with blocks of Type I corresponding to the occasions before first capture, Type II corresponding to occasions in which the animal was not captured that occur between occasions of the animal’s first and last ob- served capture, Type III corresponding to occasions after the last observed capture until the end of the study and Type IV corresponding to the scenario where an animal is not seen on any of the sampling occasions. Thus, each of these blocks corresponds to a set of missing values and the distribution of these blocks, conditional on the observed cap- ture histories can be derived (Dupuis and Schwarz, 2007). The joint distribution of the missing covariate blocks for an animal that has been captured at least once is shown to be the product of the distribution of the missing covariates in the individual blocks. The missing data are simulated conditionally on the superpopulationN and a set of sufficient statistics summarising all the marks and captures attributed to animals across all states are simulated using multinomial distributions. They note that their blocking approach to data-augmentation produces an ergodic Markov chain even when some transitions between

states are impossible. In comparison, the component-by-component data-augmentation approach (King and Brooks, 2002b) can be non-ergodic.

The posterior distribution for the superpopulation is simulated and, for the two altern- tative priors used, can be shown to be negative binomial with parameters determined by the number of animals marked during the study and derived probabilityλ of any animal being marked during the population. The value ofλis simply the probability of an animal moving into a particular state (from another state or entering from the superpopulation) and surviving summed across all states and all time periods. For posteriors of the other parameters the complete data likelihood is derived and decomposed into three separate components: one for the survival and state transition parameters, one for the entry pa- rameters and one for the capture parameters.

The imposition of constraints on parameters and the identifiability of the parame- ters is also investigated. A reparameterisation is required on the entry probabilities to allow suitable constraints to be imposed. Dupuis and Schwarz (2007) also note that there exists a correspondence between the unidentifiable parameters in the classical Jolly- Seber model (see Sections 2.2.1.2 and 2.3.1.3) and in their multistate extension. The full time-specific model has confounded initial immigration, state distribution and capture parameters which cannot be estimated separately without imposing identifiability con- straints on the parameters.

The approach developed by Dupuis and Schwarz (2007) allows the combination of a super-population approach with a Bayesian analysis. The approach extends the existing methods of analysing multistate capture-recapture studies and proposes a more computa- tionally efficient algorithm for dealing with missing covariate information within a MCMC

framework. There is no attempt at model selection but Dupuis and Schwarz (2007) note that one approach to MCMC model selection is covered in detail in King and Brooks (2002b) in which RJMCMC is used to move between a potentially vast array of alterna- tive model specifications. Also, the model could incorporate a prior covariance structure to investigate interactions between the model parameters.