Multi-State Model for Live Recaptures
The multi-state model (also known as multi-strata model) of Brownie et al. (1993) and Hestbeck et al. (1991) allows animals to move between states with transition probabilities. These models are an extention of the Cormack-Jolly-Seber model (CJS) (Cormack 1964, Jolly 1965, Seber 1965) live recapture model extended to multiple areas or states. A multi-state model with live and dead encounters can also be fitted in MARK, as well as the robust-design multistate model. In addition, the set of psi parameters (see below) can be redefined as to which transitions are estimated with the Change PIM Definitions menu choice.
Define the probability of survival for an animal during interval i on state j as S(ij), and the probability that the animal is captured during encounter occasion i on state j as p(ij). In addition, the probability that an animal moves from state j to state k during interval i is psi(ijk). If the
psi(ijk) have the constraint that
sum from k=1 to K of psi (ijk) = 1,
estimation is possible. The effect of this constraint is that animals that move off all the states in the study, i.e., move outside the study area, cause the estimates of survival to be biased in the sense that "apparent survival" is estimated. That is, emigration off all the states in the study results in "apparent survival" being survival times the probability that the animal remains on the study area. Note that a special link function, the multinomial logit link function, has been included as an option if parameter-specific link functions are used to enforce the above constraint.
A simple example will make this model clearer. Assume that 3 strata are sampled: A, B, and C (these are the state labels). Encounter histories must include the information of which state an animal was captured in. Thus, instead of using a "1" to indicate capture, we use the state label. For 5 encounter occasions, a history such as
could result. That is, the animal was initially captured in state B, captured in state C during the second occasion, captured in state A on the third occasion, captured in state C on the fourth occasion, and then in state A on the fifth occasion. The cell probability describing this encounter history is
[S(1B) psi(1BC) p(2C)] [S(2C) psi(2CA) p(3A)] [S(3A) psi(3AC) p(4C)] [S(4C) (1 - psi(4CA) - psi(4CB)) p(5A)],
where encounter occasions are separated within brackets. Note that for the fourth interval, the probability of remaining in state C is just 1 minus the sum of the probabilities of leaving state C. This cell probability demonstrates a key assumption of this model: survival is modeled with the survival rate for the state where the animal was captured, and then transition to a new state takes place. That is, all mortality takes place before movement. An animal cannot transition to a new state where a different survival rate pertains, and then die. If it dies, it must do so in the current state. If it lives, then it can transition to a new state. This assumption is critical if survival rates are different between the states. If survival is constant across the states, then the assumption is not important. Biologically, this assumption is difficult to accept, and limits the usefulness of the model.
The nastiness that results when an animal is not captured (i.e., a 0 is in the encounter history) is difficult to demonstrate without matrix algebra. However, to give you a feeling for what happens, consider the encounter history
The 0 on occasion 3 can be explained by 3 possibilities: the animal remained in C and was not captured:
S(2C) [1 - psi(2CA) - psi(2CB)] [1 - p(3C)] S(3C) [1 - psi(2CA) - psi(2CB)] p(4C) ,
or the animal transitioned to state A and was not captured, and then moved back to C:
S(2C) psi(2CA) [1 - p(3A)] S(3A) psi(2AC) p(4C) ,
or the animal transitioned to state B and was not captured, and then moved back to C:
S(2C) psi(2CB) [1 - p(3B)] S(3B) psi(2BC) p(4C) .
For both of the cases where the animal moved, it has to return to state C, because it was captured in state C during the fourth occasion.
The number of parameters in this simple example is already large. There are 3 state-specific survival rates for each interval, and 3 state-specific capture probabilities for the last 4 occasions. In addition, each interval i has the transition probabilities psi(iAB), psi(iAC), psi(iBC), psi(iBA), psi(iCA), and psi(iCB). Thus, a total of 12 PIMs are created in Program MARK to estimate these parameters. Note that these PIMs can be redefined with the Change PIM Definition menu choice to select a different psi as the value obtained by subtraction.
A trick to remember with the multi-state models is that the sum of the psi values from a state must be <= 1. The multinomial logit link, mlogit, is available to help enforce this constraint.
Another difficulty with the multi-state models is that there may be multiple maximums in the likelihood funciton. To be more sure that the solution you have found for the estimates is the global maximum of the likelihood, consider checking the solution with the simulated annealing algorithm.
At this time, only the movement model without memory is implemented in Program MARK. Brownie et al. (1993) describe more complex models where the animal remembers where it was on the previous occasion. This memory model requires a very large amount of data to provide reasonable estimates because the number of parameters grows quickly, even more so than the model considered above.
The multi-state data type has also been extended to incorporate the robust design, including all 12 of the possible closed captures data types, plus dead recoveries can be included in the multi-state data type with live and dead encounters. With the closed captures robust design extention of the multi-state models, you can obtain estimates of the population size of each state.
More details on the theory of the multi-state model are available at the WWW site http://wwwhttp://www.cnr.colostate.edu/class_info/fw663/multistrata.pdf.
Each state occurring in the data must be identified with a single character state label that identifies the state in the encounter histories. In addition, each state can have a more detailed descriptor (the state name) that identifies the state in the output.