Multistate (mult-stratum) models
Multistate models are an important generalization of CJS that allow for more flexible modelling of transitions between states. A state is simply a characterization of an animal or group of animals at some time with respect to one or more attributes that may change over time, so that individuals can change state membership. We have already seen examples of states, starting with the very simple one that animal can be either in the state alive or dead. Another example is occupancy-- sites can be either occupied or not. We don't tend to think about these as multi-state problems , because the transitions are simple-- an animal can go from alive to being dead (but not the reverse!) and we call that mortality. Age is another example of state where transitions can occur-- obviously animals age into older age classes (states) if they do not die; again though this is a simple, one-way transition ( we cannot, unfortunately, go from the old state to the young state!).
More complicated state transitions involve things like a physiological state, where animals can sometimes return to a state that they were in previously but temporarily left (e.g., I can lose weight, then gain it back). Or animals can move between multiple geographic sites (states). These more complex cases are typified by (1) existence of 2 to many states that animals can exist in, and (2) potential for multi-way transition between states over time. E.g., if an animal can be in sites A, B, or C at a given time and starts out at A at time 1, its history of state membership over 3 periods could include
AAA, AAB, AAC, ....., ACC (I come up with 9 combinations conditional on starting in A). Over each time interval state could have stayed in A, or moved to B or C. We might represent the by a table or matrix of transition probabilities
where the Psixy is the probability of moving to state x given you started in y. This of course assumes that the animal didn't die -- or move to the absorbing state (from which no return is possible) of 0!
Complicating things is the fact that we typically don't observed the animals's true states unambiguously-- we only know the animal's state if we capture it at a specific time, and we are ordinarily do not capture all animals at each occasion. So supposed we capture an animal at time 1 in state A, but don't capture it again until time 3 when it is in C. What state was it in at period 2? Well, it could have been (depending on how states are defined) A, B, or C. Now we are going to have to model: (1) the detection process, which could be different both over time and depending on the state that an animal is in, and (2) the transitions that could have occurred over intervals where we did not observe the animal. This in turn tends to make multistate models get very complicated and very parameter rich-- very quickly!
Example
The built-in example in RMark of the multistate (multi-strata) model is contained in the data object mstrata. The data are loaded and processed into RMark using familiar commands
> require(RMark)
> data(mstrata)
> mstrata.processed=process.data(mstrata,model="Multistrata")
A default design data frame is created by
> mstrata.ddl=make.design.data(mstrata.processed)
The data consist of 4 occasions in which animals are captured or recaptured and identified as being in 1 of 3 strata, labeled as "A",."B", or "C". Each data line consists of a capture history with these characters or "0" indicating "not captured". The right hand column is th frequency (number of animals) with the indicated capture history.
ch freq
1 A000 491
2 A00A 15
3 A00B 12
4 A00C 11
5 A0A0 37
For ease of interpretation assume that the states are geographic locations (sites). Thus A00C for instance denotes that the animal was captured at site A on occasion 1, not recaptured at occasions 2 or 3, and recaptured in site B at occasion 4.
Modeling proceeds by specification of each 3 parameter types in terms of variation over occasions, groups, cohorts, ages, etc. (if present). There are 3 parameter types:
S(x), the probability of survival from i to i+1 given that the animal was in state x at time i
p(y) the probability of capture at i+1 given that the animal is in state y
Psi (from:to) the probability of transition from state "from" to state "to" given survival from i to i+1.
For example, survival might be assumed to be state but not time specific; this would be represented by
>S.stratum=list(formula=~stratum)
Recapture probability might be modeled by stratum effects, time effects, and their interactions as:
>p.stratumxtime=list(formula=~stratum*time)
Finally, state movement might be specified as specific to the originating and destination states but not time or other factors
>Psi.s=list(formula=~-1+stratum:tostratum)
These 3 parameter definitions can be combined in a mark model statement to create a single model:
mod1<-mark(mstrata.processed,mstrata.ddl,model.parameters=list(Psi=Psi.s,S=S.stratum,p=p.stratumxtime))
Alternatively, we can define a list of models and use the mark.wrapper to create a model results object that can in turn be used to compute an AIC table, perform model averaging, and other functions. The attached script file provides this approach for several alternative models using this data example. The results indicate strongest support for the model
S(~stratum)p(~stratum)Psi(~-1 + stratum:tostratum)
which specifies state-specific S and p and stateXstate specific transition but no time dependency in parameters. See the script file for details.
In the default model formulation, transitions are defined as above, with the implied order events being "survive first, then move". Under this formulation x in S(x) refers to the stratum of origin. Thus overall probability of transition Phi (x,y)involves the product of S(x) the probability of surviving the interval given the initial state x, and Psi(x,y) the probability of movement from x to y. Sometimes it will make better ecological sense to reverse the order of these events, in which case Phi(x,y) is defined as the Psi(x,y)S(y), that is, the probability of transitioning to y, followed by the survival conditioned on being in y. The attached R script shows an example of how incorporate "reversal" into the multistate models via a parameter option, and compares estimates and interpretation of Psi, S, and Phi under each formulation (standard and reversed).