Multi-State Models with Live and Dead Encounters
Multi-State Models with Live and Dead Encounters
The multi-state model with live and dead encounters (Barker et al. 2005) is a generalization of the multi-state model that allows inclusion of recoveries of marks from dead animals. Format of the encounter history data is LDLD…, where the state is identified in the L portion of the encounter history, and only the value ‘1’ is used in the D portion to sigfify the death of the animal. See Data type for more details on coding multi-state encounter histories.
Assumptions
In addition to the usual assumptions of the multi-state model, this model assumes that apart from group and time effects, the reporting rate of marks from dead animals depends only on the state that the animal was in at the immediately preceeding live-capture occasion. In some applications, it may be reasonable to also assume that the state of the animal at the time of the dead recovery can be used to determine the state of the animal at the previous live-recapture occasion. This assumption is not included in the model so any such information is ignored.
Model Structure and Likelihood
If there are S states (identified by state labels), define:
fh is an SxS matrix with s,t’th element = Pr(animal alive at time h in state s is alive at time h+1 in state t),
yh is an SxS matrix of transition probabilities with s,t’th element = Pr(animal moves from s to t | alive at h and h+1),
Ph is an (Sx1) matrix with s’th element = Pr(animal alive at time h in state s is captured),
Sj is an (Sx1) vector with s’th element = Pr(animal alive at time j in state s is alive at time j+1),
r j is an (Sx1) vector with s’th element = Pr(animal in state s that dies between j and j+1 is found and reported),
D(x) = a diagonal matrix with vector x along the diagonal,
1 = a (sx1) vector of ones,
Yh is an indicator variable that = 1 if the animal was caught at time h and 0 otherwise.
Note thatfh = D(Sh)yh.
The animals in the study can be categorized according to whether their last encounter was as a live recapture or as a dead recovery
.
Animals last encountered by dead recovery
For an animal first released in state s at time i, that was found dead between samples j and j+1, and was last captured alive at in state t at time k the likelihood, conditional on the first release, is factored into two parts:
(1) Pr(encounter history between i and (including) k | first released at time i in state s) is the s,t’th element of the matrix
formed by taking the product from h=i to h=k-1:
P YhfhD(Ph+1) +(1-Yh)fhD(1–Ph+1).
We take the s,t’th element because we know that the animal was in state s at time i and in state t at time k.
(2) Pr(not caught between k and (including) j and found dead between j and j+1 | released at time k in state t) is the sum
across the t’th row of the matrix formed by taking the product from h=k to h=j-1:
{P fhD(1–Ph+1)}D(1–Sj)D(r j).
Although we know that the animal was in state t at time k, we do not know which state the animal was in at time j.
However it must have been in one of the states and therefore we can find the probability we require by taking the sum across
the t’th row of this matrix.
Animals last encountered by live recapture
For an animal first released in state s and sample i and last encountered by live-recapture in state t and sample j, the likelihood, conditional on the first release, is factored into the two parts:
(1) Pr(encounter history between i and (including) j | first released at time i in state s) is the s,t’th element of the matrix
fj-1D(Pj){P YhfhD(Ph+1) +(1-Yh)fhD(1-Ph+1)}
where the product is taken from h=i to h=j-2.
(2) Pr(Not encountered again | released alive at j in state t). This is found by finding the probability that the animal i
encountered at least once after sample j using the above expressions, and then subtracting this probability from 1.
Parameter Identifiability
If the capture occasions are indexed up to sample t and the dead recovery occasions up to sample i, then in addition to the parameters that can be estimated using the multi-state model, we can also estimate yt-1, Pt, St-1 and r j (j=1,…,t). If i > t then (complicated) confounded products of state-specific survival and reporting rates can also be estimated.
Parameter Constraints
The sum of the y parameters for transition from one state to all the others for a particular time must sum to <= 1. The MLogit link function is a powerful technique to enforce this constraint. See the link function help file for more information on using this technique.