specifying structure of transition parameters in multistate

[ctsolomon]

I am trying to figure out how to parameterize the transition parameters Psi in a multistate model of a somewhat complicated experiment.

The experiment was conducted between 2011 and 2021 on a lake that was divided into separate basins using a plastic curtain. There is some "leakage" of fish across the curtain, so estimating the transition probabilities between basins might improve our estimates of survival probability, which are our primary interest.

From 2011 through 2017, the lake was divided into an East and West basin by the plastic curtain. In 2018, an additional curtain was installed that split East into two portions. The resulting three basins of the lake, from west to east, we can now call West (which hasn't changed), East (a portion of the former East), and FarEast (the remainder of the former East).

One simple, reasonable model for the transition probabilities Psi bewteen the basins of the lake is structured like this:

Code: Select all

       transition y2011 y2012 y2013 y2014 y2015 y2016 y2017 y2018 y2019 y2020
1    West to East    b1    b1    b1    b1    b1    b1    b1    b1    b1    b1
2 West to FarEast     0     0     0     0     0     0     0 b1+b3 b1+b3 b1+b3
3    East to West    b1    b1    b1    b1    b1    b1    b1    b1    b1    b1
4 East to FarEast     0     0     0     0     0     0 b1+b2    b1    b1    b1
5 FarEast to West     0     0     0     0     0     0     0 b1+b3 b1+b3 b1+b3
6 FarEast to East     0     0     0     0     0     0     0    b1    b1    b1

This structure assumes that:

-No movement is possible from West or from East to FarEast until the 2017-2018 transition, because FarEast didn't exist until the additional curtain was installed in mid-2018.
-No movement is possible from FarEast to West or to East until the 2018-2019 transition, for similar reason.
-Otherwise, the transition probability is common for all basin-years, with two exceptions. First, there is a (presumably positive) offset for the East to FarEast transition in 2017 to 2018, because we trapped a bunch of East fish in the new FarEast basin when we installed the additional curtain. Second, there is a (presumably negative) offset for the transitions between FarEast and West starting in the 2018-2019 year, because these are non-adjacent lakes and a fish must cross two curtains to get between them.

It seems like it ought to be possible to code this model in RMark by structuring the design matrix for Psi so that it fixes Psi to 0 for some basin-years and includes appropriate indicator variables for the common transition probability (b1 in the table above) and the two offsets (b2 and b3). When I do this, however, the estimated Psi values don't have the structure that I expect - for instance, there are something like 7 unique Psi values across the basin-years, rather than the 3 that I think I asked the model to fit.

I presume that there must be some way to fix this by using subtract.stratum or altering the way I have specified the model, but I haven't yet been able to figure out the solution. I'd be grateful for any expertise/advice. I can provide code; haven't done so here because the necessity of modifying the design data makes the relevant portion a bit long.

Thanks!
Chris

[ctsolomon]

An update, after Jeff Laake graciously went back and forth with me offline to get me pointed in the right direction:

I was not accounting for the effect of the mlogit link calculations on the real parameter values for Psi, as described e.g. towards the end of section C.17 of Appendix C of the Cooch and White MARK book. Once I worked through those calculations, it was clear why the structure that I set up, with three beta values, yielded 7 different real parameter values for Psi.

It also became clear that, using the mlogit link, there was no way to set up the betas (or at least no way that I could figure out) that would yield the real parameter structure that I wanted to model. I tried switching to the logit link instead, using my original beta structure. At first this wasn't successful - MARK wouldn't converge. But if I started off by running the model using the mlogit link, and then used the estimates from that model as initial values for a model with the same structure using the logit link, I was able to get MARK to converge and give estimates for the parameters in the structure I was interested in.