Multistrata model with random effects

questions concerning analysis/theory using program MARK

Multistrata model with random effects

Postby osimmons » Thu Apr 18, 2019 7:40 am

We have a dataset with year as a random variable, individual length as a covariate, and there is some missing data. There are twelve years of data total – juveniles are measured and released and we have recorded their survival at sea in the first year, but they may stay out for a second year, which is assumed to be independent of their length. We are interested in the effect of individual length on survival. We think this is a mutlistrata model with four states: juvenile, survival to first year (phi1), survival to second year (phi2), and dead. Parameter psi is the probability of transitioning between detection sites. There are three occasions to detect the fish: seeing them at site a with probability p, seeing them at site b with the same probability p, or missing them.

States:
Juveniles 1SW MSW Dead
Juveniles 0 phi1 phi2 1-phi1-phi2
1SW 0 psi 0 1-psi
MSW 0 0 psi 1-psi
Dead 0 0 0 1

Observations:
Juveniles 1SW MSW Not Seen
Juveniles 0 0 0 1
1SW 0 p 0 1-p
MSW 0 0 p 1-p
Dead 0 0 0 1

We’ve been reading through the MARK manual and haven’t been able to find a model that corresponds exactly with this dataset. We’ve been reading about the CJS models, including the CJS random model. But our understanding is that these don’t allow for multiple states, so we expect a multistrata random model to be better (we’ve also read about robust design, Huggin’s closed population models, etc.). Does anybody know of a suitable model implemented in any of the mark recapture programs that might be used to estimate these parameters?
Thanks in advance for any advice:)
osimmons
 
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Re: Multistrata model with random effects

Postby simone77 » Tue May 07, 2019 9:18 am

There are different solutions to implement a random intercept capture-recapture model. I know it is relatively simple to do it in E-SURGE (have a look here) but I guess you may find a way also in MARK and RMark. However, before that I think you need to carefully reconsider your set of states and parameters that are still unclear.

We have a dataset with year as a random variable, individual length as a covariate, and there is some missing data.

When you say "missing data" you mean that you have not the record of individual length for all the capture events of each specimen? time-varying individual covariates are not easy to handle in capture-recapture models (you can look for that in this forum and in the MARK Gentle Introduction book). Very often people use just the individual covariate value measured at the first individuals' capture event. Missing data in individual covariates may be a problem too. Sometimes they are replaced by the average value of the population, or one may choose to censor those individuals with missing data if by doing so the sample size does not reduce too much (also in this case you can find material about that in this forum and MARK manual).

here are twelve years of data total – juveniles are measured and released and we have recorded their survival at sea in the first year, but they may stay out for a second year, which is assumed to be independent of their length.

When you say "recorded their survival", do you mean you performed capture sessions (you should give some details on your sampling scheme to make people able to help you) or did you use some kind of tracking device? "they may stay out for a second year" means that you know an individual may survive more than one year?

We think this is a mutlistrata model with four states: juvenile, survival to first year (phi1), survival to second year (phi2), and dead.

Actually these are not biological states. Juvenile and dead are states (dead is an absorbing state in the sense nothing happens, no probabilities have to be estimated, after the individual dies). However, Phi1 and Phi2 are parameters, they are state-transition probabilities that, I guess, you want the model to estimate.

Parameter psi is the probability of transitioning between detection sites.

This makes me understand that the multistrata (more often called multistate) character of your study would be like a multisite capture-recapture study that is a specific case of multistate capture-recapture modelling. If this is the case, your states would be geographical sites, nothing to do with age (I say that because you mentioned juvenile as a state) that is a dynamic state whose change is deterministic, it is not stochastic. In other words, while you expect the model estimates the probability of moving between t and t+1 from the site A to the site B (psi in your words), you do not expect the model estimates the probability of changing between juvenile and adult between t and t+1 because that age change is determined by the time elapsed.

There are three occasions to detect the fish: seeing them at site a with probability p, seeing them at site b with the same probability p, or missing them.

I think you meant "sites", not "occasions". Do you have two geographical sites where you are performing your capture-recapture study? the model allows you to test whether the recapture probabilities differs among your sites and to get them estimated (they do not need necessarily to be the same).

Code: Select all
      Juv   1SW   MSW   Dead
Juv   0   phi1   phi2   1-phi1-phi2
1SW   0   psi   0   1-psi
MSW   0   0   psi   1-psi
Dead   0   0   0   1


Formatting tables with the BBCode is a pain in the neck (at least for me), the more elegant solution would probably be attaching it as a figure. Anyway, this is your transition matrix I guess. I think this probabilistic matrix is incorrect because your states are wrongly defined.

I think you should define two transition matrices (relative to what happens between one capture session and the subsequent one, t->t+1), one for the probability of surviving and one for the probability of moving between sites.

Apparent Survival (phi):
Code: Select all
         aliveSiteA   aliveSiteB   Dead
aliveSiteA      phi      0      1-phi
aliveSiteB      0      phi      1-phi
Dead            0      0      1


Movement (psi):
Code: Select all
         aliveSiteA   aliveSiteB   Dead
aliveSiteA      1-psi      psi      0
aliveSiteB      psi      1-psi      0
Dead            0      0      1


For the detectability part of the model, you would have:
Recapture (p):
Code: Select all
         notDet   DetSiteA   DetSiteB
aliveSiteA      1-p      p      0
aliveSiteB      1-p      0      p
Dead            0      0      1


Note that the Recapture parameter refers to what happens at time t+1. At that time an individual that has survived the interval t-> t+1 is either at site A (its biological state being aliveSiteA) or at site B (its biological state being aliveSiteB) which entails that it can only be captured (with probability p), respectively, at site A and at site B.
simone77
 
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