Multiple multi-state-variables modeling?/unobservable states

[leanna_jones]

Hello! I have been working through the Gentle Intro to MARK (and extremely grateful for such helpful, extensive documentation!), but I am having some difficulty sorting out how to approach my dataset. I believe that I need a multi-state live and dead recapture model. I am interested in looking at how exposure to a parasite affects survival; we have about 20 years of data from about 600 animals. I have several groups (sex, study site) and I also need to look at age. The issue with age is that, while about 50% are first captured as cubs, the other 50% are caught as adults, and can be anywhere up to 17 at first capture. So I cannot rely on time since first capture as a reliable proxy for age. Since exposure increases with age, it is important that I account for simple age effects in looking for survivorship effects of exposure. So I have three issues, I think:

1) Multiple multi-states?: I believe that I need one multi-state variable to account for exposure status, but I think I also need another multi-state variable to then account for age. (So far as I understand it, that is the main way to account for individual co-variates that change over time?) Is that possible in MARK? Or is there another way that I can specify age-at-first-capture as an individual covariate and then have age calculated from there?

2) Unobservable points: I am only including individuals in my study for which we have bloodwork testing (and exposure results) at some point; however, we have MANY more capture points than bloodwork results. I am guessing I need 4 states: positive, negative, not caught, caught but untested (indicating at least that they are still alive). However, I will have a huge number of caught but untested datapoints...how much of a problem is this? I feel like those points are important for overall survivorship estimates for the individual...I may also consider individuals in the positive state at all points after they get an initial positive (as it is generally a permanent state), but I will still have a lot of caught but unknown status for individuals prior to or who never get a positive result...

3) Most long-term, collared study animals are caught every 3 years; yearlings are caught if they can, but only some end up being followed long-term. Many animals are collared, so we also have resight data on those. So, in theory, I can indicate that as a sort of "capture" even in years they were not actually captured (to indicate they were known to still be alive). However, other animals are not collared, so I would not have that data for them. So I guessing that I should ignore that data even if available, to have the same approach to all animals...? Again, for my purposes, I am less interested in the overall population and capture dynamics, and more in sorting out any survivorship effects of exposure...

Any help is greatly appreciated! My apologies if I missed any key answers to these questions in the documentation!

[ganghis]

Hi Leanna,
There are a number of interesting questions here - I'll do a little cherry picking.

1. In MARK, I think you'd need to specify each combination as a state. E.g. diseased-age 1, diseased-age 2 etc. However, it's clear that there is some uncertainty at age when animals are first captured, so you'd likely need to use a model that allows for this (traditional multistate models don't). One alternative would be to look into ESURGE (here you can also decompose the state transition processes into a number of steps - as with disease and age status - but this requires some knowledge of matrix algebra and some patience). Another would be to look at Bill Kendall's new robust design formulation for hidden Markov models that I think is in MARK now. Additionally, I remember seeing some new papers about unknown age analysis, but haven't looked at them. I suggest you do.

2. In general, it is not permissable to restrict analysis to a subset of individuals for which the covariate of interest is dependent on the number of times an animal is captured. In your case, this is because animals that are retained in the study will tend to be the ones that survive long enough to have bloodwork done. One solution is to delete all encounters for animals up until the time they are first tested. If there is heterogeneity in survival (some animals are just more fit than others), this may still bias things a bit, but including all observations leading up to the first time bloodwork is done will clearly bias things. Another solution is to include all animals in the analysis, but now you have to estimate the probability that an animal that is initially caught is in one of the disease states. This puts you back in the ESURGE or Bill Kendall model framework.

3. I think it would be possible to include collar data and recoveries, but this really depends how complicated you want to get. If you know the theory and can program up your own models, anything is really possible here. The issue with canned software is that it's hard to get to far out of the box. You might be able to finagle a way to get ESURGE to handle these with a complicated detection model. I'd suggest starting simple and slowly adding capabilities - maybe focus on the disease/age state process with captures only and then see if you can figure out how to incrementally include band recoveries and collared animals.

Sounds like a neat study.

Regards, Paul

[leanna_jones]

Thanks so much for your reply, Paul! I really appreciate it. I am afraid that matrix algebra is probably a bit beyond my capabilities, unless I find a co-author wanting to run the analyses. For most individuals we know age at first capture within 1-2 years certainty, so hopefully that would not be a major issue, though I will look into the unknown age analysis. However, if I combine disease and age states, is there any way to later tease apart the age and disease effects? Thanks again for your help!

[ganghis]

Hi Leanna,
Yes, you could structure the design matrix so 2 year olds always turn into 3 year olds the next year, etc. so the only transition parameters you're really estimating are for disease state (these can be constrained however you want them to (age independent transition rates, age dependent transition rates, etc. - anything you can do with a design matrix anyway).

Afraid I don't have time to help out with any extra analyses at the moment, but good luck!