Multistate model to account for many dropped transmitters

[kquail]

I have radio-transmitter tracking data for chicks until 35 days old (at which point they are considered independent). Each chick was not tracked each day (typically 2-3 days between checks, but this was not consistent within or across individuals). Even on days where chicks were tracked, they were not always found (including individuals that were found alive again later), thus p < 1. There are also several circumstances (>50% of individuals) where the transmitter was dropped before the chick reached 35 days. I have one grouping variable (2 groups) and one individual covariate (mass). I am looking for an approach that can help me take these dropped tags into account.

My current plan is to analyze this data using a multistate model with states Alive (A), known dead (D), and unknown (U)--the last state being for chicks with dropped transmitters, with p=0 and transition probabilities to other states = 0. My questions are about the general analysis, but I plan to use RMark in case that is relevant.

1. Should birds with dropped transmitters just be given a 0 in the encounter history when they dropped the transmitter and for the rest of the study? Or does it make sense for 'dropped transmitter' be its own state as I described above? If it's its own state, is it unobservable or observable? I know if I find the transmitter and it's not on the bird....but of course I can't actually observe the bird in that case...

2. How should I treat the missing occasions in the encounter history (i.e. days the chick was not tracked)? In my data the missing tracking occasions differ across many individuals (I have them aligned by age rather than date, such that at time 1 all chicks are 12 days old regardless of hatch date). Should I just code these missing occasions as 0 or would it make sense to create an additional state for missing occasions and set p=0? My thought is that this creates an issue because it will add additional parameters for that new state that need not be estimated.

3. In examples such as Devineau et al 2014 (DOI: 10.1002/jwmg.660), birds are censored with "." after the initially being marked as 'newly dead' (they are newly dead for just the sampling occasion that the death was discovered). Is this appropriate for my case or is it better to mark them as state D for the remainder of the study?

Any suggestions would be greatly appreciated. I am still new to survival analysis and would welcome any suggestions on reading materials.

[jlaake]

No one has chimed in here, so I'll provide some ideas. While I was searching about multistate models on the web, I saw a paper that purported that a large transmitter loss rate can lead to bias in demographic rates, so you may want to look into that. I didn't read it so it could be referring to transmitter failure rather than a tag that falls off but continues to transmit. Either way, if a transmitter falls off I guess you are assuming that the bird could be alive or dead. Obviously the best solution would be to reduce the loss rate.

Beyond that I think what you are proposing seems reasonable. It is not impossible to set p=0 by individual. It would be very cumbersome to do in the MARK interface but with R, you could easily code it to fix the appropriate p parameters by individual quail but you would want to use quail ID as a grouping variable so you get design data for each quail. Then you could use a dataframe with quail id and times that are missing and run through that with the code to set fix=0 for p for that id and for those times.

With regard to your question about state D and U and what happens afterward, it would seem the easiest thing to do would be to have all 0's afterward and use -1 as the frequency which is a loss on capture and I believe that removes any further use of data beyond the last encounter (ie last D or U). But maybe Gary White or Bill Kendall can chime in here to say if that works or not. On the occasion that they are found dead or a lost transmitter is found, you would want to record D / U for that occasion and then 0 afterward.
Both D and U will be absorbing states with Psi=0 for movement to any other state. Psi for A to D is your mortality rate.

Since you are locating radios, it would seem reasonable to set p to be constant across strata (A,D,U) unless a dead quail or lost transmitter is harder to detect because it doesn't move. However, I don't think there will be sufficient data in D or U to estimate a separate p. Presumably these lost transmitters are still working and just fell off. If transmitters stop working you are kind of screwed.

[kquail]

Thanks a bunch for your suggestions, Jeff.
I ran a multistate model, but ended up only using alive and dead states. I figured I didn't need the U state if I am going to censor those individuals anyway. I censored, using the freq column, both dead and dropped-transmitter individuals starting from the next sampling occasion after found dead or with a dropped transmitter.

I was able to individually fix p=0 for the specific occasions where each individual was not searched for. I have a follow-up question though: in RMark, does the occasion value in the design matrix for p actually equal i or i+1? More specifically, if my encounter history is "A.AAAAAA.000" where the dots were occasions where the individual was not searched for, would I set p=0 for occasions 2 and 9 or for 1 and 8? I believe it's the latter because there are only 11 occasions for p in the design matrix (despite 12 sampling occasions total in ch) and we can't calculate p for the first encounter of an individual?

[jlaake]

I ran a multistate model, but ended up only using alive and dead states. I figured I didn't need the U state if I am going to censor those individuals anyway. I censored, using the freq column, both dead and dropped-transmitter individuals starting from the next sampling occasion after found dead or with a dropped transmitter.

But aren't you interested in the rate of loss (transition to U from A)? Also isn't it possible for it to transition to U before you have seen it since p<1. How are you censoring it as a U? At the last A? That wouldn't be correct. All you really know is that it was in U when you find the transmitter. It could have happened at a prior occasion and you didn't find it or it could have happened during a gap when you weren't searching for them.

I was able to individually fix p=0 for the specific occasions where each individual was not searched for. I have a follow-up question though: in RMark, does the occasion value in the design matrix for p actually equal i or i+1? More specifically, if my encounter history is "A.AAAAAA.000" where the dots were occasions where the individual was not searched for, would I set p=0 for occasions 2 and 9 or for 1 and 8? I believe it's the latter because there are only 11 occasions for p in the design matrix (despite 12 sampling occasions total in ch) and we can't calculate p for the first encounter of an individual?

You could have answered your question by looking at the design data for p. Design data aren't fixed in concrete. I created the default design data for my thinking which isn't always the best, nor may you think the way I do. So, you can change them. Also, design data times/occ will vary depending on the parameter. For p they relate to the time/occasion for the capture event. I created occ (occasion) because time can be a non-integer value and may be a year etc.

Here is the first few records of the design data for p for the mstrata example

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> ddl$p
   par.index model.index group cohort age time occ occ.cohort stratum Cohort Age
1          1          19     1      1   1    2   1          1       A      0   1
2          2          20     1      1   2    3   2          1       A      0   2
3          3          21     1      1   3    4   3          1       A      0   3
4          4          22     1      2   1    3   2          2       A      1   1
5          5          23     1      2   2    4   3          2       A      1   2

As you can see, the first p is for time=2 which I labelled occ=1 (being the first occasion for a p but it is actually the second capture occasion). So you can use time or occ as you see fit. But recognize that if you change begin.time then the time value in the design data changes but occ is fixed unless you change it. You may decide to add 1 to each occ value so it refers to the capture occasion. It is a numeric value so that would work fine. Recognize that time is a factor variable. If you are ever concerned use str() function to see the types (eg, str(ddl$p)).

Now for S the design data differ because it relates to the interval between occasions and I use the first occasion in the interval.

Code: Select all

> ddl$S
   par.index model.index group cohort age time occ occ.cohort stratum Cohort Age
1          1           1     1      1   0    1   1          1       A      0   0
2          2           2     1      1   1    2   2          1       A      0   1
3          3           3     1      1   2    3   3          1       A      0   2
4          4           4     1      2   0    2   2          2       A      1   0


For S time and occ have the same values.

Whenever you have a question like this, look at the design data and think about the model. MS models are based on CJS concepts which only model the events for occasions after the first "capture/release occasion". In comparison JS models use the entire capture history including 0's prior to first observation. It is important to understand the model you are using. Read the section in Cooch and White for your model (eg Multistrata).

If you let me gripe here for a second. When I write posts like this, I feel like I'm rewriting parts of Appendix C in Cooch and White and the Workshop notes for RMark where issues like this are discussed. While it is not perfect, I spent a lot of time writing the documentation, so please take some time and read it.

[kquail]

Regarding the inclusion of the unknown (U) state, I had convinced myself (since writing the original post) that it wouldn't add anything important to the model that wouldn't be accounted for by just marking an individual as 0 or excluding it by censoring. I don't think I want to estimate the rate of loss (A to U) because I'm only really interested in calculating survival as accurately as possible, despite the transmitter loss. I also thought that it might just be making the model unnecessarily complex and adding parameters that aren't super informative. (Not to mention other questions it brings up like what should p(U) be? Same as p(A) and p(D)? Allowed to vary separately on its own?)

You are definitely right, though, that I might be censoring individuals too early if I censor them at last A. Part of the issue is that sometimes the transmitter was just never found, so there could have been 3 occasions in a row where the individual was searched for but not found before it was considered a loss. It seems arbitrary in that case where to begin the censor anyway. I can see two potential solutions, one being the addition of the U state, where the placement of the U isn't necessarily super accurate for each bird, but it at least indicates where to begin the censor. The other option being to use the "." to censor the ends of encounter histories instead of 0s and the freq column (so that 0s before the censor still represent occasions the individual was searched for and not seen). Is there some reason why the latter approach would produce less accurate estimates or be less appropriate?

Thank you also for answering my question about the design matrix. I appreciate your explanation and I will look back over the Cooch and White book again. I just wanted to make sure that I was interpreting what I saw in the design matrix correctly (I had change the begin.time from the default, but really that shouldn't have changed the interpretation). Thanks so much for your time. I'm sure everyone on this forum appreciates that you are still active on here.

[jlaake]

Regarding the inclusion of the unknown (U) state, I had convinced myself (since writing the original post) that it wouldn't add anything important to the model that wouldn't be accounted for by just marking an individual as 0 or excluding it by censoring. I don't think I want to estimate the rate of loss (A to U) because I'm only really interested in calculating survival as accurately as possible, despite the transmitter loss. I also thought that it might just be making the model unnecessarily complex and adding parameters that aren't super informative. (Not to mention other questions it brings up like what should p(U) be? Same as p(A) and p(D)? Allowed to vary separately on its own?)

I realize your interest is survival but you need to handle the loss correctly to do so. And if you might do this again, I'd think you might want to understand when loss is occurring.

You are definitely right, though, that I might be censoring individuals too early if I censor them at last A. Part of the issue is that sometimes the transmitter was just never found, so there could have been 3 occasions in a row where the individual was searched for but not found before it was considered a loss. It seems arbitrary in that case where to begin the censor anyway. I can see two potential solutions, one being the addition of the U state, where the placement of the U isn't necessarily super accurate for each bird, but it at least indicates where to begin the censor. The other option being to use the "." to censor the ends of encounter histories instead of 0s and the freq column (so that 0s before the censor still represent occasions the individual was searched for and not seen). Is there some reason why the latter approach would produce less accurate estimates or be less appropriate?

This is a question for Gary because I don't completely understand what he does with dots in MARK. I don't use them. But given your description it seems like your loss is more than just the transmitter off the bird and the transmitters may be failing as well if you can't find them.

Thank you also for answering my question about the design matrix. I appreciate your explanation and I will look back over the Cooch and White book again.

So maybe you don't understand the terminology or I'm misunderstanding what you are saying but your question about occasions was about the design data and not the design matrix. The design data are data that are associated with the PIMS. A formula is applied to the design data to get a design matrix.

Also Cooch and White will help you understand the models but you really need to read the RMark workshop notes to understand how RMark works to build models in MARK. If you have not found it, the link for the RMark documentation appears the first time in each session that you enter library(RMark).

[kquail]

I realize your interest is survival but you need to handle the loss correctly to do so. And if you might do this again, I'd think you might want to understand when loss is occurring.

Is the only concern, if not using the U state, the ability to keep informative 0s before censoring (which I think can be solved with "dot" censoring, see below)? Or is there some other reason why having the U state helps to more accurately estimate psi(A to D)? My thought is that it would be more informative to the model if I censor individuals after a A/0 (given I don't include the additional state) rather than after a U, because at least it would tell the model the most recent state.

This study was actually conducted back in 2011/2012 and the method for attaching transmitters has been improved since then, so modeling transmitter failure from the old method won't matter as much at this point.

This is a question for Gary because I don't completely understand what he does with dots in MARK.

In case anyone reading this wants to know: Based on Cooch and White (Chapter 4, pp29-34), it looks like a "dot" is automatically treated as an occasion where p=0 without having to set the detection parameter manually. The example in the book mentioned needing to then set survival equal for the interval before and after the "dot," but this is a case where all individuals have the same missing occasion. In my case, censoring with the "dot" at the end of encounter histories would vary by individual. Based on the book example, the estimates are not affected, only the SEs, so at the very least for me it would be important to keep in mind that parameter estimates (if they vary with time) for the last several intervals will have biased/meaningless SEs. Although someone can chime in if they believe otherwise.

So maybe you don't understand the terminology or I'm misunderstanding what you are saying but your question about occasions was about the design data and not the design matrix.

Oops, I did use the wrong term.

[jlaake]

Whatever you decide is fine with me but since p<1 then it is possible that it is still alive after the last A and thus alive. But what you assume about A to U transition probability will affect that. I was presuming that this was transmitter loss and you could eventually find it but it sounds like it is transmitter failure as well.

I'm confused regarding the dots because you initially referenced a post which said that dots would not work and you needed to set p=0.

[kquail]

you initially referenced a post which said that dots would not work and you needed to set p=0.

I wonder if you mean something that I have since edited out of my very first post? (Perhaps I could have added it as an update instead of removing it, but it was before anyone had responded.) I had originally asked about using the dots as individual covariates for missing occasions on an individual-by-individual basis, but I later found the sidebar in Cooch and White explaining why not to do that. What I am talking about now, though, is to use the dots as the method of censoring the ends of encounter histories.

Thanks again for your help, Jeff!

[jlaake]

Ok. My mistake.

But I don't know if dots will work as you propose for censoring. I would have thought you also need to set S to 0 but then why wouldn't you use negative frequency. Like I said I don't use dots so have never looked into them. Gary or Evan or Bill will have to respond. Sorry.