Non-estimability

[brp]

I am working with the Huggins Closed Captures data type with 4 occasions. I am interested in estimating population size.

MARK shows the fully time-dependent model {p(t) c(t)} as having 6 parameters (but it should be 7) and the standard errors of the 4 capture rates (p) are all zero. Are any of the estimates of p or c correct? Or is it only the last p that is not estimable (i.e. not correct)?

Thanks for your help,
Becky

[gwhite]

There is not parameter identifiability for the model {p(t) c(t)}. You definitely need to go read the chapter on closed models in the Gentle Introduction.

Gary

[cooch]

I am working with the Huggins Closed Captures data type with 4 occasions. I am interested in estimating population size.

MARK shows the fully time-dependent model {p(t) c(t)} as having 6 parameters (but it should be 7) and the standard errors of the 4 capture rates (p) are all zero. Are any of the estimates of p or c correct? Or is it only the last p that is not estimable (i.e. not correct)?

Thanks for your help,
Becky

You need to thoroughly read Chapter 14 (the chapter on closed abundance estimation) -- especially the sections on the required constraints for time-dependent models. You need to place some sort of constraint on the final estimate of p (see, for example, section 14.3.1).

[brp]

I ran the predefined model {p(t) c(t) PIM}. The number of parameters was 6 (not 7). The last p was not estimated as 1 and the none of the N-hats were estimated as M_{t+1}.

In Section 14.3.1, it states:

"... the last p parameter is not identifiable unless a constraint is imposed. So, for example, in model M_t, the constraint of p_i = c_i is imposed, providing an estimate of the last p from the last c."

Does that sentence mean set p=c or just p_4=c_4?

Thanks,
Becky

[cooch]

I ran the predefined model {p(t) c(t) PIM}.

Don't -- those models assume you know what you're doing at a high level. They probably should be dropped from MARK (IMO), but they do save time when applied correctly. Doing so for closed models is tricky. So, don't (in fact, I *never* recommend pre-defined models for much of anything).

In Section 14.3.1, it states:

"... the last p parameter is not identifiable unless a constraint is imposed. So, for example, in model M_t, the constraint of p_i = c_i is imposed, providing an estimate of the last p from the last c."

Does that sentence mean set p=c or just p_4=c_4?

Thanks,
Becky

Either would suffice.

[Eurycea]

Is there any generally accepted advice on exactly how to constrain the last estimate of p? Is making p_4=c_4 better than fixing p, from say, an average of p's from a p(t)=c(t) model? It seems like people generally avoid the full p(t) c(t) model in the first place, is that correct?

[cooch]

Is there any generally accepted advice on exactly how to constrain the last estimate of p? Is making p_4=c_4 better than fixing p, from say, an average of p's from a p(t)=c(t) model? It seems like people generally avoid the full p(t) c(t) model in the first place, is that correct?

Better, in terms of what? The point of the exercise (with closed abundance estimation) is to get as good an estimate of abundance as is possible, given the data. Interest is rarely focussed on interpretation of the model structure for p and/or c.

You'll likely achieve this larger objective (i.e., estimating N) by model averaging, so whether you run p(t)=c(t), or a model constraining the last p and c only, generally makes precious little difference in practice. If p(t)=c(t) is biologically more plausible than a model that simple constrains the final p and c values, then include the former, and not the latter.

[brp]

I am a novice but I just read the chapter 14 (closed captures) and it appears that the additive offset approach is recommended. See the M_tb suggestion in section 14.3.1. The design matrix using the additive offset approach is the last figure in section 14.5.1.

Becky

[Eurycea]

Is there any generally accepted advice on exactly how to constrain the last estimate of p? Is making p_4=c_4 better than fixing p, from say, an average of p's from a p(t)=c(t) model? It seems like people generally avoid the full p(t) c(t) model in the first place, is that correct?

Better, in terms of what? The point of the exercise (with closed abundance estimation) is to get as good an estimate of abundance as is possible, given the data. Interest is rarely focussed on interpretation of the model structure for p and/or c.

You'll likely achieve this larger objective (i.e., estimating N) by model averaging, so whether you run p(t)=c(t), or a model constraining the last p and c only, generally makes precious little difference in practice. If p(t)=c(t) is biologically more plausible than a model that simple constrains the final p and c values, then include the former, and not the latter.

Say you didn't care about N, for example, and were concerned more with testing whether p(t)=c(t) was a better model than p(t) c(t), having no prior biological information to support either model.

[cooch]

Say you didn't care about N, for example, and were concerned more with testing whether p(t)=c(t) was a better model than p(t) c(t), having no prior biological information to support either model.

Then simply fit those models. At least 2 suggestions were made earlier in this thread -- either put some sort of constraint on terminal p and c, or perhaps try an additive model.

As an aside, showing some time-variation in say p or t, relative to a model where p(t)=c(t), is of marginal interest (as are time-dependent models in general). The question of interest is *why* do things vary over time -- not that they do. Of course they vary over time. They may not vary in a biologically significant way, or in a way detectable given the limits of a particular data set, but they do vary. So, suppose you find support for a temporally varying model versus a more constrained parameter reduced model. The next question would be "so what?". By itself, such a demonstration is of little value. As mentioned, it is *why* things vary over time that is of interest. So, a Huggins model with estimates constrained by relevant covariates (contrasted against other models) might be a good start to getting at the *interesting* question.

As a further aside, I'd have to work pretty hard to imagine why there would be much interest in whether or not probability of initial capture (p) differed from probability of subsequent capture, conditional on having been previous captured (c). Suppose they did (say, if individuals were trap happy following initial capture). What would you use this information for? I could see some uses in terms of modifying/improving/understanding sampling structures, but beyond that. I can't even recall a publication using closed abundance estimators where focus was in p/c, and not on N. Of course, only JDN has read all the papers, so my not having seen one should be taken with several grains of salt.

Further (and here is where it can get tricky), I can imagine Huggins models where you try to improve precision of modeling of p and c by putting constraints on N (the modeling isn't tricky in terms of mechanics, but thinking of plausible constraints might be -- there are logical relationships such that imposing a constraint on N will impose some potentially unintended constraints on the other parameters; there is an analogous situation with Pradel models). Still can't think of anyone who has ever tried such an approach (basically, since I don't know of an example where the focus wasn't on estimating N).