Closed-population c vs. CJS p

questions concerning analysis/theory using program MARK

Closed-population c vs. CJS p

Postby stshroye » Tue Jun 28, 2022 10:21 am

If a population were truly closed (phi = 1), is there any reason why the estimates of c from models Mt or Mtb would not be the same as the estimates of p from the constrained CJS model phi(.) p(t)? Or, estimates from models M0 or Mb vs. phi(.) p(.)? Thanks.
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Re: Closed-population c vs. CJS p

Postby jhines » Tue Jun 28, 2022 1:14 pm

If Phi=1, then p from M0 should equal p from phi(.)p(.). The estimate of phi should be 1.0. If you fix phi=1, then even if phi isn't really 1.0, the two models should give the same result for p.

Model Mt should give the same p's as phi(.)p(t), with phi constrained to 1.

This assumes you have enough data (enough occasions and enough captures) so the parameters are identifiable.
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Re: Closed-population c vs. CJS p

Postby stshroye » Tue Jun 28, 2022 3:51 pm

jhines wrote:If Phi=1, then p from M0 should equal p from phi(.)p(.). The estimate of phi should be 1.0. If you fix phi=1, then even if phi isn't really 1.0, the two models should give the same result for p.

Model Mt should give the same p's as phi(.)p(t), with phi constrained to 1.

This assumes you have enough data (enough occasions and enough captures) so the parameters are identifiable.


What about closed-population models Mb and Mtb? Shouldn't those also agree with the corresponding CJS models with phi constrained to 1, since the CJS models are based only on recaptures?

Assuming the data are adequate, is it correct to interpret any substantial differences in estimated recapture probabilities between closed and CJS models (with phi unconstrained) as evidence for violations of closure? Obviously if estimated phi is substantially < 1 then there is evidence for violation of closure, but I'm thinking that comparisons of estimated recapture probabilities between closed and open models of the same data could be informative as well.
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Re: Closed-population c vs. CJS p

Postby cooch » Tue Jun 28, 2022 4:42 pm

stshroye wrote:What about closed-population models Mb and Mtb? Shouldn't those also agree with the corresponding CJS models with phi constrained to 1, since the CJS models are based only on recaptures?


Model M(b) and M(tb) are both parameterized in terms of 2 structural parameters (without and with time variation, respectively) - p (initial encounter) and c (conditional, subsequent) encounter. There is no structural CJS equivalent.

Assuming the data are adequate, is it correct to interpret any substantial differences in estimated recapture probabilities between closed and CJS models (with phi unconstrained) as evidence for violations of closure? Obviously if estimated phi is substantially < 1 then there is evidence for violation of closure, but I'm thinking that comparisons of estimated recapture probabilities between closed and open models of the same data could be informative as well.


Interesting idea, but not one that will work. If you're simply looking for evidence of lack of closure, then perhaps the most clever/straightforward approach is to use the Pradel model approach (survival and recruitment, applied to putative closed encounter data). This approach (invented and re-invented on a couple of occasions), is introduced briefly in the -sidebar- on p. 6 of chapter 14 of the MARK book.
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Re: Closed-population c vs. CJS p

Postby stshroye » Tue Jun 28, 2022 6:07 pm

cooch wrote:
stshroye wrote:What about closed-population models Mb and Mtb? Shouldn't those also agree with the corresponding CJS models with phi constrained to 1, since the CJS models are based only on recaptures?


Model M(b) and M(tb) are both parameterized in terms of 2 structural parameters (without and with time variation, respectively) - p (initial encounter) and c (conditional, subsequent) encounter. There is no structural CJS equivalent.

Assuming the data are adequate, is it correct to interpret any substantial differences in estimated recapture probabilities between closed and CJS models (with phi unconstrained) as evidence for violations of closure? Obviously if estimated phi is substantially < 1 then there is evidence for violation of closure, but I'm thinking that comparisons of estimated recapture probabilities between closed and open models of the same data could be informative as well.


Interesting idea, but not one that will work. If you're simply looking for evidence of lack of closure, then perhaps the most clever/straightforward approach is to use the Pradel model approach (survival and recruitment, applied to putative closed encounter data). This approach (invented and re-invented on a couple of occasions), is introduced briefly in the -sidebar- on p. 6 of chapter 14 of the MARK book.


Thanks for the input. I am aware of the Pradel model approach and I am using that, but I'm also trying to understand and interpret differences in estimates of closed-population c and CJS p for the same data. I know there is no equivalent of closed-population p in a CJS model, but since p in a CJS model is actually the recapture probability, I still don't understand why c in model M(b) or M(bt) is fundamentally different than p in CJS models phi(=1) p(.) or phi(=1) p(t). I also don't understand why differences in estimated recapture probabilities between closed and open models are uninformative about violations of closure, since emigration would reduce probabilities of recapture.
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Re: Closed-population c vs. CJS p

Postby cooch » Tue Jun 28, 2022 7:08 pm

I still don't understand why c in model M(b) or M(bt) is fundamentally different than p in CJS models phi(=1) p(.) or phi(=1) p(t).,,


Simple -- c is the probability of subsequent encounter, conditional on a first encounter. There is an equivalence in CJS for closed models where there is no distinction between first and subsequent encounters -- which is why you can write M(t) or M(0) as a CJS model with phi=1. Buts, its also why you can't structure a CJS equivalent of M(b), for example, where you need to make a distinction between first and subsequent.

And, don't forget, there is no p(1) in a CJS model, while there is in a closed model.
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Re: Closed-population c vs. CJS p

Postby stshroye » Wed Jun 29, 2022 9:22 am

cooch wrote:
I still don't understand why c in model M(b) or M(bt) is fundamentally different than p in CJS models phi(=1) p(.) or phi(=1) p(t).,,


Simple -- c is the probability of subsequent encounter, conditional on a first encounter. There is an equivalence in CJS for closed models where there is no distinction between first and subsequent encounters -- which is why you can write M(t) or M(0) as a CJS model with phi=1. Buts, its also why you can't structure a CJS equivalent of M(b), for example, where you need to make a distinction between first and subsequent.

And, don't forget, there is no p(1) in a CJS model, while there is in a closed model.


In a CJS model, p is the recapture probability that is conditional on a first encounter, because animals have been captured and released. Any trap response has already occurred. So how is this different from c in M(b) or M(tb)?
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Re: Closed-population c vs. CJS p

Postby cooch » Wed Jun 29, 2022 9:46 am

In a CJS model, p is the recapture probability that is conditional on a first encounter, because animals have been captured and released. Any trap response has already occurred. So how is this different from c in M(b) or M(tb)?


Nope -- but you can keep spending you time trying to find some way to twist CJS into being equivalent to closed estimators (if it were possible, it would have been done long ago).
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Re: Closed-population c vs. CJS p

Postby stshroye » Wed Jun 29, 2022 4:13 pm

cooch wrote:
In a CJS model, p is the recapture probability that is conditional on a first encounter, because animals have been captured and released. Any trap response has already occurred. So how is this different from c in M(b) or M(tb)?


Nope -- but you can keep spending you time trying to find some way to twist CJS into being equivalent to closed estimators (if it were possible, it would have been done long ago).


I'm not trying to "twist CJS into being equivalent to closed estimators," I'm just trying to fully understand why closed-population c is from M(b) or M(tb) is different from CJS p. I have not been able to find the answer elsewhere, which is why I am asking on this forum. My last question still has not been answered.
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Re: Closed-population c vs. CJS p

Postby cooch » Wed Jun 29, 2022 4:21 pm

I'm not trying to "twist CJS into being equivalent to closed estimators," I'm just trying to fully understand why closed-population c is from M(b) or M(tb) is different from CJS p. I have not been able to find the answer elsewhere, which is why I am asking on this forum. My last question still has not been answered.


Because c is estimable only given information (i) conditional on the first encounter, given that (ii) leading 0's in the encounter history are informative, where in the CJS likelihood, leading 0's are not. In a closed population, the leading 0's in a history (say, '001001') mean something. Why? Because in a closed population, the encounter of the individual on occasion 3 means they *were* there on occasions 1 and 2, and simply missed. On the other hand, in a CJS modeling context, the initial 0's are entirely ambiguous -- could have been there and missed, or wasn't there at all and entered between occasions 2 and 3.

In short, closed means inference based on entire encounter history, whereas open isn't. This is what allows you to pull apart p and c in the former.
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