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.
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?
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.
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.
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).,,
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)?
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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