Estimating total population size using robust design

questions concerning analysis/theory using program MARK

Estimating total population size using robust design

Postby Will » Mon May 30, 2011 10:58 pm

I am using the closed robust design with Huggins closed capture estimator to estimate population size of whales at a breeding area over 5 years. We know that not all the whales come to the breeding area each year, hence using the robust design to estimate temporary emigration probabilities. As I understand it, N-hat is the population size in the study area in each primary sampling period. I am also interested in the total population size, i.e. how many whales are likely to be outside the study area each year as well as those available to be sampled. In the Kendall et al. paper from 1997 there seems to be a way of estimating the total population size under random emigration:
N0i = Ni/(1-gamma)
My best model has Markovian temporary emigration for one group and random for another. Is there a way to estimate the total population size for the group with Markovian temorary emigration?
Many thanks.
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Re: Estimating total population size using robust design

Postby cooch » Tue May 31, 2011 6:42 am

Will wrote:I am using the closed robust design with Huggins closed capture estimator to estimate population size of whales at a breeding area over 5 years. We know that not all the whales come to the breeding area each year, hence using the robust design to estimate temporary emigration probabilities. As I understand it, N-hat is the population size in the study area in each primary sampling period. I am also interested in the total population size, i.e. how many whales are likely to be outside the study area each year as well as those available to be sampled. In the Kendall et al. paper from 1997 there seems to be a way of estimating the total population size under random emigration:
N0i = Ni/(1-gamma)
My best model has Markovian temporary emigration for one group and random for another. Is there a way to estimate the total population size for the group with Markovian temorary emigration?
Many thanks.


There are ways you can get there from here, but only under strong assumptions. Moreover, even the estimator for random emigration makes strong assumptions about the effective closure about the super-population (i.e, that the population consists entirely of 2 states: available for detection, and not available for detection, but not permanently emigrating from the super-population). It isn't hard to come up with biologically plausible scenarios where estimates of the super-population size are pretty silly.

As part of your thinking, might be worth also having a look at the Jolly-Seber chapter (13). Carl spends a fair bit of time on the concept of 'super-population'.
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Re: Estimating total population size using robust design

Postby murray.efford » Sun Sep 11, 2011 7:51 pm

After talking with Will, I think there is some point in reviving this thread. Evan suggests strong assumptions are needed, but I think strong assumptions may be justified here. If the Markovian migration parameters are constant and superpopulation size is constant, isn't the equilibrium probability of being available given by
Pr(available) = Pr(entering available subpopn) / (Pr(entering available subpop) + Pr(leaving available subpop))
or (1 - GammaPrime) / (1 - GammaPrime + GammaDoublePrime) (using RMark parameter names)?
That would lead to a super population estimate of Ni-hat/Pr(available). I haven't thought about how survival comes into this, but at a pinch Will might be able to assume 100% survival. Any comments from those who know about the robust design?

Also, I see a fundamental distinction between the temporal and abstract 'superpopulation' of Schwarz & Arnason and POPAN, and the more biological and spatial superpopulation of Kendall (1999) and the RD implementation in MARK. The two seem interchangeable only in peculiar conditions. Am I missing something?

Murray
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Re: superpopulation and Markovian temporary emigration

Postby Eurycea » Tue Dec 05, 2017 4:44 pm

I've been thinking about this problem and found this old thread. The problem is of trying to estimate a superpopulation size when Markovian emigration is present for the RD model of Kendall, Nichols, and Hines 1997. Cooch stated that it is possible to get there but I have yet to see anyone explain how. I couldn't find anything doing some lit searches. The formula by Murray Efford is the only attempt I've seen. So I thought I'd post my thought process about it here to see if anyone can point out the flaws or what I'm missing exactly.
I’ll distinguish between two subpopulations here, which sum to total the superpopulation, N^0. One is the “local” population, which is N. The other is the portion of the superpopulation unavailable for capture, N^0 - N. The superpopulation during the following period is the sum of individuals that entered the local population (from somewhere in the superpopulation) (X_t), individuals that were unavailable (Y_t), and individuals that were recruited into the local (B^X_t) and unavailable B^Y_t populations.
That is,
N_t^0 = X_t + Y_t + B^X_t + B^Y_t,
where

X_t = (1-\gamma_t`)(N^0_t_-_1 - N_t_-_1)(\phi_t) + (1-\gamma_t")(N_t_-_1)(\phi_t)

and

Y_t = (\gamma_t`)(N^0_t_-_1 - N_t_-_1)(\phi_t) + (\gamma_t")(N_t_-_1)(\phi_t).

This leaves us with the problem of how to estimate recruitment (new individuals that were not in the superpopulation during the previous time period). Recruitment is not modeled directly by the Kendall et al. 1997 method. However, this should just be the number of individuals of the sub population minus the contribution from the previous subpopulation

(B^X_t = N_t - X_t).

If we calculate a superpopulation size assuming random temporary emigration, the following relationship appears to hold: \frac{X_t}{Y_t} = \frac{B^X_t}{B^Y_t} = \frac{(1-\gamma)}{\gamma}. The consequence of this, I think, is that the ratio of recruitment into the superpopulation is assumed to be equivalent to the ratio of recruitment into the local population. This may be a strong assumption, but if we make it for the Markovian case, we can solve for B^Y_t. However, we need to assume what the superpopulation size was to have a starting point (N^0_t_-_1). Would this work for periods where gamma' happens to equal gamma" and we can use the N/(1-gamma) estimator? Perhaps this would work using something like Murray Efford's solution?

If the Markovian migration parameters are constant and superpopulation size is constant, isn't the equilibrium probability of being available given by
Pr(available) = Pr(entering available subpopn) / (Pr(entering available subpop) + Pr(leaving available subpop))
or (1 - GammaPrime) / (1 - GammaPrime + GammaDoublePrime) (using RMark parameter names)?
That would lead to a super population estimate of Ni-hat/Pr(available).


What is odd is that in screwing around with some estimates from a model of real data, changing the N^0_t_-_1 around doesn't seem to affect the subsequent estimates by much at all. It may not be even appropriate to change a single value to see how it affects the rest because I would guess that there is some interdependence among the model parameter estimates.

In any event, this is kind of where I'm at with it and would love to see if anyone (particularly the experts) has any additional insight or could even spell out a more general approach to this problem when you have time-varying Markovian temporary emigration and want to get at an estimate of the superpopulation.
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Re: Estimating total population size using robust design

Postby Eurycea » Mon Dec 18, 2017 6:46 pm

Well it looks like they discuss some of this indirectly in this paper, with a hierarchical Bayesian model:

https://www.frontiersin.org/articles/10 ... 00025/full

When they condition on the full capture histories, they model a recruitment process and provide three strategies, discussed in more detail in the Appendix:

https://www.frontiersin.org/articles/fi ... f/2/177816
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