Delta method

questions concerning analysis/theory using program MARK

Delta method

Postby wijw » Wed Mar 24, 2021 5:36 pm

Hello,

After using the POPAN formulation, I got the following gross population estimates for my study population. I want to estimate population size at the end of the sample period. Let's say I got the following summary.

To get the population size I added the N*-hat from group 1 to 6. But what about the confidence intervals? How can I calculate that? Any help is appreciated.

95% Confidence Interval
Grp. Occ. N*-hat Standard Error Lower Upper
---- ---- -------------- -------------- -------------- --------------
1 0 21.000000 0.0000000 21.000000 21.000000
2 0 49.811024 5.2074010 40.604884 61.104427
3 0 67.798339 7.0878514 55.267758 83.169915
4 0 67.798339 7.0878514 55.267758 83.169915
5 0 65.031060 6.7985514 53.011931 79.775225
6 0 78.867455 8.2450516 64.291066 96.748677
wijw
 
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Re: Delta method

Postby cooch » Wed Mar 24, 2021 6:21 pm

wijw wrote:To get the population size I added the N*-hat from group 1 to 6. But what about the confidence intervals? How can I calculate that? Any help is appreciated.

95% Confidence Interval
Grp. Occ. N*-hat Standard Error Lower Upper
---- ---- -------------- -------------- -------------- --------------
1 0 21.000000 0.0000000 21.000000 21.000000
2 0 49.811024 5.2074010 40.604884 61.104427
3 0 67.798339 7.0878514 55.267758 83.169915
4 0 67.798339 7.0878514 55.267758 83.169915
5 0 65.031060 6.7985514 53.011931 79.775225
6 0 78.867455 8.2450516 64.291066 96.748677


You titled your question 'Delta method', so you already know the answer. You apply the Delta method to the sum of non-independent variable (taking the covariance among samples into account). Using the Delta method to calculate the variance of a sum, or difference is easy, and described in lots of places (for example, Chapter 6, sidebar beginning on p. 72 -- simply replace the word 'difference' with 'sum', and go from there.) Note that estimate 1 (above) is a garbage estimate (SE = 0 indicates its meaningless) -- you shouldn't include it.

And, I'll urge you to think carefully about N*-hat means, and whether simply adding up N(2) -> N(6) gives you what I suspect you think/hope it does.
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Re: Delta method

Postby wijw » Wed Mar 24, 2021 10:14 pm

Thank you for the quick reply.

I believe the concern is about the individuals that could have entered and left the population in between occasions..
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Re: Delta method

Postby keevilm » Thu Mar 25, 2021 8:24 am

I think that you don't want N* if you are not modeling hidden recruitment of individuals that die in their birth interval and if you don't intend to count individuals that a not present simultaneously in the population. The original motivation for N* is to estimate the total size of a salmon run when doing multiple intermittent sampling within a single spawning season. There's not too many other situations where the N* (or the net super population) is biologically meaningful. Maybe think of the superpopulation as the soul population because it includes every 'soul' that was ever alive over the length of the study (but they are never all alive at the same time) so that for any one occasion it includes individuals that are alive, individuals that are dead, and all the individuals waiting to get born. I expect that it is more relevant to consider the population size as the number that are alive at the same time, and since JS is an open population model, that quantity is not the same from one occasion to another so it only makes sense to consider one occasion at a time, and these are the N-hat values.
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