POPAN N lower than total animals included in dataset

[jlaake]

One way is to use the derived values and their vcv matrix and computing the std error of N1-N2 (difference of two abundances): Sqrt(var(N1) + var(N2) -2 cov(N1,N2)). An approximate z test would be z=(N1-N2)/ se(N1-N2).

[eecrone]

Thanks! That's a good idea.

In my real data, there are more than 2 groups being compared, but I'll look up how to calculate an F-statistic from means, variances and covariances of group means. There is probably a way.

After posting this, I also started to wonder if there is a way to hack the models to reconstruct equal N (true abundances) from f0 (the "N" parameter in RMark's POPAN model). I'll add it here in case there is interest in continuing the conversation:

In the real data, the number of observed nests in each of 7 years is 15, 14, 9, 9, 2, 20, 20. I created a model with:

Code: Select all

bees.ddl$N$adj = 20 - c(15,14,9,9,2,20,20)


and identity links for N:

Code: Select all

N.dot = list(formula = ~1, link = "identity")
N.adj = list(formula = ~1+adj, link = "identity")
N.year = list(formula = ~0+year, link = "identity")


If the slope for the N.adj model differs significantly from 1, then the years differ significantly in abundance. If I could figure out how to fix the slope value for "adj" to 1, then maybe I could create a null model with equal abundance in all years. [This would mean fixing the value of the slope, or, equivalently, adding an offset, unlike the more common need to fix the real value of a parameter.]

Cheers,
Elizabeth

[jlaake]

Hmm. I didn't realize that my message posted. As I wrote that response, I realized that you would want to work on log scale because we assume N follows a log normal distribution so that log(N) follows a normal distribution
What i suggested could be done with a delta method variance on log(N1)-log(N2).

But since you have more than 2 abundances that won't help. What you propose will not work because you are modelling f0 even though it is stated as N. N can be constant and f0 may not be if p varies. Likewise, f0 can be constant but N may not be.

What may be possible is to form a regression over time of the derived values but you will have to account for the var-cov structure. Not exactly sure how you would go about doing that. Maybe someone else on the list can help.

BTW, please do not refer to these as RMark models. Im seeing this too often. These are MARK models. RMark is only an interface for building MARK models.

Jeff

[jlaake]

One further thought. The regression idea I proposed was in response to your suggestion for a model with POPAN. Obviously a regression is testing for a trend. Whereas the F test you were suggesting is testing for any difference in the set of values. Both may be appropriate depending on your interest.

One final thought would be to look into the variance components assessment. Essentially, asking the question is there more variation in abundance across years than just sampling variance. This allows estimation of a mean abundance and process variance or a trend over time and process variance. It can be done by exporting your model to MARK and using code there or with var.components and var.components.reml in RMark. Sometimes the code in MARK will work better than var.components if you have problems. This would have to be done on the derived values of N and preferably after converting values and var-cov matrix to log scale. I'm sure Evan has a section on variance components in Cooch and White and you may want to read it first.

[cooch]

...

One final thought would be to look into the variance components assessment. Essentially, asking the question is there more variation in abundance across years than just sampling variance. This allows estimation of a mean abundance and process variance or a trend over time and process variance. It can be done by exporting your model to MARK and using code there or with var.components and var.components.reml in RMark. Sometimes the code in MARK will work better than var.components if you have problems. This would have to be done on the derived values of N and preferably after converting values and var-cov matrix to log scale. I'm sure Evan has a section on variance components in Cooch and White and you may want to read it first.

That would be Appendix D in the MARK book. I would agree with Jeff that considering a V-C/RE approach might be best. I would aso add that there are multiple challenges in trying to model N directly. It is far simpler to model the components/processes that lead to temporal change in N, in a TSM (time-symmetric model) -- either the 'Pradel' or 'Link-Barker' models (being equivalent, except that Link-Barker correctly handles any losses on capture). While it might not be inuitively obvious, modeling change in N can be very rigorous, whereas a lot of the estimation and modeling of N derived from open population data can get kind of iffy. TSM models are covered in general in Chapter 13. In context of a V-C/RE approach, Appendix D has a fully worked example of a such an analysis (including fitting a 'trend' model) - starting on p. 37.