Output from POPAN model - what is N?

[JonL]

1. I run the POPAN model and cannot quite figure out what N in the output lists means. I get different values for N in "View estimates of real parameters" window and "View derived estimates" window. N in the former window (last line) surely is an estimate of the superpopulation (all animals that ever entered the study site)? The latter window list "Population Estimates of....N hat" which I figure is the number estimated to be present at the different capture occasions and "Gross Population Estimates of .... N* hat" which I figure also should be the "superpopulation" (excluding or not excluding those that entered and exited during one capture interval, never available for capture?).

2. When I run a full [phi(t), p(t) pent(t)] model I always get both outputs of N, gross N tends to be larger. I also run models where I reduce the number of parameters by fixing phi for the first few capture occasions to 1 and pent for the last few to 0. This is biologically reasonable (these toads enter the breeding site succesively and depart succesively. Most are present at the height of breeding. It is also supported by the estimates from the full model. However, now Gross population is often (I run different years and both sexes) reported as "Not a number". Is this worrying?

I have specified log link function for N and Mlogit(1) for the pents (inkluding those fixed to 0).

[cschwarz@stat.sfu.ca]

Some of these are covered in the Gentle Introduction to MARK on the Jolly-Seber model.

Yes, there are several "N"s floating around.

N = super-population size = total of all animals that entered study site.

N_i = abundance at time of sampling at the ith occasion.

N* = total gross population size. New entrants to the Jolly-Seber model are only counted if they survive to the next sampling occasion, i.e. entrants between sampling occasions 1 and 2 are only counted if they survive to sampling time 2. In some case, animals could die after entering the study area, but before having a chance to be sampled. If you assume uniform entry between sampling occasion and constant survival rate between sampling occasions, you can derive expressions for the gross number of entrants between sampling occassions INCLUDING those that entered and died. See Schwarz et al, 1993, Cdn J of Fisheries and Aquat Sci, 50, 1181-1191. If you add the gross entrants you get the gross super-population size.

So N* > N.

If phi=1, then gross entrants = net entrants. If you restrict pent=0 that shouldn't case a problem.

The NaN sounds like a division by zero. The "correction factor" for uniform entry and uniform surviva is log(phi)/(phi-1). At phi=1 this gives 0/0 which is a problem. However, the limit as phi-> 1 is found using l'Hopital's rule as derivate top / derivative bottom = 1/phi / (1-0) = 1/ phi = 1 which is correct. The code in MARK may not do the check for the division of 0/0.


However in this case with phi=1, Gross Births=Net births so N*=N.

[talia_umass]

This was a very helpful clarification, thanks Carl!

As a follow up question: Is N the only parameter that is actually estimated, the other two (N_i and N*) are derived after the model itself has finished?

Thanks again!

[cschwarz@stat.sfu.ca]

Yes, the superpopulation parameter N = total of all animals that enter the system is in the likelihood function. All other population sizes (N_i = population size at sampling occasion i, and gross population numbers) are derived parameters.

[pstevens]

When I read the numerical exmple in Schwarz and Arnason (1996), I notice that the N = sum of B^i from i=2 to i=8.5. However, when I do a similar summation for the numerical examples in Ch. 13 of "A Gentle Intro..", N NOT = B^i from i=2 to i=k-1. Is this because of a difference in the estimation procedures between the original paper and MARK's implementation?

[cschwarz@stat.sfu.ca]

The superpopulation size (N) is the total of all animals every present for capture.

Theoretically it is defined as
N = B_0 + B_1 + ... B_{k-1) = N_1 + B_1 + .... B_{k-1}

In the p(t) models, the population size at the first occasion (B_0=N_1), the number of births between occasion 1 and 2 (B_1), and the number of births at the end (B_{k-1}) are not estimable and the estimate of the super-population size is then biased.

In simpler models (e.g. p(.)), this isn't a problem.

In practical cases, you hope that you start sampling when there are only a few animals present and you could "interpolate" from N_2 back by simple linear interpolation. Similarly, at the end of the chain, you hope to continue to sample until nothing much is left is coming in and again you could add a simple linear interpolation. All are ad hoc, but you do what is necessary to get a sensible estimate.

The differences between the 1996 paper and MARKS implementation of POPAN are likely due to slight differences in treating this ad hoc case. MARK simply adds the estimates. In 1996, I think I tried to do a linear intepoation. I'll have to check in more detail.

Carl Schwarz

[nburgmei]

I've just started using MARK after having read through the manual and much of this site and I'm still having a little difficulties with N.

My N always equals the number of individuals I marked in the study except when I set (p.) as constant and then it gives me an estimate. AICc ranks this as the lowest model. My N*-hat stays within 1 individual for the other models, but again jumps with the constant (p.). I initially reported the N*-hat from the derived estimation as the population estimate for a poster presentation, but now that I'm delving a little deeper into it I'm skeptical.

Thanks,

Nick Burgmeier

[cschwarz@stat.sfu.ca]

If N equal the observed number of animals, this usually indicates an overparameterized model or very sparse data.

[nburgmei]

Thanks for the quick reply.

I've run all possible combinations of models, just to check, and they all give me a N=observed.

I'm working with a low density species; thus far we've captured 56 individuals with only 16 recaptures over 3 rounds of sampling. We're about to run 3 more rounds of sampling. Is this likely the cause of my problems? Also, is N*, which I do get an estimate for, a meaningful number to report? I don't want to report anything at this presentation that isn't particularly useful.

[cschwarz@stat.sfu.ca]

With only 3 sample times, the ordinary Jolly-Seber model likely isn't very useful for you because too many parameters are non-estimable even with "perfect" data. If you data is sparse, then more parameters are non-estimable.

You can basically only estimate phi_1, p_2, and N_2 - all of the other parameters are confounded with each other.

The super-population size likely isn't very meaningful in this experiment.

Do you need a fully open model that allows for both births/immigration and deaths/emigration?