Parameters at the boundaries in POPAN parameterization

[TGrant]

I have some parameter estimates that I'm trying to tell if they are correct, didn't converge, hit the boundary, or something else.

I'm using the POPAN parameterization on 5 yrs of mark-recapture data for snakes. The data are relatively sparse. Passive traps were run every day for much of the spring and fall. I collapsed the data into 1 week capture occasions. Survival was modeled as constant during each of the four seasons, with fall and spring seasons defined as weeks during which there was sampling. PENT was also modeled as constant during each season. I also used a yearly covariate of precipitation on survival and PENT that worked well. Identifiability shouldn't be an issue since survival and PENT were constrained to be the same each season, except as they varied by the covariate.

Survival estimates for fall and winter are my concern. For fall, phi was estimated as 1.0000000, SE=0.1248158E-04, CI = (0.3718319E-03, 1.0000000). I ran a profile likelihood CI on this parameter and got '* * WARNING * * Error number 3 from VA09AD optimization routine. ' but the phi SE and CI were different: phi=1.0000000, SE = 0.5685929E-05, CI = (0.9964474, 1.0000000). What should I conclude from this? It seems clear to me that the survival is virtually 100% in the fall, but I'm not sure if I can use the P.L. CI since there was an error. Curiously, the beta parameters also changed and so some other real parms were also slightly different.

I ran a profile likelihood on winter survival and it was 1.0, SE = 0.0, CI = (1.0,1.0). Normally if SE = 0.0, I would think there was a problem, but since it's at the boundary I think it's correct.

[cschwarz@stat.sfu.ca]

Hi Tyler.

Before answering your questions, there are a number of other potential problems that you need to address.

You will need to give more details on exactly how you are coding your data. For example, you have 5 years of data, condensed to weekly samples which occur in the spring and fall. So this gives you a total of 50-60 capture occasions?

You modelled survival as "constant" during the occasions within each fall/winter - ok.

But they you modelled the PENTS as also being constant. I think you may wish to rethink this. The PENTS are the proportion of the super-population (total number of entrants over the course of the study) that enter between each sampling occasion. It seems unlikely that the PENTS would be constant over these sampling occasions - e.g. births are more likely to be concentrated in certain parts of the years. Immigration may not be constant over time. I'm not sure what you are trying to do by modelling this in this way?

You may wish to consider a robust-type of design with spring/fall being the primary periods and weeks within the seasons as the secondary periods?


You likely go the error message below (with the confidence interval of 0->1) because you used the sin link with very sparse data. In this cases, the sparse data give estimates with very poor precision and the sin link just gets confused if the estimates are close to the boundary with very large standard errors.

However, you should likely step back a bit and rethink a bit of your analysis approach.

Carl Schwarz

[TGrant]

Dr. Schwarz,

There are 90 capture occasions (weeks in which some level of trap effort was exerted). The number of occasions per fall or spring varied somewhat. Typically trapping occurred from approx Apr 1 to June 31 (12 weeks) and Sept 1 to Oct 15 (6 weeks). Some days were not trapped because of inclement weather. I used calendar weeks as capture occasions. To account for the fact that not every day was sampled every week/occasion, I used a covariate on p of the number of days trapped in that week.

So there are 90 capture occasions spread over 5 years, 103 snakes, and a total of 389 captures. Only adults were included in the analysis. Pretty good for snakes, but still relatively sparse.

Just to be clear, I modeled PENT as constant within each season, but different among seasons. So in the DM for PENT parms there were betas for the intercept, 3 seasons, and the precipitation covariate. That seemed reasonable to me. The precip covariate allows it to vary between years (assuming it is informative). Does that still seem wrong to you? It assumes snakes are entering the population at a constant rate over the season, but at least its different among seasons. I had some more questions on the PENTs I was developing after talking to the species experts and haven't posted yet. In short, the PENTS were non-zero for summer and winter, and zero for spring and fall, which didn't seem to make a lot of sense.

I considered robust design but for some reason I wasn't thinking of it that way. I was thinking the closure assumption would apply to the entire season. I'll have to look at that again. We wanted N ests to look at the trend over time, but we would get those from RD as well. So maybe I should get my primary and secondary sampling occasions and time intervals straight and throw it in RD and see how that works.

Tyler Grant

[cschwarz@stat.sfu.ca]

Thanks for the extra details on the study design.

Basically the experimental set up is:
Year 1 Year 2
sw1 .... sw12 fw1...fw6 sw1..sw12... fw1...fw6 ...

where sw1..sw12 are the sampling weeks in the spring/summer and fw1..fw6 are the sampling weeks in the fall.

The p parameter refers to the probability of capture in (the midpoint) of each week. The phi parameters refer to survival between weeks. Note that the interval betweens sw1...sw12 is 1 week; between sw12 and fw1 is 9 weeks; betwee fw1...fw6 is 1 week; between fw6 and sw1 is 28 weeks? I don't know enough biology of the snakes you are studying to know where most of the mortality occurs, or if your snakes hibernate etc, but a model with constant survival PER UNIT time may be suitable. The PER UNIT TIME would enforce the weekly survival rates to be equal across the wider gaps, but would still allow for a "lower" survival rate for the longer intervals.

You mentioned that you are studying adult snakes. So new additions to the population are immigrants from outside the study area + recruitment of younger snakes? The immigration number from outside the study area may be approximately constant over time, but is it is sensible to think that juvenile recruitment is constant (in numbers) over time as well?

The PENTS also refer to the intervals between sampling occasions, i.e. some refer to a 1 week interval and some to a 9 week interval and some to a 28 week interval. So making the PENTS equal in each season (presumably for the intervals sw1->sw2... sw11->sw12) may work, but what did you do for the longer intervals? Did you force them to be equal to the PENTS for the shorter intervals? The MARK code could likely adjust for equal entry per UNIT TIME (I think I checked this in the relevant chapter of the GIM manual), but you need to check to see if the estimates are sensible.

Adjusting for a covariate (precipitation) is tricky because the restriction that pents must sum to 1. I don't know how to interpret the resulting estimates in this case and am surprised that some of the PENTS were at 0 then as the logit(0) = -infinity which usually doesn't work very well in computers.

But... you mention that you are really interested in the TREND in abundance over time. It turns out that estimating trend in abundance is "easier" (i.e. less prone to biases caused by heterogeneity in catchability) than estimating raw abundances. I would suggest that you look at the Pradel-lambda model (that estimates population growth directly without estimates of population size) or the Link-Barker models (with phi and f="new recruits per member of the population"). The lambda term in Proadels approach would estimate the population growth between occasions=phi+f in the Link-Barker model Have a look over the JS chapter in the GIM manual for examples of the POPAN, Pradel-lambda, and Link-Barker models fit to the same data set and the equivalences between them.

Carl Schwarz

[TGrant]

Dr. Schwarz,

You've got the experimental setup correct. Weekly survival was modeled as constant within seasons, i.e. phi for sw1 to sw12 were all equal. But if 1sw1 means sampling week 1 in year 1, 1sw1...1sw12 were all equal, as were 2sw1...2sw12, but 1sw1 was not equal to 2sw1, because phi varied by annual precipitation.

You're correct on the immigration and recruitment. I'm sure recruitment varies each year. I was using the annual precip as a covariate on PENT to help model this annual variation in recruitment. The model of constant PENT over years was selected by AICc over a model where there was an effect of year on PENT (model PENT(season) vs model PENT(season+year) - PENT still varied by season for both models). The constant across years model (PENT(season))is kind of unrealistic so I was taking a clue from a polar bear paper (Macdonald et al?) and using a covariate to better model, estimate, and understand year to year patterns in recruitment and abundance.

The PENTs were modeled basically the same as the phis. For the summer and winter intervals, I used the time interval facility in MARK so that estimates of PENT and phi for summer and winter are still weekly estimates, MARK just takes the root of the estimate for the season so the output is in weekly survival and weekly PENT. As described in the MARK book briefly on p 2-5, then more on pp. 4-27 to 4-29.

PENTs don't work well with covariates? I said zero but they weren't exactly zero, they are pushing the boundary at around e-4. I did use a separate multinomial link for both groups, males and females.

I guess it makes sense to run it in Pradel-lambda. I read the JS chapter in the GIM, but people easily understand abundance when you show that to them, and it's harder to explain how the lambda in Pradel is estimated and why it's better. I'll have to talk to the biologists who collected the data, but I think trend is the most critical thing they want to know since it's a listed species. Especially if the PENTs aren't working right and right now they are making me suspicious.

Thanks for all your help.