bk, buffer size, sigma and density

[Andrea]

Hi all - 

I'm looking for some feedback on the effect bk is having in my model. I am using secr 4.1.0 and am working with camera trap data from wolverines. 77 detections of 7 animals. 

I have run a series of fairly basic models, and the model D~1, g0~bk, sigma~1 is identified as the most parsimonious model by AICc. 

                      model                                   detectfn     npar    logLik           AIC          AICc         dAICc   AICcwt

m2                D~1 g0~bk sigma~1             halfnormal    4      -387.0837    782.167    802.167    0.000        1

m7                D~1 g0~bk + T sigma~1       halfnormal    5       -385.2195   780.439    840.439   38.272       0

m5                D~1 g0~bk + Type sigma~1 halfnormal    5      -386.0878    782.176    842.176   40.009       0

m1                D~1 g0~b sigma~1               halfnormal    4      -440.6641    889.328    909.328  107.161      0

m0                D~1 g0~1 sigma~1               halfnormal    3      -454.8092    915.618    923.618  121.451      0

m4                D~1 g0~T sigma~1               halfnormal    4      -452.8202    913.640    933.640  131.473      0

m3                D~1 g0~Type sigma~1         halfnormal    4      -453.3662    914.732    934.732  132.565      0

In all models I used a mask that included a 40 km buffer around trap locations. The suggest.buffer function recommended a 31 km buffer. RPSV(CHBM, CC = TRUE) = 6814.49, so 4 times that recommends ~28 km buffer. Other literature had used 40 km, so I rounded up and went with a 40 km buffer to be safe. Spacing was 2500.

I then used the code:

mask.check(m2, buffers = c(30000, 40000, 50000), spacings = c(1500, 2500, 3500), LL = FALSE, file = "m2.maskcheck")

to evaluate the influence of buffer size on density, and ideally confirm that I had used an appropriate buffer size. 

With those 3 buffer sizes, buffer has an impact:

I then used esa.plot to investigate things further and compared esa.plot of m2 to the null model. 

Clearly, there's a pretty big difference between these plots. I looked into other models and a buffer of 40 km is sufficient for all other models without the bk covariate. For models with bk it seems that a buffer size of about 80 km is needed before density estimates stabilize. The bk covariate also has a big effect on density and sigma estimates. I would have expected bk to increase density estimates, but instead they are lower. Sigma is much higher - which would explain the need for a larger buffer. 

Null model parameters, D~1, g0~1, sigma~1

             link     estimate           SE.estimate          lcl                         ucl

D          log     1.398627e-05    5.551335e-06      6.608876e-06      2.959895e-05

g0         logit   2.330294e-02    3.947773e-03     1.669913e-02       3.243214e-02

sigma   log     7.589314e+03   5.797354e+02     6.535445e+03     8.813124e+03

m2 model parameters, D~1, g0~bk, sigma~1

$`bk = 0`

           link      estimate             SE.estimate           lcl                        ucl

D          log     5.758101e-06     2.344267e-06       2.672592e-06     1.240583e-05

g0         logit   4.181946e-03     1.749903e-03      1.839603e-03      9.478440e-03

sigma   log     2.448864e+04     3.472138e+03    1.857276e+04     3.228886e+04

$`bk = 1`

            link     estimate             SE.estimate          lcl                        ucl

D         log      5.758101e-06     2.344267e-06      2.672592e-06     1.240583e-05

g0        logit    1.943967e-01     6.816278e-02      9.323484e-02     3.615555e-01

sigma   log     2.448864e+04    3.472138e+03     1.857276e+04    3.228886e+04

I reran the model using a mask with a larger buffer (90 km). Under this model, confidence intervals of parameter estimates increase. 

$`bk = 0`

            link     estimate              SE.estimate          lcl                         ucl

D         log      7.070365e-06     6.128653e-06       1.630192e-06      3.066515e-05

g0        logit    3.548623e-03     1.512013e-03       1.538013e-03      8.166162e-03

sigma   log     1.923345e+04     7.228713e+03     9.432973e+03     3.921623e+04

$`bk = 1`

             link     estimate             SE.estimate          lcl                        ucl

D          log      7.070365e-06    6.128653e-06      1.630192e-06     3.066515e-05

g0        logit    1.504353e-01     8.382900e-02      4.667432e-02     3.904027e-01

sigma   log     1.923345e+04    7.228713e+03     9.432973e+03    3.921623e+04

I've never encountered this before and have struggling to think about why the addition of bk on g0 would so substantially increase sigma and the required buffer. I do not have any outliers in the detection data. The impact on density is also the opposite of what I would have expected. I'd be curious if anyone has thoughts on what might be going on here. 

Thanks

Andrea

[Ben Augustine]

I'm sure Murray will have good input about this, but to me, it looks like sigma is on the edge of identifiability under the null model due to the small trapping array extent relative to individual movement. Then, when you include the bk effect, the drop in the first capture p0 estimate leads sigma to not be identifiable without the constraint imposed by the state space buffer. So the sigma estimate depends arbitrarily on the buffer size.

Ben

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[Andrea]

Hi Ben - 

Thanks for your thoughts. I'm wondering why you think sigma is on the edge of identifiability in the null model? The estimate and confidence intervals seem reasonable to me based on the literature. I'm just not totally following your explanation so if you have time to add a few more details I'd certainly appreciate the help - Thanks!

Andrea

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[Ben Augustine]

Hi Andrea,

I'm not 100% sure this explains the behavior you are seeing, but I believe my explanation is consistent with all the evidence you present here. Hopefully others will chime in, too.

Here is an enumerated list of points that I think are relevant.

1. First of all, looking at your plot of spatial recaptures on the trapping array suggests to me that sigma may not be identifiable. This is just an intuition based on my prior experience. The scale of individual movement is pretty close to the size of the trapping array and I've seen identifiability go either way in this case. It turns out, though, that sigma is identifiable with this data set under M0.

2. Here is a result I'm pretty confident in, though I haven't confirmed it or seen it confirmed in the literature. When the extent of individual movement is approaching the extent of the trapping array, under M0, the true value of p0 will play a role in whether or not sigma is identifiable. If you lower the true p0 enough, sigma will no longer be identifiable, because capture events at the tail of the detection function become increasingly rare.

3. When you include a bk effect when there is trap happy behavior, p0.p (first capture p0) goes down considerably. In your case, it goes down an order of magnitude.

4. Under trap happy Mb, the first capture events (all data up to the first capture at each trap) overwhelmingly inform the estimation of sigma when you have many traps with no captures. The majority of individual by trap by occasion data are possible or actual first capture events. All data for traps with no captures are possible first capture events that did not happen. Therefore, the p0.p estimate has a high "leverage" on the sigma estimate.

Combine 2 - 4. The lowering of first capture p0 under M0 (which is just p0 because all events have the same baseline detection probability) to that under Mb (p0.p), an order of magnitude drop, made sigma no longer identifiable.

Do the parameter estimates ever stabilize as you increase the buffer? If not, I think sigma is not identifiable under Mb for this data set, whether my explanation is correct or not.

Ben

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[Andrea]

Hi Ben - 

Thanks for the further explanation. That seems like a plausible explanation here. Indeed, the trapping array is small relative to the scale of individual movement. 

All parameter estimates in the bk model do stabilize once the buffer size reaches 80 km. 

buffer size	esa	LL	D-hat	se(D-hat)
70000	1001077	-388.176	6.99E-06	6.31E-06
80000	991395.9	-388.179	7.06E-06	6.16E-06
90000	990039.8	-388.179	7.07E-06	6.13E-06
1.00E+05	989887.6	-388.179	7.07E-06	6.12E-06
110000	989874.9	-388.179	7.07E-06	6.12E-06

Parameter estimates from model with 90 km buffer:

bk = 0 

	link	estimate	S.E. estimate	lcl	ucl
D	log	7.07E-06	6.13E-06	1.63E-06	3.07E-05
g0	logit	0.003548623	0.001512013	0.001538013	0.008166162
sigma	log	19233.45219	7228.713188	9432.973321	39216.23335

bk = 1

	link	estimate	S.E. estimate	lcl	ucl
D	log	7.07E-06	6.13E-06	1.63E-06	3.07E-05
g0	logit	0.150435277	0.083828995	0.046674316	0.390402696
sigma	log	19233.45219	7228.713188	9432.973321	39216.23335

However, if I plot the models, the upper confidence intervals go to infinity in the models with the larger buffer size (plots are original 40 km buffer vs 90 km buffer).

This plot would suggest to me that the bk model is not identifiable with these data. If I'm understanding your suggestion correctly, this behaviour is likely a result of the trapping array being too small relative to the scale of individual movement in conjunction with the order of magnitude drop in g0 in trap happy models (bk). Does that seem plausible?

Again, really appreciate your insight. 

Thanks

Andrea

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[Murray Efford]

Ben's explanation makes a lot of sense to me. The informative dataset is possibly very thin (7 individuals and few re-detections except repeats at initial sites?) coupled with small array relative to natural movements. The bk model alerts us to some difficult biology that probably invalidates the null model estimate, without providing a solution.

Murray

[Ben Augustine]

Yes, that sounds right to me. I don't know what happens in general if you continue to increase the buffer far beyond the point where sigma is identifiable, e.g. will the point estimates stabilize like you're seeing. Perhaps so. I believe if you convert the 20km sigma estimate to a 95% home range area, it will be much larger than that expected for a wolverine and also much larger than the trapping array area (though I can't infer the dimensions from the info in these emails), suggesting the 20km estimate is due to a trapping array that is too small for Mb.

Ben

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[Andrea]

Hi - 

Murray, I looked into this a bit more. For 2 of the individuals, all of the re-detections occurred at the initial site. For the 5 remaining, it was a bit more mixed, but even then 2 of them had almost all of their re-detections at the initial site. So, yes, I would have to agree that the informative dataset is indeed pretty thin. 

Ben to your point, a sigma of ~20km translates to an estimated home range of ~7500 km^2 which is indeed much larger than a wolverine home range as well as the trapping array - supporting what you've written that the estimate is due to a trapping array too small for Mb.

Thank you both. I really appreciate your help sorting through this. 

Andrea

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