Sampling Area and Sigma

[mgoode2]

I’m running DENSITY 4.4.6.1 in Windows 7 to analyse white-tailed deer from scat collection.

I am concerned about my sampling area and the calculation of sigma.
We sampled an area of 1-km2 which is about half the size of the average homerange for white-tailed deer. So our measure of movement between traps is confined by an area less than the average homerange. If sigma is calculated from our trap data and in turn used to adjust the buffer (3sigma) should I be concerned that the buffer may be to small and do you believe this will change our density estimate? No habitat mask was used around the area.

Basically, can sigma be underestimated if your sampling area is smaller than the average area used by the species?

[murray.efford]

I'm not clear how you are sampling the population - is it by searching a polygon for scats, or with some sort of trap? Density 4.4 does not deal with polygon data, but new ML methods suitable for scat collections will be documented in the next release of 'secr'. You're right to be concerned about the relative size of home ranges and search areas, and although SECR is fairly robust, scat collections in which each individual is detected only in a polygon less than the home range size do not contain enough information needed to estimate the spatial scale of movements. This is not an issue of buffer size - if you make that large enough it should not matter. Marques et al. preprint for Ecology have an example of poor performance of an equivalent Bayesian estimator when (by my calculation) 95% home ranges were 3x plot size. The ML method performs slightly better - both depend partly on how much data you have. It would be interesting to check by simulation for your particular case - you could wait a few weeks for secr 1.6 and do it yourself or send me some data offline.
Murray

[mgoode2]

We searched 150 10-m radius plots within the 1-km2 sampling area for each capture event.
5 capture events and the 10-m plots were randomly generated for each capture event.
So we used the GPS location of the plot as a "trap" location.
I would be happy to send you our input files if you are interested.

Thanks,
Matt

[murray.efford]

Matt
OK, so your data can be analysed with a 'proximity' detector in Density 4.4, and you can ignore my excitement about polygons. There's still most likely a problem if the plots together span less than one home range, as implicitly we rely on the localisation of each individual's detections (found in some detectors and not others) to infer the home-range scale, sigma - as you recognised. There's no absolute rule - estimates just get worse as the ratio of sampling scale to home range decreases - hence the desirability of running some simulations to see what you should expect in this particular case. As this concerns established models I'd suggest you try it yourself (plug in values of density, g0 and sigma in the PowerAnalysis module).
Murray

[sixtystrat]

I am revisiting this deer data set and I ran a simulation to see if our sampling grid (1 km2) was too small to estimate deer density when their home range diameter is about the size of the grid. I used a density of 0.127, go=0.05, and sigma=560(CV=0.35).
Repl MLLoglk MLesa MLDens MLseDen MLg0 MLseg0 MLSigma
1 -18196.926 584.416 0.2781 NA 0.04987 0.00167 7.12
2 -20008.88 239.405 0.6615 0.2685 0.04713 0.00141 18.21
3 -21068.953 1174.701 0.1215 0.0111 0.05131 0.00166 475.07
4 -21405.975 515.412 0.3072 0.4728 0.04847 0.00146 10.65
5 -20318.87 1167.022 0.1304 0.0112 0.04883 0.00133 530.62

It looks like the estimates are ok for 2 of the 5 replicates. For the other 3, I'd guess that there was a convergence problem. Would I conclude that, if convergence is reached, the estimates are okay or do I need to scrap the whole thing? Thanks for any assistance!
Joe Clark

[murray.efford]

Hi Joe

These home ranges are definitely in the danger zone: as I calculate it they are about 5x the size of the grid, or in terms of the table in Efford (2011 Ecology), the grid has area about 3 sigma^2. So substantial bias and uncertainty can be expected. This will be increased if I understand correctly and you have allowed sigma to vary between individuals, so some have even larger ranges.

I doubt lack of convergence can be blamed for the erratic estimates: all the g0-hat look good. I suspect the simulations with bad sigma-hat (and hence bad D-hat) simply had bad movement data (few or no recaptures at a different site), but I cannot tell. If you specify the number of detectors, the number of occasions, the habitat configuration (did it extend beyond the grid?) I may be able to say more. I'm a little puzzled by the large likelihood values, but maybe you had a lot of plots or a lot of occasions.

Murray

[sixtystrat]

Thanks Murray. Yes, there were a ton of detectors (701), 5 occasions, and I used a homogenous habitat configuration around the grid with a 1500-m buffer. How did you calculate the ome range size relative to the grid? I was thinking that sigma was analagous to the HR radius...
BTW, I ran suggest.buffer and it suggested a 500-m buffer but the density estimates were a lot different with the 1500 buffer so I went with that. The detectors were about 20 m apart and my results were pretty erratic when I ran mask.check. Thanks for the quick response Murray.

[murray.efford]

Joe - when I simulate data according to your description (but with CV(sigma)=0)) I get

Code: Select all

deergrid <- make.grid(nx = 26, ny = 27, spacing = 39, detector = 'proximity')
CH <- sim.capthist(deergrid, popn = list(D = 0.127, buffer = 1500),
    detectpar=list(g0 = 0.05, sigma = 560), noccasions=5)
summary(CH)
Object class      capthist
Detector type     proximity
Detector number   702
Average spacing   39 m
x-range           0 975 m
y-range           0 1014 m
Counts by occasion
                    1   2   3   4   5 Total
n                 117 122 119 122 119   599
u                 117  21   8   6   2   154
f                  17  16  19  17  85   154
M(t+1)            117 138 146 152 154   154
losses              0   0   0   0   0     0
detections        906 866 868 939 896  4475
detectors visited 508 510 494 524 518  2554
detectors used    702 702 702 702 702  3510

Does this look like your actual data, or have I missed something? I'm worried there may have been a mix-up over units (that's a lot of detections!)

The relation between sigma and HR is a bit subtle. In the past we have used the multiplier 2.45 to convert sigma to a 95% HR radius, but when it's g0 (not lambda0) that is half-normal (the default detectfn in Density and 'secr') the multiplier should be 2.24 (ms in review). So 95% range is approx pi * (2.24 * sigma)^2.

Murray

[sixtystrat]

Yes that looks like my simulated data but not the real data. There were only 140 detections in the real data. It does look like a mixup. Here are the real estimates:
secr.fit( capthist = DeerCH, model = list(g0 ~ 1, sigma ~ t + h2), mask =
AAFBhabmask, buffer = 1500, CL = FALSE, detectfn = 0, groups = sex, method =
BFGS, trace = TRUE )
secr 2.3.2, 17:19:00 07 Nov 2012

Detector type proximity
Detector number 701
Average spacing 21.08923 m
x-range 584715.9 586079.1 m
y-range 115052.8 116039.4 m
Usage range by occasion
1 2 3 4 5
min 0 0 0 0 0
max 1 1 1 1 1
N animals : 33
N detections : 140
N occasions : 5
Mask area : 1359.956 ha

Model : D~1 g0~1 sigma~t + h2 pmix~h2
Fixed (real) : none
Detection fn : halfnormal
Distribution : poisson
N parameters : 9
Log likelihood : -695.198
AIC : 1408.396
AICc : 1416.222

...
Fitted (real) parameters evaluated at base levels of covariates

session = 1, h2 = 1, t = 1
link estimate SE.estimate lcl ucl
D log 0.06344873 0.01873779 0.03599820 0.11183171
g0 logit 0.04974865 0.00857506 0.03539103 0.06951121
sigma log 147.02180601 33.38283945 94.73957364 228.15609795
pmix logit 0.82577428 NA NA NA

session = 1, h2 = 1, t = 1
link estimate SE.estimate lcl ucl
D log 0.06344873 0.01873779 0.03599820 0.11183171
g0 logit 0.04974865 0.00857506 0.03539103 0.06951121
sigma log 147.02180601 33.38283945 94.73957364 228.15609795
pmix logit 0.82577428 NA NA NA

session = 1, h2 = 2, t = 1
link estimate SE.estimate lcl ucl
D log 0.06344873 0.01873779 0.03599820 1.118317e-01
g0 logit 0.04974865 0.00857506 0.03539103 6.951121e-02
sigma log 650.74380313 147.75818972 419.33364941 1.009858e+03
pmix logit 0.17422572 NA NA NA

session = 1, h2 = 2, t = 1
link estimate SE.estimate lcl ucl
D log 0.06344873 0.01873779 0.03599820 1.118317e-01
g0 logit 0.04974865 0.00857506 0.03539103 6.951121e-02
sigma log 650.74380313 147.75818972 419.33364941 1.009858e+03
pmix logit 0.17422572 NA NA NA

[murray.efford]

It's worth playing with the simulations until they generate data that looks like the real thing (before doing the laborious model fitting). I can see three differences - I didn't allow for incomplete usage, whereas not all sites were checked on all 5 occasions; you seem to have doubled the density estimate; your simulation sigma (560) is much larger than the weighted mean of the mixture-model estimates. Not sure if these differences are enough to explain the difference between 140 and 4475...
Murray