effect of high mobility of some individuals in estimates of density
Pablo Lau
We are modelling the density of jaguars at several sites in South America. Data from some localities have misidentifications that cause two distant individuals to be mistaken for the same individual. The data then appear to represent a highly mobile individual. A posteriori correction of the identification leads to a higher density estimate compared to data with errors. My question is: there is a theoretical reason to expect a reduction in density estimation if some individuals have higher than average mobility (detected in traps more distant from each other than average) .
Thanks in advance
Murray Efford
Pablo Lau
Thanks very much, Murray and Jeff. Now I 'm more confident about methods in secr. I Will compare estimates of ESA in several analysis as suggested by Jeff .. Again, Thanks for your comments.
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Murray Efford
Pablo Lau
Here are the Jeff's posts.
1) Hello Pablo. It sounds to me that your sigma estimates are inflated due to the misidentifications resulting in larger estimated activity areas and thus the lower densities (greater area per individual).
2) Sigma calibrates the spatial relationship between captures and the inferred animal distribution. As you know, sigma scales the detection process in relation to the distance from the activity centers. This also means that sigma defines the spatial scale of animal movement relative to the trap array, and thus allows SCR models to infer the effective sampling area around traps. A larger sigma implies animals range farther, requiring a larger effective sampling area, so that given a fixed number of detections, a larger effective sampling area (implied by a large sigma) would lead to a lower density estimate. Did you estimate the effective sampling area (ESA) of the trap grids? The analysis with the misidentifications should have a considerably larger ESA.
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Pablo Lau
To contribute to the discussion, I am attaching a comment from David Borchers, whom I had contacted some time ago with the same original question from this group and just sent me his response:
"you are correct in saying that you don’t need to estimate ESA explicitly to estimate abundance (and in fact when you do not have constant density in the survey area, ESA does not appear in the likelihood and may not be a very useful quantity).
I’ll do my best to explain:
First, a “density surface” is like a heatmap showing where there tend to be more animals (hotspots) and where less. If, instead of plotting the map with hotter colours showing higher density areas, you plot it as a kind of 3D topographic map, with higher elevations indicating higher density areas, then you have a “density surface”.
If you assume constant density, i.e., that this topographic map is just a flat plane, then the concept of and ESA is useful. Otherwise it can be misleading.
For now assume the map is flat, i.e. that density is constant. Suppose also that the survey region has area A. In this case, the proportion of the population in any subregion of the same size within the survey area is the same. So, if you knew what fraction of the survey region your traps sampled then, then you know what fraction of the population you sampled. An intuitive estimator of the number of animals present is the number of animals you detected divided by the fraction of the population you sampled. For example, if you sample half the population and detected n=10 animals, you’d estimate there to be 10/0.5 = 20 animals there. If you sample a quarter of the population and detected n=10 animals, you’d estimate there to be 10/0.25 = 40 animals there, and so on.
The fraction of the area you sampled can be written as ESA/A, since ESA is the area you effectively sampled. And when density is constant, this is also the fraction of the population you sampled. Hence you’d estimate abundance to be n/(ESA/A) animals. And the smaller ESA is, the bigger the abundance estimator n/(ESA/A) is, i.e., for any observed number of animals n, you get a bigger abundance estimate if ESA is smaller. And ESA is smaller when sigma is smaller.
In your example, the misidentifications make sigma too big, so ESA is too big, so you get too small an abundance estimate.
Things are not so simple when density is not constant, i.e. when there are some hotspots with lots of animals and some cold spots with few animals. In this case, the number of animals you detected divided by the fraction of the population you sampled is still a good estimator of abundance, BUT the fraction of the population you sampled is not necessarily the same as the fraction of the area you sampled (ESA/A). You could have sampled half the area, but the half you sampled might all be in a cold spot of density, so that you have sampled much less than half the population.
Moral of the story: If you forget about ESA and just maximise the likelihood, you get unbiased density estimates, whether or not density is constant. If density is constant, then ESA/A is a useful indicator of the fraction of the population you sampled, but otherwise it may not be.
That said, whether or not density is constant, a smaller estimated sigma will tend to give a larger estimated abundance. This is because smaller sigma implies that you have sampled a smaller fraction of the population, whether or not density is constant. Including the misidentifications will tend to make sigma larger, and abundance smaller, whether or not density is constant.
One final comment: If you have a randomised design ( i.e. randomised method of placing arrays of detectors) that gives equal probability of arrays being anywhere in your survey region, this ensures that on average (over all possible realisations of the design) the fraction of the area sampled is the same as the fraction of the population sampled, even if density is not constant in the survey region.
David.