Detailed question and supporting
calculations regarding computations involving animals occurring
in groups. The fundamental challenge with your work is the
choice of data set. Indeed this data set does include animals
(minke whales) that occur in groups, but with the added
complication that the survey employed stratification. Hence the
computations include not only details involved in group size,
but also details involved in stratified estimates. I think it
is the later that is the cause of the discrepancies you
Sprinkled below are some modifications to your calculations along with some narrative and supporting documentation
mod.cs <- ds(ClusterExercise)
thesummary <- summary(mod.cs)
Expected.S <- thesummary[["dht"]][["Expected.S"]][["Expected.S"]] # mean group size
effort <- thesummary[["dht"]][["clusters"]][["summary"]][["Effort"]] #effort
abundance.indiv <- thesummary[["dht"]][["individuals"]][["N"]][["Estimate"]] #abund indiv
abundance.groups <- thesummary[["dht"]][["clusters"]][["N"]][["Estimate"]] #abund group
Question 1 (expected group size)
For the study area, expected group size is
the ratio of estimated individual abundance to estimated group
abundance.indiv / abundance.groups
Question 2 (precision of estimated expected group size)
From comments in the function dht.se in the mrds package:
This computes the se(E(s)). It essentially uses 3.37 from Buckland et al. (2004) but in place of using 3.25, 3.34 and 3.38, it uses 3.27, 3.35 and an equivalent cov replacement term for 3.38. This uses line to line variability whereas the other formula measure the variance of E(s) within the lines and it goes to zero as p approaches 1.
Here is Eqn. 3.37
Question 3 (density in study area)
Because the study area was stratified, the overall density is a weighted mean of stratum-specific densities, with the weighting factor being the sizes of the respective strata. See Buckland et al. (2001) Sect 3.7.1, Eqn 3.122 or equivalently (below) Buckland et al. (1993), Sect 3.8.1
density <- thesummary[["dht"]][["individuals"]][["D"]][["Estimate"]] #density
density.studyarea <- (area/area * density) + (area/area * density)
Question 3b (precision of overall density estimate)
Variance (or other measures of precision)
of the density estimate from a stratified survey is described in
Section 3.7.1 of Buckland et al. (2001) detailed in Eqns. 3.123
through 3.126; equivalently Buckland et al. (1993), Sect. 3.8.1
Perhaps others can provide more detailed
answers to your questions.
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I would approach your problem in a slightly different coding way. It is as if you've conducted multiple surveys in the same area. I would combine the three grids into a single data set and use "grid" as a covariate in the detection function model and also as you stratification criterion.
Using the function 'dht2' in the Distance package, it can perform the effort-weighted computations you are performing by hand. The function will also properly handle the computation of uncertainty. The scenario is depicted thus
For your second question about the
bootstrap, the function `bootdht` in the Distance package can
perform bootstrap computations, but it is not yet capable of
bootstrapping complex situations such as yours.
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Use of "grid" as a covariate in the detection function will make the scale parameter (sigma) of a key function differ for each grid. Catch being that you must assume that all grids share the same key function (no mixing of hazard rate and half normals between grids).
You can contrast the grid-specific
detection function model against a model without the grid
covariate--that model then assumes the detection function is the
same across all grids.
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Not sure what's happening here, not surprised you are concerned. Note that the first table of results (Summary statistics). I'm guessing the encounter rates are minute (because they are detections per snapshot). The cv.ER values in that first table seem plausible, so those zeros are likely just formatting problems.
But the line you have highlighted does not
seem right--the upper and lower confidence interval bounds are
equal, implying SE=0, which clearly isn't right. What code did
you use to get this result?
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I haven't spotted problems with your
code. If you want to send more details to me offline, I might
be able to have a look tomorrow.
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