Analysis of populations in clusters in R

[Mei-Ling Bai]

Hello everyone,  

 

I am using Distance package in R to analyse bird transect data where birds could occur in groups.  It seems that I could only use cluster size as a covariate in the detection function (by specifying formula = ~size in the ds function) to deal with size bias . Is there some way to deal with the problem in R with "regression method", as the default in the Distance software?

In addition, how to apply the derived ESW in the estimation of population density? When animals are not in clusters, we use D = n/(2*ESW*L). When they are clustered, the n should be the number of clusters or the number of animals?

Many thanks in advance!

Mei-Ling

[Eric Rexstad]

Mei-Ling

The preferred manner for resolving the size-bias issue (the mean size of your observed groups is a positively-biased estimates of the size of groups in the population because observers are likely to miss small groups at large distances) is via the covariate approach you describe.  This approach recognises detectability is a function both of distance from the transect and group size.

You could write your own code to perform the regression method (as I did in a question to the list back in 2017).  You can see the example here

https://groups.google.com/g/distance-sampling/c/2W7DJtaB82Q/m/4Ri2MLiiAAAJ

The difficult part of incorporating the result of the regression is to produce a measure of precision for mean group size; which is needed to propagate variability in estimated group size into the uncertainty in density of individuals.  The code in the link above takes a simple-minded approach to the uncertainty issue.

Regarding the formula you include for estimating density; that formula will produce an estimate of the density of *clusters* when detections are made of groups.  In that case, the 'n' in the formula would be the number of groups.

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Eric Rexstad
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[Mei-Ling Bai]

Eric, 

Thank you very much!

So I treated the size-bias issue via the covariate approach. But I am still not quite clear about how to calculate density with ESW when size is included in the model. 

I made two models:

M1 <- ds(dt, key = "hr", adjustment = "herm")
M2 <- ds(dt, formula = ~size, key = "hr")

and acquired ESW of each by  mean(predict(M1 or M2,esw=T)$fitted)

for M1, I could get

M1$dht$clusters$D$Estimate = n / (2*ESW*L)

M1$dht$individuals$D$Estimate =  n / (2*ESW*L) * mean size

However, these equations did not hold for M2.

What is the right way to use ESW in estimating density when size is included in the Model?

Best greetings,

Mei-Ling

Eric Rexstad <er...@st-andrews.ac.uk> 於 2020年12月22日 週二 下午4:57寫道：

[Eric Rexstad]

Mei-Ling

The "M2" approach you describe does not work, because when a covariate enters the detection function, there no longer is a single ESW.  Instead there is a different ESW for each group size; in other words, ESW is a function of group size, because detectability is a function of group size.

With group size as a covariate, a Horvitz-Thompson-like estimator is employed to estimate density of individuals.

Examine slide 14 of this set of lecture materials on covariates in the detection function

https://workshops.distancesampling.org/online-course/lecturepdfs/Ch5/L5-4%20MCDS.pdf

To extend this framework to the case where group size is a covariate, examine slide 2 of

https://workshops.distancesampling.org/online-course/lecturepdfs/Ch5/L5-5%20MCDS%20Complications.pdf

You could also listen to the lectures narrating those slides at the same website.

Recognise the interdependence between ESW and PaP_a.  Recognise ESW=Pa⋅wESW = P_a \cdot w, where w is truncation distance.  The PaP_a formulation is used by `ds()` and it fits smoothly into the Horvitz-Thompson-like framework.