Community N-mixture model with detection history from sub-counts

[Tom]

Dear All,

 I am hoping folk could share their thoughts regarding the legitimacy of a community N-mixture model I want to try.

First some background: We have bird abundance data collected at 190 point count locations grouped in 18 transects, located in a range of habitats. We visited each location 8 times over the course of a year and conducted 10-minute, unlimited radius counts. 

Point count locations were >200m apart and we were very careful to avoid double counting, so I think it is ok to assume populations were closed during each visit. However, because visits were separated by several weeks, populations were unlikely to be closed between visits. I base this on some earlier N-mixture modelling, which suggested very low detectability and high estimates of abundance. My understanding from AHM1 is that this is a likely consequence of violating the closure assumption.

However, I hope all is not lost. Inspired by Alldredge et al (2007) “Time-of-Detection method for estimating abundance from point-count surveys” The Auk, when we collected the data, each visit (i.e. total 10-minute count) was divided into four 2.5-minute sub-counts. So we have detection history for each individual in the four sub-counts of each visit (e.g. 1,0,1,0; or 0,0,1,0 etc.). Could I use this detection history to estimate detectability during each point count, and treat each visit as a separate “season” without the expectation that populations would be closed between visits? Does this sound like a legitimate approach?

My proposed model is below, which is largely based on the DRY model in Chapter 11 of AHM1. If I incorporate the detection histories from the sub-counts, can I just run the same model, but with sub-count detection rather than visit detection? And then replace the current “season” parameter with “visit”?

I would really appreciate anyone’s thoughts about this.

Many thanks,

Tom

  

# Count of each species i at site j during visit k. Can I just reframe the model so that k = sub-count detection rather than visit detection?

Counti,j,k ~ Binomial(Ni,j, pi,j,k)

 # Probability of detection, with a habitat-specific intercept, and species-specific slopes based on time of count (minutes after dawn) and season. Can I replace “season” with “visit”?

Logit(pi,j,k) = alpha0habitat + alpha1i x timej,k + alpha2i x seasonj,k 

 # True abundance as a function of landscape structure, with data augmentation

Ni,j ~ Poisson(phii,j, lambdai,j)

 # Landscape structures A and B, with a random effect of transect gamma to account for possible spatial autocorrelation

log(lambdai,j) = beta0i + beta1i x landscapeAj + beta2i x landscapeBj + gammatransectj 

[Morgan Tingley]

Hi Tom,

Yup, the general strategy you propose will work. You can use the internal-survey repeats to estimate per-survey detectability. The question, then, is how you structure the model with respect to the multiple visits. If you believe that you have closure violations, then you should allow occupancy to change from "season" to "season" (i.e. visit to visit). In your current model, you do that only by treating the same site as independent in different visits. That's a problem, because they are not independent at all -- this lack of independence will inflate parameter estimates.

A better solution would be to allow for temporal autocorrelation. My preference in that case would be to do something like this:

Counti,j,k,t ~ Binomial(Ni,j,t, pi,j,k,t)

where t = visit number

then for visit 1:

log(lambdai,j,t=1) = beta0i + beta1i x landscapeAj + beta2i x landscapeBj 

(I'm presuming that gamma*transect term is no longer needed)

and for visit t >1

log(lambdai,j,t) = beta0i + beta1i x landscapeAj + beta2i x landscapeBj  + phi x Ni,j,t-1

Where the lambda at visit t is correlated, via phi, with the true N at time t-1. This creates a temporal autocorrelation at your sites from visit to visit, and corrects for the lack of independence, all while allowing lambda to change from visit to visit via immigration / emigration.

If you're curious, I recently worked with the wonderful Flavia Montaño‐Centellas on just this sort of model – a multi-season community N-mixture model – which we published in Ecography in the context of community assembly in Bolivia. See here:  https://onlinelibrary.wiley.com/doi/full/10.1111/ecog.05379  (PDF is open access, but email me if you for some reason can't access it). 

[Tom Bradfer-Lawrence]

Hi Morgan,

Thanks so much for your response, and the link to the Montano-Centellas paper. It's really useful. 

If you don't mind, please could you clarify a couple of things? In your email, for visit t > 1 you include the terms phi x Ni,j,t-1. In the paper this is phii,t-1 x Ni,j,t-1. I think it should be the latter? 

From my understanding of the paper, this phi term ties each species' abundance at Nt-1 to Nt ? So I assume that this phi is separate to the data augmentation phi which I included in my model: Ni,j ~ Poisson(phii,j, lambdai,j), and I just need to give the latter another name to distinguish between them?

Many thanks,

Tom

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[Morgan Tingley]

Hi Tom,

For phi as a temporal autocorrelation term, yes I was using a short-hand.  Phi should be species-specific, so it should be indexed by species (i). Whether phi varies from time period to time period, depends. In our case (Montano-Centellas et al.), our time periods were each in different seasons, so we figured that the temporal autocorrelation may not be constant. In your case, given how close your visits are in time, you could perhaps better assume that phi is constant from visit to visit. In that case, phi would just be a vector of length n.species, indexed in your model as phi[i]. 

For phi in your original model, yes, sorry for the confusion, that's a different term (just same greek letter, yikes). 

Although to be honest, I'm not exactly sure what "data augmentation" means in the context of your equation for true abundance, N. A Poisson distribution takes one term, and only one term, lambda. In your equation:

Ni,j ~ Poisson(phii,j, lambdai,j)

I don't know what that means, or even how it would work. If I gave that equation to JAGS, I'm pretty sure JAGS would give me an error. So this is an aside from your original query, but I'm not sure what you mean by "data augmentation" in this context.

cheers,

Morgan Tingley

[Tom Bradfer-Lawrence]

Hi Morgan,

Thanks for the clarification. That's a good point about whether the temporal autocorrelation should vary or not among visits. I will have to give it a bit of thought whether it is appropriate in our case. Perhaps I will vary it between the two halves of the year, which correspond to dry and wet season in our landscape.

Just to explain about "data augmentation". I realise I made an error before regarding the data augmentation phi, I'm getting my Greek letters confused. Kery and Royle's AHM1 uses omega as data augmentation to account for never-detected species in the regional pool. Later on K&R use phi to account for zero inflation, but state that a model could have both data augmentation and zero inflation. Anyway, currently in my JAGS script for the abundance portion of the model (very similar to K&R's script) reads:

for(i in 1:nspec){

    for (j in 1:nsite){

    a[j,i] ~ dbern(phi[i])

    N[j,i] ~ dpois(a[j,i] * lambda[j,i])

log(lambda[j,i]) <- beta0[j] +                      # Habitat-specific intercept

    beta1[i] * LandscapeA[i] +                     # Species-specific slopes

    beta2[i] * LandscapeB[i]

}

}

With the phi term coming from a distribution of phi[i] ~ dunif(0,1) to account for zero inflation. The model does work in its current form. However, given the new approach with the temporal autocorrelation, I think I will start again from scratch and assess whether I need to include a zero inflation component at all. I also suspect that I don't need to include data augmentation for the never-detected species. I guess it depends on the study questions, but I think these rare (or rarely detected) species are likely unimportant for our work.

Many thanks again for your help with this.

Kind regards,

Tom