[R-sig-eco] question about appropriate model for abundance studies

[Gavin Simpson]

On Fri, 2011-03-18 at 14:14 +0100, Sacha Viquerat wrote:
> hello! I did a count survey on a tropical fish species! Well, I didnt do 
> it, Im just helping at the statistics-stage. We recorded some water 
> parameters alogside each catch, such as no3, no2, po4 etc. as the data 
> are count data (and require error terms due to spatial 
> pseudoreplication), the glm with poisson error structure should be the 
> method of choice. however, I do fear that in doing so, I will not be 
> able to model fish abundance correctly. In my opinion (and, as far as I 
> remember, in the opinion of those who gave ecology classes), the 
> abundance of species should be sort of bell shaped, since there will 
> always be, for example, an optimal temperature, pH-level and so forth. 
> However, I have not yet seen such a discussion arise on one of the many 
> forums. Am I missing the obvious??

I think you are confusing the error structure with the model or
systematic structure? Yes, Poisson errors are expected with count
abundance data, or if there is more dispersion than expected under the
Poisson, then a negative binomial error might be appropriate or failing
that ZIP, ZINB or burdle models.

To fit a bell shaped response curve, you need to fit a quadratic
equation in the response; say no3 + no3^2. The Guassian model of species
abundance is IIRC correctly specified if a Possion error distribution
with a log link is specified.

Say you have species abundances and you wish to model them using a
Gaussian response model, then in R you could do:

mod <- glm(abund ~ no3 + I(no3^2), data = dat, family = poisson)

where `mod` will contain the fitted model, `abund` is the variable with
the single species counts, `dat` is the data frame containing `abund`
and `no3`.

This is fine if you want to use this model to extract the optimum, max
fitted abundance and tolerance of the response, but if you are just
interested in the fitting the response curve then using orthogonal
polynomials should be more computationally robust. 

You can test whether the bell shaped response is a better fit to a
sigmoid (monotonic) increasing or decreasing response (say where you
have only sampled a part of the gradient of relevance to a particular
species or the gradient space/environment is truncated in some way), by
comparing the quadratic model with one that includes the no3 term, eg:

mod2 <- glm(abund ~ no3, data = dat, family = poisson)

then

anova(mod2, mod, test = "Chisq").

HTH

G
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