[R-sig-ME] Negative binomial in glmmadmb

[Ben Bolker]

On 14-07-17 02:41 AM, Daniel Moreno Fernández wrote:
> Dear Mr. Bolker,
> 
> My name is Daniel Moreno-Fernández. Currently, I am working with
> forest regeneration data. I would like to assess how the
> environmental factors evaluate the influence of environmental
> factors on natural regeneration of forest species. Thus, we
> installed several plots in the forest and each plot was subdivided
> into subplots. We counted the number of young trees (seedlings) in
> each subplot. Due to the variance is higher than the mean we used a
> negative binomial. In addition, the hierarchical desing makes 
> necessary to enter a plot random effect. We assumed a constant
> value of the dispersion parameter ( family="nbinom"). We fitted
> following model using the glmmadmb function in R:
> 
> model1<-glmmadmb(Seedlings~ph+calcium +(1|Plot), data=reg, 
> zeroInflation=FALSE,   family="nbinom",  link="log")
> 
> I am worried about some doubts about the the random effect and the
> residual error (I have hardly worked with Generalized linear
> models):
> 
> - Are the random effect and the residual error within the
> log-link? - Concerning the properties of the the random effect and
> the residual error, I expect that both follow a N ~ (0, var), am I
> right?
> 
> Yours sincerely,
> 
> Daniel
> 

  [cc'ing to r-sig-mixed-models]

  You might be a little bit confused about negative binomial models.
The model fitted by glmmadmb in this case is

   Seedlings ~ NegBinom(mu,theta)         (1)
   mu = exp(b0+b1*ph+b2*calcium + eps_i)  (2)
   eps_i ~ Normal(0,sigma_p^2)            (3)

The effect of the log link shows up as the exponential (inverse-link)
in eq. 2.  'Residual error' is a bit hard to define for GLMMs, but we
can say that the _conditional distribution_ of the response (i.e.
conditional on the plot-level random effects) is negative binomial.

   Ben Bolker