[R-sig-ME] Help - I have an underdispersed glmm :(

[Ben Bolker]

  Can you say a little more about why you expect q1 to be
Poisson-distributed, or more generally what mean-variance relationship
you expect?  Is there a mechanistic/theoretical framework for the
distribution of this variable?  Some reason not to find a transform that
makes the responses reasonably homoscedastic and linear?

  In general, it *only* makes sense to compute/test dispersion for model
families where the variance has a fixed relationship with the mean
(binomial, Poisson, ...), not when there is an estimated scale parameter
(Gaussian, Gamma, ...)

  Ben Bolker

On 18-01-19 08:29 AM, Juan Dueñas wrote:
> Dear all,
> 
> 
> I wish to describe the relationship between the diversity of soil fungi
> and the application of different nutrients (fertilization). My response
> variable is the exponentiated Shannon index of diversity (q1). The
> explanatory variable has four levels. Each of the treatment factors was
> applied at the plot level and there are four replicates of each factor
> per elevation. Six randomly distributed soil cores were taken within
> each of the plots.
> 
> For the GLMMs I used lme4 package version 1.1-15, and vegan 2.4-4 to
> estimate q1.
> 
> One of the problems I have is that q1 takes decimal values, therefore it
> would be inappropriate (or impossible?) to fit my response variable with
> a poisson probability distribution. Therefore I tried gamma for the
> model specification with a log link function. I performed model
> selection with pairwise likelihood ratio tests.
> 
> I then checked my favored model for over-dispersion (which is depicted
> in the output below). It seems, that the model is under dispersed! I was
> checking the literature for solutions to this issue, but I could only
> find some vague notions, namely that some level of underdispersion is
> tolerated. In the case of overdispersion, it is recommended to use
> quasilikelihood, but apparently this solution has been disabled a while
> ago in lme4.
> 
> Generalized linear mixed model fit by maximum likelihood (Laplace
> Approximation) ['glmerMod']
> Family: Gamma ( log )
> Formula: q1 ~ Treatment + (1 | Elevation) + (1 | Elevation:Plot)
> Data: dat
> Control: glmerControl(optimizer = "nlminbw")
> 
> AIC BIC logLik deviance df.resid
> 1523.6 1547.9 -754.8 1509.6 231
> 
> Scaled residuals:
> Min 1Q Median 3Q Max
> -2.0938 -0.6378 -0.0694 0.5815 3.1634
> 
> Random effects:
> Groups Name Variance Std.Dev.
> Elevation:Plot (Intercept) 0.02632 0.1622
> Elevation (Intercept) 0.01366 0.1169
> Residual 0.17924 0.4234
> Number of obs: 238, groups: Elevation:Plot, 47; Elevation, 3
> 
> Fixed effects:
> Estimate Std. Error t value Pr(>|z|)
> (Intercept) 2.63742 0.16504 15.981 <2e-16 ***
> TreatmentN -0.08395 0.13284 -0.632 0.527
> TreatmentNP -0.15163 0.12964 -1.170 0.242
> TreatmentP -0.12925 0.12998 -0.994 0.320
> ---
> Signif. codes: 0 '***' 0.001 '**' 0.01 '*' 0.05 '.' 0.1 ' ' 1
> 
> Correlation of Fixed Effects:
> (Intr) TrtmnN TrtmNP
> TreatmentN -0.412
> TreatmentNP -0.418 0.522
> TreatmentP -0.417 0.524 0.535
> 
> 
> chisq ratio rdf p
> 38.4696552 0.1658175 232.0000000 1.0000000
> 
> 
> My concrete questions are: Should I be concerned that my model is
> underdispersed? Will the coeficients of the fixed terms be reliable in
> this scenario?
> 
> 
> I appreciate any help on this regard.
> 
> 
> Best regards,
> 
> Juan F. Dueñas
> 
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