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

[Juan Dueñas]

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