# How to assess overdispersion in Poisson GLMM, lmer( )

I have a GLMM with Poisson distribution and random spatial block. My experimental design is 2x2 factorial, with 4 blocks, resulting in 16 total data points. Here is the specification of the model in R using the lme4 package.

lmer(rich ~ morph*caged + (1|block),
family=poisson, data=bexData)


When I call summary on this object, I am returned

   AIC   BIC logLik deviance
18.58 22.44 -4.288    8.576
Random effects:
Groups Name        Variance Std.Dev.
block  (Intercept)  0        0
Number of obs: 16, groups: block, 4


I have left out the fixed effect parameter tests and correlations for brevity.

Here are my primary questions:

1. Can you use this output to calculate overdispersion?

• I have read that overdispersion can be calculated as the residual deviance divided by the residual degrees of freedom. Is that 8.576 / (16 - 4)? (Zuur et al., Mixed Effects Models)
2. If this calculation is correct, the estimator phi = 0.715. This indicates that there is not overdispersion in my data.

• Does this indicate that there is underdispersion?
• Is this a problem?
• Can anybody offer advice as to thresholds for over/underdispersion at which corrections to the models should be made? Zuur has said in one book that 5 is a common cutoff. Do people agree with that?
• How can such corrections be made?
3. I've also noticed here that the variance for the random effect is 0.

• Does this mean that there are precisely no error correlations between data points within my blocking factor?
• If this is so, why would a generalised linear model of the form shown at bottom have an AIC substantially higher, around 55?
• is AIC a reasonable method for choosing GLMM over GLM (as suggested by Zuur)?

.

glm(rich ~ morph*caged, data=bexData,
family=poisson)


• See here. Looks like summary() will do it in lmer Oct 5, 2012 at 14:37