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I am working with count data and trying to understand if my model fit is acceptable for this poisson Generalized Linear Mixed Model:

Richness.glmer<-glmer(Richness ~ Unit.type + plot.type + (1|NFI.Point), family = poisson, data = PartA.data.Birds )

R output for GLMM

I ran a dispersion test using overdisp_fun and got the following result:

chisq = 55.90, ratio = 0.65, p = 0.995, logp = -0.00489

With a ratio of 0.65, does this mean my model is acceptable? Or is it underdispersed? I also made the following residual plot of my data, which appears to be okay, but I am not sure if the combination of this residual plot that the 0.65 dispersion ratio indicates a good model fit or not

enter image description here

I understand what I would do if I was running a Generalized Linear Model that had overdispersion issues (which seems to be the most common problem). But in my case with my GLMM I am uncertain of:

1). If my model fit is acceptable, and, if not 2). What to do about it.

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A ratio below 1 suggests that you have under-dispersion. However, 0.69 is not very small and might be due to sampling variation, particularly since you have only 24 observations in total, so I would not be too concerned at this point.

Under-dispersion can arise from a poorly specified model. If you had more data I would suggest trying a random slopes model and possibly a model with an autocorrelation structure. Bootstrapping is another option, but again, with so little data, it may not be very reliable.

Note also that your random intercept has very low variance, so you might try comparing the model with a regular glm() and also using Conway-Maxwell-Poisson regression, available in the compoisson package for R, which specifically handles under-dispersed count data.

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    $\begingroup$ Thank you @RobertLong. This is good food for thought (although I have some work to do before I will understand all of it). Also, you picked up an error in my code. I re-worked it and edited my post so you'll see I have 90 observations, not 24 as noted earlier. I also re-did my residual plot. I am curious to know if this changes any of your ideas on this. I have a difficult time understanding what qualifies as a "not very small" dispersion ratio. Are there any rules of thumb for interpreting these things? $\endgroup$ – tri Jul 21 '16 at 16:48
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    $\begingroup$ @tri I'm not a fan of rules of thumb. They have a tendancy to become de facto rules, like p < 0.05 or rmsea < 0.05 or TLI > 0.9 etc etc. With 24 groups and 90 observations your sample isn't enormous, but you could get a parametric and/or semi-parametric bootstrap estimate of the confidence interval for the ratio, and see if it's upper bound is over 1 in which case you can be happy. Also try a CMP model as I suggested and compare it with the poisson glmm and a regular poisson glm $\endgroup$ – Robert Long Jul 21 '16 at 20:18

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