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I'd like asking your help to understand a statistical issue from my data set. I ran a GLM with proportional data, using a binomial distribution. However, I've found underdispersion in my model and I don't know how to deal with that. I'm aware that a solution for overdispersion is fit a model using a quasibinomial distribution, but I couldn't find a solution to my problem in the literature.

I'm comparing the differences between continuous forest sites and fragments regarding the proportion of richness and abundance of specialist species. So, the models are:

M1 <- glm(prop_rich_speci ~ LandscapeBin, family = binomial, weights=rich_total_sp, 
          data = envir.all)
M2 <- glm(prop_abu_speci ~ LandscapeBin,  family = binomial, weights=abu_total_sp, 
          data = envir.all)

I think using a quasibinomial distribution I can solve my problem (underdispersion), as Ben suggested.

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    $\begingroup$ What do you mean by proportional data? Do you mean the outcome is a proportion, as in continuous but bounded between 0 and 1? $\endgroup$ – robin.datadrivers Jan 24 '15 at 20:22
  • $\begingroup$ If it's a continuous proportion then you might find a beta model rather than a binomial more suitable. That said, the variance function of a quasibinomial should work well in that case (in that the beta variance is a scaled version of p(1-p)), and will work for underdispersion as Ben suggests below. If your data aren't typically close to 0 or 1 it may do very well. $\endgroup$ – Glen_b Jan 24 '15 at 23:26
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My answer from http://article.gmane.org/gmane.comp.lang.r.general/316863 :

short answer: quasi-likelihood estimation (i.e. family= quasibinomial) should address underdispersion as well as overdispersion reasonably well

If you just want to assume that $\textrm{variance} = \phi \cdot N p(1-p)$ with $\phi < 1$, quasi-likelihood estimation will work fine. Depending on the source of your underdispersion, how much you're concerned about modeling the details, other aspects of your data, you might want to look into ordinal or COM-Poisson models (both of these approaches have R packages devoted to them).

There is generally less concern about underdispersion than overdispersion; I speculate that two of the reasons are

  • overdispersion is probably the more common problem
  • underdispersion leads to conservatism in statistical inference (e.g. decreased power, lowered type I errors), in contrast to overdispersion which leads to optimism (inflated type I error rate etc.), so reviewers etc. tend not to worry about it as much
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