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To check for overdispersion in GLM with a Poisson distribution one can compare the residual deviance with the residual degrees of freedom. If they are equal the Poisson error assumption is appropriate (The R Book, Crawley, 2007). If the residual deviance is bigger than the residual degrees of freedom overdispersion is indicated. Crawley indicates that overdispersion for Poisson distributions can be corrected with the quasi-Poisson distribution.

My question is how can I check for overdispersion with the Gaussian distribution and how can I correct for it?

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how can I check for overdispersion with the Gaussian distribution and how can I correct for it?

The Poisson and the binomial have a variance that's a fixed function of the mean. e.g. for a Poisson, $\text{Var}(X)=\mu$, so it's possible to have some count data which has $\text{Var}(X)>\mu$, i.e. more dispersed than would be expected for the Poisson. There's no corresponding situation for the Gaussian. [If variance were some fixed value, like $1$, then a sample with larger variance would be overdispersed, but in the Gaussian family it's just another Gaussian.]

Since the Gaussian has a variance parameter, more dispersion will just be a larger variance parameter... so you don't have overdispersion with the Gaussian.

So there's nothing to correct. (On the other hand, changing dispersion would be an issue to deal with)

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  • $\begingroup$ To this can be added the fact that the variance and mean of a Gaussian typically don't even have the same dimensions or units of measurement. So comparing variance and mean is usually nonsensical. In a way, this is the same point from another perspective: not only is there no relationship; there could not be. (The exceptions would be when the variable being considered has no units.) $\endgroup$
    – Nick Cox
    Apr 2, 2015 at 12:37

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