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I am analyzing data that is gamma distributed. Hence, an ANVOA was a good choice but a GLM with gamma distribution worked well. To report the data I want to compare all groups to the control treatment and use the Holm method for adjusting the p-value. For this I used the glht function in r (multicomp package) and this also worked well.

My concern is the following:

Is it correct to use the glht or similar package (such as emmeans) if the data are not normally distributed or should I use a different way of performing the post-hoc tests?

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There is no problem using those packages. They have methods that recognize the model was fitted with glm() instead of lm() and adjust appropriately.

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