I have been using the White.adjust=T argument for Two-Way Anovas before and been advised by others that this can be used (it was also discussed here in one long post on heteroscedasticity). It computes coefficient covariances by hccm(). If this is not okay it would be great for me to know. R did not show any warning or error when I used it

I think I will just report Anova results then. In my specific case it only makes sense to make all comparisons or no specific comparisons. Just for my understanding, if it would happen to be a significant effect of factor 1 , significant effect of factor 2 and a significant interaction between factor 1 and 3, I would run a post hoc test like this: response~factor1 + factor 2 + factor1 x factor3 ? (purely hypothetical)

Hi Dale70, Thanks so much for your reply! This is exactly what I had in mind and what seemed logical to me. However, could I then still use a letter design to indicate signficant differences, given that I am not actually comparing all possible combinations with each other?

I have, however, a follow up question in terms of how to display the statistical results in a boxplot or barplot. A common thing after a post hoc test would be to add letters over the, in this case, 4 bars. But as I didn't do all the individual comparisons I guess I cannot do that. What is a common way to do that?

Sorry that I didn't answer that but I actually do not know how to accept an answer.

Many thanks for that fast reply and the clarification. That does make a lot of sense. I must have misunderstood this.

Yes. Sorry. I should have mentioned that!