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I am using lmer in R to run LMM. My DV is continuous and my IVs are categorical. Many statistician said if the three-level parameter is significant, I cannot interpret the two-level parameter. Does it also include when the parameters are not contradict to each other too?

For example, in the two-level, Group2:color2 is significant, but in three-level, Group2:color3:male is significant. Does it mean that I can interpret both levels as they are not contradict to each other, i.e. The first one is telling me that group2 and the other group is different in color2 than the baseline color, whereas the second one is telling me that the effect of group2 and color3 is higher in male compared to female. Or should in just ignore the two-level interaction regardless of whether than they contradict to each other or not?

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You need to take all levels into account when interpreting 3 level interactions, which is why they are so hard to interpret.

It often helps to look at predictions for all combinations of the three variables. Alternatively, and in some sense equivalently, you can use some tricks discussed here. This refers to Stata, but I would not be surprised if similar tricks are easy to implement in R.

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  • $\begingroup$ Thanks Maarten Buis. I am quite surprised while some people said I should just interpret three-way level and ignore the lower levels if three-way interaction is significant, you were saying that we should interpret from all levels. $\endgroup$
    – user3288202
    Commented Apr 10, 2014 at 14:43
  • $\begingroup$ The third level interaction term is the difference in difference in difference. It is hard to make sense out of that without considering the lower levels. $\endgroup$
    – Maarten Buis
    Commented Apr 10, 2014 at 15:12
  • $\begingroup$ Yes that is also true. But that might mean that I will have to change the baseline a lot of times, doesn't it? $\endgroup$
    – user3288202
    Commented Apr 11, 2014 at 8:09
  • $\begingroup$ That is one way, but not the way I would choose. Instead I would use the methods I proposed in my answer. Regardless, understanding such a model is a lot of work and presenting the results such that your audience understands it quickly (as they don't want to invest the same amount of time) is even more work. It can be done, but it cannot be done quickly. $\endgroup$
    – Maarten Buis
    Commented Apr 11, 2014 at 8:25
  • $\begingroup$ Ok I see you point. Thank you very much for your replies. $\endgroup$
    – user3288202
    Commented Apr 11, 2014 at 10:08

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