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I plan to apply a LMM with the following structure: Y ~ A1 + A2 + A3 + A1:A2 + A1:A3 + (1|subject). I expect the two interactions to display opposing patterns (approximately like in the picture). However, I am unsure how to test for this opposing pattern and the negative relation between A2 and A3. My initial idea was to test for a negative correlation between the difference scores A2_1 - A2_2 with A3_1 - A3_2 but I would like to test this hypothesis within the LMM structure, too. Another idea I had was to test for a negative correlation between the two regression weights of A2 and A3 on Y, but as I have only single coefficients in the end I am not sure how to do this. To me it also looks like a three-way interaction but this is not true, as A2 and A3 do not interact with each other. I would be very thankful if anyone could help me with this question.

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    $\begingroup$ Sure $A_2$ and $A_3$ interact with each other: it's just that their coefficient is either not distinguishable from zero or is assumed to be zero. Regardless, you are dealing with a three-way interaction and that means you need to include all related two-way interactions in its interpretation. It's OK that one or more of the latter might be zero. $\endgroup$
    – whuber
    May 2 at 14:20
  • $\begingroup$ Thank you for your comment. So you mean that I should include the three-way interaction in my regression (and all according 2-way interactions) and it should be signigicant? How would I interpret this interaction? Do you maybe have an example paper or a source of someone who did something similar? $\endgroup$
    – rbeginner
    May 2 at 14:27
  • $\begingroup$ This seems to be the same as your question here. If so, please choose one to delete, to help minimize duplications on this site. $\endgroup$
    – EdM
    May 2 at 14:32
  • $\begingroup$ Significance is a separate question from interpretation! The former concerns whether your data enable you to detect an effect, while the latter concerns how to relate your estimates in a meaningful way to your application. $\endgroup$
    – whuber
    May 2 at 14:51

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Your sketches of hypothetical combinations of the 3 binary predictors seem to show 8 different outcome values. That would require a 3-way interaction among those 3 individual predictors to give 8 different outcomes, as there are 8 combinations of 3 binary predictors.

To evaluate that possibility you should include that 3-way interaction along with all of the lower-level interaction and individual terms. It's possible that some of those lower-level terms might be "statistically insignificant," in that you can't prove that you can distinguish them from values of 0. But they all need to be in the model for the 3-way interaction term to make sense.

It's very dangerous to think about "significance" of a lower-level term involved in higher interactions, anyway. With usual treatment coding of predictors, the coefficient for such a lower-level term represents a situation when the terms with which it interacts are at their own reference levels. With an interaction, changing the reference level of some other predictor can thus affect its own apparent "significance."

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