How to test for an "interaction" between two interactions

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.

• 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.
– whuber
May 2 at 14:20
• 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? May 2 at 14:27
• This seems to be the same as your question here. If so, please choose one to delete, to help minimize duplications on this site.
– EdM
May 2 at 14:32
• 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.
– whuber
May 2 at 14:51