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I was wondering in anova whether it is possible to have a three-way interaction without having any two way-interaction. Does it make any sense to hypothesize this? Does it make any sense to hypothesize that there is an intercation but no main effect?

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Sure. It's easier to write about your last question (interaction with no main effect) so I will do that, but the logic is the same.

One way for this to happen is if one independent variable is dichotomous and the other continuous and the relationship between the continuous IV and the DV is in opposite directions for the two levels of the other IV.

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  • $\begingroup$ Would it be reasonable to say that, even though interaction terms without main effects (and higher order equivalents) are possible in principle, the type of mechanisms we see in science and social science tend not to show this behaviour, such that one should be extra weary of overfitting when this is observed? $\endgroup$
    – LBogaardt
    Jun 20, 2020 at 16:58
  • $\begingroup$ I think you should look at it on a case by case basis. $\endgroup$
    – Peter Flom
    Jun 20, 2020 at 17:36

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