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One question about three-way interaction terms. Let's label each variable:

  • $A$: main variable
  • $B$: 1st moderator
  • $C$: 2nd moderator

I'm interested in hypothesizing the relationships $A$-$B$ and $A$-$B$-$C$. Should all two-way ($A*B$, $A*C$, $B*C$) and three-way interaction terms ($A*B*C$) be included in a regression model and result or would be it fine to include some of interest ($A*B$, $A*B*C$) only?

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  • $\begingroup$ Welcome to Cross Validated! Your question seems to be an extension of this question about 2-way interactions. The same principles will apply: for inference it's usually important to keep all lower-level terms in a model with interactions. There can be exceptions in some circumstances, discussed on that page. Please read that page and then edit your question to ask specifically about what's still unclear. $\endgroup$
    – EdM
    Commented Feb 6 at 21:07
  • $\begingroup$ Thanks for your quick reply. Should including all the two-way interactions be essential empirically even when some of them are not hypothesized? Otherwise, would the model be under-specified? $\endgroup$
    – Sangcheol Song
    Commented Feb 7 at 1:28

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The general rule is to include all the lower level effects from an interaction. So, you include all the main effects, and all the two way interactions when you have a three way interaction. This is known as the hierarchy principle. See Chapter 9.6 of Penn State's book "Regression Methods" also see this thread (the thread is mostly about including main effects when there is a 2 way interaction, but the same principles apply). In particular, note the somewhat different answers by Frank Harrell and Whuber, two of our highest-reputation contributors.

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  • $\begingroup$ Thanks for your quick reply. Should including all the two-way interactions be essential empirically even when some of them are not hypothesized? Otherwise, would the model be under-specified? $\endgroup$
    – Sangcheol Song
    Commented Feb 7 at 1:07
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    $\begingroup$ In my opinion, you should generally include them because it makes interpretation easier. But see the discussion in the earlier thread for some more views. $\endgroup$
    – Peter Flom
    Commented Feb 7 at 1:32

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