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I have a question about the interpretation of lower order interaction terms in the presence of a significant higher order interaction effect.

Suppose I have a 2 (factor $A$) $\times$ 2 (factor $B$) $\times$ 2 (factor $C$) design where the highest order interaction ($A\times B\times C$) is significant and a lower order interaction term ($A\times B$) is also significant. Does the significant $A\times B\times C$ interaction render the $A\times B$ interaction uninterpretable (much like how main effects are rendered uninterpretable in the presence of a significant interaction)?

Under this kind of circumstance, should I run a set of post-hoc/planned comparisons to check how different conditions are different?

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It's not that lower order interactions or main effects are completely uninterpretable when there's a higher order interaction. It's that their interpretation is qualified. For example, a main effect of A just means that overall, within the range of the IVs you have chosen, the levels of A differ; but that the magnitude, or possibly direction, of the difference really does vary across the other variables. So, not completely uninterpretable, just qualified. Upon a study of the data you may find you really do believe there is the main effect in general that's just varying in magnitude. Or, you may find that the main effect only exists for a limited range of conditions.

In your case You have an AxB interaction and the magnitude of the interaction is dependent upon C. It's possible the direction is as well but that's relatively unlikely. The AxB interaction you have suggests how to examine the three way. Make 2 2x2 AxB interaction plots, one at each level of C. Your 3-way interaction tells you that the differences in these two interaction plots are worth noting.

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  • $\begingroup$ It bears noting that the interpretation of a main effect in the presence of an interaction depends on the type of sums of squares calculated. As John Fox notes, type II sums of squares obey marginality and assume that interactions to which a main effect is marginal are zero. As such, they are not meaningful when interactions are non-zero. $\endgroup$ – Marcus Morrisey Aug 6 '12 at 19:01
  • $\begingroup$ @john Thanks for the explanation. It's very helpful. $\endgroup$ – user11392 Aug 6 '12 at 21:35
  • $\begingroup$ Yes, good point Marcus. To really interpret the interaction you need to remove the main effects. It might be a good idea to explicitly mention that an interaction plot would have those removed. If I get some time later to update the answer I might add how to do it. Note that I'm only discussing the main effect to give a simpler context for discussing lesser order interactions. $\endgroup$ – John Aug 7 '12 at 1:22
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You might want to run two 2-way, AxB ANOVAs, one for each level of the C variable. That approach will let you look at the "simple" two way interaction effects unaffected by the 3 way interaction. Since there was a statistically significant 3 way interaction, we expect that the two simple 2 way interactions will not look the same.

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We're looking for long answers that provide some explanation and context. Don't just give a one-line answer; explain why your answer is right, ideally with citations. Answers that don't include explanations may be removed.

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    $\begingroup$ Why would you do that? $\endgroup$ – John Aug 6 '12 at 7:17
  • $\begingroup$ I've tried to clarify to respond to your logical question, @John. $\endgroup$ – Joel W. Aug 6 '12 at 12:44
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    $\begingroup$ OK, but I think what you're really saying is look at the two lower order interactions to see how they change. Individual ANOVAs of the two interactions won't tell you they're different from each other, that's what the 3-way interaction is for. For example, if you found that the two way was significant at one level of C and not at the other you've got no new information, just new tests. To clarify, only the 3-way tells you that difference in significance is different, and you already have that. You really just have to look at them. $\endgroup$ – John Aug 6 '12 at 16:18

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