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I have a significant three-way interaction (with no significant two-way interactions and no significant main effects).

To begin with I have followed this up with two two-way ANOVAs (one at each level of the third variable). The first part of my question is whether this is an appropriate follow-up.

This showed no significant main effects or interactions at either level of the third variable...is this the end of the analysis?

If so, how would I interpret the three-way interaction?

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A three way interaction means that the interaction among the two factors (A * B) is different across the levels of the third factor (C). If the interaction of A * B differs a lot among the levels of C then it sounds reasonable that the two way interaction A * B should not appear as significant. This could be the case of your data.

To put it another way: A two way interaction A * B exists in reality (not statistically) along with a three order interaction A * B * C only if the way that the factors A and B interacts among the levels of the factor C is similar.

So, use a table or an appropriate error chart in order to visualize the way that the interaction of A, B differs between the levels of C and try to interpret those findings.

If you want to emphasize the differences that you will notice then you may apply standard statistical methods (t - test, Kruskal Wallis etc) and confirm the differences with a statistical test. Keep in mind that in that case it is a good idea to make a Bonferroni correction for the rejection level.

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The three-way ANOVA is used to determine if there is an interaction effect (independent variables interact if the effect of one of the variables differs depending on the level of the other variable) between three independent variables on a continuous dependent variable.

Therefore you would only be interested in the significance value of ABC. If it is significant, you would report that there is a 3-way interaction which means that at least one of the 2-way interactions changes across the third independent variable. If ABC is not significant, it would be better to apply 2-way ANOVA.

Because when interaction effects are present, it means that interpretation of the main effects or underlying lower level interactions is incomplete or misleading. Hence the significances of A, B, C, AB, BC, A*C are not important.

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