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I hope I can clearly articulate my stats troubles! I am having difficulty interpreting findings related to a significant three-way interaction. My study includes one between-subject variable (Group 1 and 2) and three within-subject variables (B with 3 levels, C with 2 Levels, and D with 2 levels). The DV is accuracy. I have found significant main effects of A, B, C, D, a significant 3-way interaction between B, C, and D, and the following two-way interactions: B X C and B X D. Based on the SPSS output for the 3-way interaction, the contrasts show that Variable B at level 1 is significantly different from B @ level 2 and 3 (but B at Level 2 and 3 is not significant) across C and D. This pattern of results is consistent with the 2 significant two-way interactions (I.E., B X C AND B X D). Since I cannot examine simple effects by splitting files on SPSS, can I interpret the three-way interaction using the significant main effects and two-way interactions? If not, how would I go about interpreting this three-way interaction? thanks in advance for your help!

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    $\begingroup$ Could you provide us with the model formulae you have used, and maybe with an excerpt of your data table and the types (continuous, factor) of your variables? $\endgroup$
    – frank
    Jul 21 at 6:57

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A significant 3-way interaction (here among all of the within-subject variables) means that you can't properly assess the association between any one of those predictors and outcome unless you know the values of the other 2 predictors. Thus the "significant main effects" are potentially misleading, as each represents either the association of a predictor with outcome at baseline levels of its interacting predictors or an average over all levels of the interacting predictors, depending on how the model is structured.

The way to "interpret the three-way interaction" is to recognize that it represents the additional association with outcome beyond what you would expect based on the individual coefficients ("main" effects) and their two-way interactions. That can be hard to explain in words. One good way to proceed would be to present the overall table of ANOVA results and then illustrate the 3-way interaction by showing representative predictions from the model.

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