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I ran a two-way ANOVA and got a significant interaction. I ran a Tukey.HSD() post-hoc test in R and no pairwise comparisons were significant. Is this an error on my part? My adviser insists that this is not possible. If it is possible, why does this happen?

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It is certainly possible and does happen quite frequently, especially if there are many pairwise comparisons (which is likely the case if you're investigating an interaction term).

The Tukey procedure controls the Type I error rate and requires a larger difference to declare significance compared to if no adjustment was used. The ANOVA F-test uses MSE in the denominator which borrows information from all the data and is not affected by this adjustment.

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