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I have conducted a 2x3 mixed factorial ANOVA, my interaction effect has come out as significant but when I go on to do an independent t-test to determine where the differences are my t-tests all come out as non-significant. My question is how would I interpret this? what does it mean?

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  • $\begingroup$ Welcome to CV. It would help to know which post-hoc test you are using. Regardless, it is likely more stringent than an omnibus F-test for at least 1 parameter being significantly different from zero. $\endgroup$
    – Mike Hunter
    Nov 20, 2015 at 18:40
  • $\begingroup$ Add the means in your conditions to your question. $\endgroup$
    – John
    Nov 20, 2015 at 18:58

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Without further details it is difficult to know your exact issue but you're operating under a mistaken belief that there must be a significant simple effect when you have a significant interaction. As an aside there is no guarantee of any significant simple effect in an ANOVA with more than 2 levels (in which case it's just an alternative to a t-test).

Interactions are about differences among differences. None of the individual differences needs to be significant, or even close to significant, for the differences between the differences to be. Consider that one effect is +5 and the other is -5. The difference between those effects is 10 which is much larger than either simple effect. Therefore, it would be very easy to not have a significant effect of 5 but one of 10.

Further, with something as simple as a 2x3 design you probably should not be conducting any post hoc tests because the important effects are probably obvious. Take your 2 level variable and get the effect for every level of the 3-level variable. Look at the 3 effects. Whatever stands out as differences among those three effects is the important component of your interaction. Don't do a post hoc test unless you want to address something more complex than that.

(You also might want to look at Gelman & Stern (2006) for further exploration into what you are likely doing wrong)

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