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I am using linear mixed models to identify important factors, and it turns out that:

  • A: significant
  • B: not significant
  • A×B: significant

Does it mean that because A×B shows that the effect of A depends on the effect of B, only the effect of A is not actually significant?

I have read many sources, and they seem to suggest that if the effect of A×B is significant, then we cannot interpret that the effect of A is significant on our dependent variable. Am I understanding right?

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What you read is correct. If the interaction is significant, interpreting either main effect, whether significant or not, is basically pointless (and misleading). The reason is that when $A$ and $B$ are involved in an interaction, the coefficient for $A$ is the effect of $A$ when $B=0$; in other words, the effect is conditional on the value of $B$, and is not a main effect. Similarly, the coefficient for $B$ is the effect of $B$ when $A=0$.

The fact that $A$ is significant merely means that $A$ has an effect when $B=0$. Similarly, the fact that $B$ is not significant merely means that $B$ doesn't have an effect when $A=0$, though it probably does have an effect for other values of $A$; this is precisely why the interaction is significant.

What you would need to do is look at simple slopes, which shows the significance of the $A$ effect as a function of the $B$ variable. You need to determine at which values of $B$ does $A$ have an effect, and vice-versa. Kris Preacher provides an online tool to decompose 2-way interactions in linear mixed models.

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  • $\begingroup$ Sorry so it means that I cannot interpret A as having a main effect for DV? In other words, if the results are just like this, I should ignore the main effect of A? $\endgroup$ – user3288202 Mar 23 '14 at 20:24
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    $\begingroup$ You should decompose the interaction. You can interpret the significant A effect as "A has an effect when B=0". If you want the significance of the A effect for other values of B (which you surely do), then you need to look at simple slopes (i.e. decompose the interaction). You can do this by using the online tool that I link to in my answer. $\endgroup$ – Patrick Coulombe Mar 23 '14 at 20:26
  • $\begingroup$ Ok I think I get what you mean. The effect of A is conditional by B. Thanks very much Patrick Coulombe. $\endgroup$ – user3288202 Mar 23 '14 at 20:30
  • $\begingroup$ Oh Patrick Coulombe, could I ask if I would like to know only effect of A to DV without any condition, then I will have to remove B and B*A from this model, right? $\endgroup$ – user3288202 Mar 23 '14 at 20:58
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    $\begingroup$ You will have to remove BA (and not necessarily B). If you retain B without BA, the interpretation of the A effect is the effect of A controlling for B (holding B constant). In that case, the A effect is interpretable as a main effect (its interpretation is not conditional on the value of B). $\endgroup$ – Patrick Coulombe Mar 23 '14 at 21:01

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