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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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marked as duplicate by Nick Stauner, gung, Glen_b, Nick Cox, mpiktas Mar 24 at 7:22

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up vote 6 down vote accepted

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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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? –  user3288202 Mar 23 at 20:24
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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. –  Patrick Coulombe Mar 23 at 20:26
    
Ok I think I get what you mean. The effect of A is conditional by B. Thanks very much Patrick Coulombe. –  user3288202 Mar 23 at 20:30
    
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? –  user3288202 Mar 23 at 20:58
    
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). –  Patrick Coulombe Mar 23 at 21:01

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