Why does a model with an interaction has to include the interaction main effects?

[at474]

Hello fellow stats talkers,
If we are only interested in an interaction, I was wondering why a model with a significant interaction also has to include the interaction main effects.

I was always told that if there is a significant interaction term, you cannot remove the interaction main effects from the model, whether significant or not. Why is model-A = DV~Var1+Var2+Var1:Var2 adequate but not model-B = DV~Var1:Var2?

Additionally, isn't it the case that if an interaction term is significant, we can ignore the main effects anyway? Then why not test a model with only an interaction term, like model-B, from the get-go?

When determining whether an interaction term is significant in an ANCOVA with Weight as the DV, Age as a covariate and Gender as a factor (male and female), the main effects do not enter in the interaction sum of squares equations:

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statistics for male
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SSX(male) = sum(Weight_males^2)-sum(Weight_male)^2/N_cell_observ

SSXY(male) = sum(Age_male*Weight_male)-sum(Weight_male)*sum(Age_male)/N_cell_observ

SSRegression(male)=SSXY(male)/SSX(male)

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statisitcs for females
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SSX(female) = sum(Weight_females^2)-sum(Weight_female)^2/N_cell_observ

SSXY(female) = sum(Age_female*Weight_female)-sum(Weight_female)*sum(Age_female)/N_cell_observ

SSRegression(female)=SSXY(female)/SSX(female)

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statistics for males+females
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SSX(male+female)=SSX(male)+SSX(female)

SSXY(male+female)=SSXY(male)+SSXY(female)

SSRegression(male+female) = SSRegression(male)+SSRegression(female)

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statistic for the common regression
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SSRegression(common)=[SSXY(male+female)]^2/SSX(male+female)

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statistic for the interaction term
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SSRegression(difference)=SSRregression(male+female) - SSRegression(common)

As you can see, determining whether there is a significant interaction between Age and Gender does not involve the main effects. Therefore, if we are interested in determining whether there is an interaction, it seems superfluous to include main effects in a model.

Thank you for your time!

Antoine
Post-doc,
Georgetown University

[Blaz]

Well, I'm hoping I understood your question correctly: It might be convenient to try to look at the problem from a regression point of view. In order to test the significance of the interaction term one needs to include both original terms in the analysis before the interaction. In this way one tests the significance of the interaction term when both of the original variables are controlled for. The reason is that interaction in itself is a nonsense multiplicative variable, and it's interpretative meaning stems directly from the fact that influences of both original terms (or main effects) have been partialised from the DV.

Hope this helps.