I'm actually reading the book "A handbook of statistical analysis using SAS"

When talking about an ANOVA analysis the author discuss the results of a model in which a dependent variable is analyzed with the three independent variables diet, biofeed and drug:

The diet, biofeed, and drug main effects are all significant beyond the 5% level. None of the first-order interactions are significant, but the three-way, second-order interaction of diet, drug, and biofeedback is significant. Just what does such an effect imply, and what are its implications for interpreting the analysis of variance results? First, a significant second-order interaction implies that the first-order interaction between two of the variables differs in form or magnitude in the different levels of the remaining variable. Second, the presence of a significant second-order interaction means that there is little point in drawing conclusions about either the non-significant first-order interactions or the significant main effects.

I can't understand the interaction explanation. I know that the interaction coefficient is statistically significant when the magnitude of the effect vary across the values of the covariates but what about the second order interaction term?


1 Answer 1


First, consider the main effects and what significant two-way interactions mean. If drug and diet had a significant interaction it would mean that the average effect of diet differed by levels of drug... maybe at low drug levels the diet has a positive effect but at high drug levels, the diet has a negative effect. Thats a two-way interaction and thats why it doesn't make sense to interpret the main effects in the presence of an interaction (diet doesn't have an 'average' effect (main effect) at all, rather the effect depends on the the drug). Now consider the three-way interaction of diet x drug x biofeedback. You've just uncovered a two-way interaction between drug and diet; the presence of the 3-way interaction tells you that at different levels of biofeedback, the nature of the 2-way interaction is different. So maybe diet has a positive effect at low levels of drug but a negative effect at high levels (your 2-way interaction) but what if that's only true at low levels of biofeedback? Maybe this interaction effect is only true at low levels of biofeedback, while at high levels of biofeedback there's no difference between diet effects by drug. That would produce a 3-way interaction. So, since the 2-way interaction depends on the 3rd variable, it's not appropriate to talk about it as an interpret-able 2-way interaction.


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