Assume I have a simple model with one dependent variable $Y$, two moderators, $W,Z$, one predictor $X$. To keep it simple, I want to model 3-way interaction effects.

My question is if we can model 3-way interaction effects while dropping lower level interaction effects.

For instance, based on the previous example we have:

$Y = b_0 + b_1X + b_2Z + b_3W + b_4XZ + b_5XW + b_6ZW + b_7XZW$ (model 1)

This is the case with two moderators, which leads to the interaction effect $XZW$.

Could you create a model that does not contain all of the above interaction effects, but only the chosen ones stated below? From my understanding you cannot (you need all of the above effects), that is, the following is wrong:

$Y = b_0 + b_1X + b_2Z + b_3W + b_4XZ + b_5XZW$ (model 2)

But someone recommended me to keep it simple and drop some terms, so I want to be sure.


1 Answer 1


In practice it will be possible to put them in a model, however it does not make much sense since any higher-dimension interaction will be forced to account for the effect of the lower-way interaction.

In your example, the interaction WX is missing. Any interaction between this will be modeled indirectly through XZW.

If you are looking to interpret a model with interaction, I would suggest to create one level for each cross interaction instead. Grouping different levels based on background knowledge or inference, will make much more sense.

  • $\begingroup$ thanks for the reply, but I am not sure yet if I can follow you. I added labels to the models, so we are clear which model we are speaking about. So you are saying that WX is missing in model 2, but that does not matter, because it will be indirectly in XZW? And you final suggestion is still to use model 1 in any case (one level fo each cross interaction)? But then you are also saying grouping levels based on background knowledge makes more sense, which would go more in the direction of model 2. $\endgroup$
    – EtoAls
    Jun 16, 2021 at 19:01
  • $\begingroup$ Exactly. If you have a 3-term interaction, you assume that there is an effect of x, w, and z. Thus you indirectly assume tat there is an interaction between x and w. So the model without the term doesnt make sense, because you assume an interaction but don't model it. $\endgroup$
    – Kirsten
    Jun 17, 2021 at 9:20
  • $\begingroup$ If you group variables, you estimate the effects differently, by simply creating a new variable v containing all relevant levels of x, w and z. Note that now I assume that both w,x and z are discreate/grouping variables. Based on prior knowledge or test results you can now combine levels of the variable v as relevant. $\endgroup$
    – Kirsten
    Jun 17, 2021 at 9:23

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