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This may be a silly question, but here it goes:

When modelling data, is it valid to add to the model 4 main effects but only 1 specific interaction between two of the variables? or one should look into all the interactions between the 4 main effects in order to create a valid model?

I commonly see that when researchers add interaction in a model, they usually add all possible interactions (often limiting it to 2-ways).

Thanks for the answers (references are well-appreciated)

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It is definitely valid. If theory posits only one interaction, but multiple main effects, then there is no reason to "stuff" the model just to have a kind of symmetry.

Note that the model with all interactions would be (much) more complex than one with fewer interactions. As such, I would say that the burden of explanation lies on those who propose using the more complex model, especially if they have no theory to guide them. Numquam ponenda est pluralitas sine necessitate.

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  • $\begingroup$ Thank you for your answer. In my case, the theory does not specifically suggest that only one interaction is important but I used a sequential approach to model my data (e.g. adding one variable at the time) and I found out that when adding only one interaction the model yields a better fit overall. Moreover, all the other interactions are non-significant, therefore, I was wondering whether I can just use the model with one interecation since it has the best fit and it is the only significant interaction. $\endgroup$
    – John Walk
    Jun 26, 2020 at 11:34
  • $\begingroup$ Oy. Stepwise model fitting invalidates all p values and significances. P values presuppose a single model built on a given piece of theory. See here: stats.stackexchange.com/a/20856/1352 $\endgroup$
    – Stephan Kolassa
    Jun 26, 2020 at 11:55
  • $\begingroup$ Thank you for redirecting me the relevant discussion. I'm using AIC and log-likelihood as selective criteria. If that is an invalid process, what would be the optimal approach? Considering that I'm looking into the predictive nature of 4 variable and there is no clear theoretical background, how do I know what is the right model for my data without using a stepwise approach? Thank you in advance, this is very helpful. $\endgroup$
    – John Walk
    Jun 26, 2020 at 12:16
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    $\begingroup$ Hm. If you want to do prediction, then you don't need significance at all, except possibly as a guide - what you are interested in is which model predicts best. (You did mention "predictive nature".) If you are interested in inference, then you should split your sample in half. Use the first half to build a model. Here you can run wild. Do stepwise regression using AIC or p values or whatever. But don't snoop at the second half. Only once you have built a model you feel confident with, apply it to the other half of the data and derive p values, without another model selection step. $\endgroup$
    – Stephan Kolassa
    Jun 26, 2020 at 13:54

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