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Suppose I'm building a logistic regression model y~ax1 + bx2. How do I know when I may want to think about adding the interaction term, i.e. y~ax1+bx2 +(a*b)x3? Is it purely domain knowledge, or are there some visuals or statistic tests I can try?

My understanding of interaction is when 2 variables combined produce a new affect. i,e: sleeping pills make you sleep, alcohol make you drunk, but sleeping pills * alcohol will make you dead, which is the new affect. Is this right? Does this "new effect" need to be as dramatic as that?

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V1 might affect V3 V2 might affect V3

V1 might also affect the effect of V2 on V3 V2 might also affect the effect of V1 on V3

The effect of V1 on the effect of V2 on V3 is a first order interaction between V1 and V2 on V3

The effect of V2 on the effect of V1 on V3 is a first order interaction between V1 and V2 on V3

If your objective is to test a theory, use domain knowledge to include or not such and such variable in your model before any measurements be made.

And stick to it until your article is published.

If your objective is to find the optimal model for the phenomena you observe, use information criterion such as AIC or BIC, to select the "best" model among many "candidate models", including and not including interactions. Often the best will be the one that summaries the maximum of your data's information with the minimum number of parameters.

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In general, interaction effects are much weaker than main effects. In my experience, main effects in the presence of interactions capture more than 90% of explained variance with interactions capturing the rest. This suggests that they should be used sparingly.

The question of knowing when to use them can be decomposed into two buckets: confirmatory vs exploratory.

Confirmatory interactions are driven by strong theoretical domain knowledge and refer to testing carefully developed and specific a priori hypotheses in the context of a formal model.

Exploratory interactions are developed when there are more than a few variables and theory is weaker, if it exists at all. They can be automated combinations of 2-way or higher interactions. The problem with a shotgun approach like this is that, even with a small number of variables, say around 10 limited to 2-way interactions, the possible combinations explode into a very large number. One rule of thumb to reduce this to a more manageable number is to use only those variates whose main effects are large.

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