Let's say I want to fit a model that relates a target variable $Y$ to a set of predictors $X_1, ...X_n$. Let's also assume that two of them ($X_1$ and $X_2$ for example) are correlated but allow for some variance (we can think of the height and weight of a person for instance).

As it's a common problem in linear regression models, if I just fit a multivariate linear model, there will be no way to distinguish between the effect of $X_1$ and $X_2$ height and weight. My question is, if I add an interaction term, what would I be achieving exactly? On one hand, I can now distinguish the effect from either of them from the effect of both combined, but on the other I've made the original problem even worse since now I have three collinear variables

  • $\begingroup$ In the presence of interaction terms it's standard practice to ignore the main effects used in creating them and focus on their relationship. $\endgroup$
    – user234562
    Aug 12, 2020 at 12:55
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    $\begingroup$ @user332577 I have never seen that practice before: could you supply a reference? $\endgroup$
    – whuber
    Aug 12, 2020 at 13:11
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    $\begingroup$ @whuber Ha! Although I'm confident of having read that claim somewhere, at the moment, I can't directly support it. Specifying interactions is, as you know, a subject of considerable controversy. The 1991 Sage book Multiple Regression is a bit dated now but it contains a rigorous and thorough discussion of specifying and interpreting interaction terms backed up with numerous applied examples. What they do recommend is centering the Xs to a mean of zero before creating any interaction as a way of reducing collinearity down to "essential ill-conditioning" (pgs 32-36) $\endgroup$
    – user234562
    Aug 13, 2020 at 12:08
  • $\begingroup$ The common conflation of "correlation" with "interaction" may play a role here. Some people say that you should include interaction in the case of correlated variables, and I think their logic was that they are identical concepts. This is, of course, horribly misguided. Also, to amplify user332577, the phrase "I can now distinguish the effect from either of them from the effect of both combined" is dubious. A "main effect" is really just the effect of a variable when the other variable is equal to 0. And the "combined effect" is really an effect of one variable on the effect of the other. $\endgroup$ Aug 15, 2020 at 17:47


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