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Conceptually, what would be the difference between interacting two variables in a regression and combining these two variables into one via a similarity measure? I'm looking at top managers and boards in terms of their functional backgrounds (outcome is internationalization). In one model I interact the two while in another one I combine both variables into one using either cosine or jaccard similarity (so TMTs and Boards with similar backgrounds get a score closer to 1, while dissimilar groups one closer to 0). I'm just not sure how to interpret both models (one is significant, the first one, and the other one isn't). Would it be fair to say that an interaction looks at complementarity between these two groups while a similarity measure at, sorry for the redundancy, 'similarity"?

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An interaction term between two variables used as predictors in a regression is the product of their individual values. A significant interaction means that the association between one variable and outcome depends on the value of the second variable. It doesn't say anything directly about how well the two variables are associated with each other.

That's a good deal different from what you get by trying to combine the variables by similarity. Your similarity measures look specifically at how well the two variables are associated with each other, and uses the strength of that association as the predictor for outcome in the regression. It throws away any information about how their associations with outcome depend on each others' values.

Although interactions are standard in regressions, the similarity approach might make more sense for your problem. That decision about how to proceed would depend on your understanding of the subject matter.

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