Say we have a phenomenon where we want to see if x is related to y where x is continuous. Further, there is an opposite effect of x on group 1 compared to group2. Say for group 1, increasing x is associated with increasing y, for group 2, increasing x is associated with decreasing. (this is not realistic but think medication that works very well for one group but is poisonous for another.(adsbygoogle = window.adsbygoogle || []).push({});

So if I do regression ##y\sim x+\epsilon## I would expect to get 0 association. Since they would cancel out on average.

So I know more appropriate model is ##y\sim x+I(G2)+x*I(G2)+\epsilon## where I(G2) is an indictor for belonging to group 2 with a value of 1 and 0 if it belongs to group1.

But what if I want to interpret the association between y and x while holding the group constant? then the equation would be ##y\sim x+I(G2)+\epsilon## since in regression, that is what one does. But in this case, how would that make sense?

I would interpret ##\hat{\beta_{x}}## as the difference in mean of y holding the group status constant? What does that even mean in this case? How can we get an unique estimate for ##\beta_{x}## when holding the group constant when we don't even know what group it is? Does this mean that ##y\sim x+I(G2)+\epsilon## is just invalid as a model?

I know that ##y\sim x+\epsilon## is valid because it just averaged over group. that is,## E(y|x)=E_{group}(E(y|x)|group))## and is just the model that produces the marginal interpretation. And from the interaction equation, ##y\sim x+I(G2)+x*I(G2)+\epsilon## we can get valid interpretations as well.

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# A Concept of Interaction

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