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In linear regression we can use A:B to show the first order interaction of variable A and B. But it is hard to know what effect was caused by A and what was caused by B. So I want to understand how exactly A:B worked

Firebug
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user162385
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2 Answers2

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it is hard to know what effect was caused by A and what was caused by B

Actually it is not hard, but impossible. First, interaction term in regression tells you on effect of A and B together, rather then about their individual effects. Second, regression per se does not tell you anything about causality.

Tim
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In R syntax A:B includes $A \times B$ in the regression model so

lm(y~A+B+A:B,data=mydata)

is fitting $$ Y=\beta_0+\beta_1A+\beta_2B+\beta_3AB+\epsilon $$ There is a discussion of this in the book "An Introduction to Statistical Learning" by James et al.

PM.
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    More precisely: it's $A\times B$ or if both A and B are discrete, it's all combinations of pairs of values of A and B that were found in data. The resulting model matrix is obtained through model.matrix function. – Tim May 23 '17 at 08:17