# How do I interpret the results of a regression which involves interaction terms?

If I have a linear regression of the form:

$$y \sim 1+\beta_1x_1 + \beta_2x_1x_2,$$

where $x_2$ is a Boolean variable depending on another variable $x_{2cont}$, a positive variable:

$$x_2 = \begin{cases} 1 & \mbox{if } x_{2cont} > 5 \\ 0 & \mbox{if } 0 < x_{2cont} \le 5. \end{cases}$$

And I got regression results. How shall I interpret the regression results?

esp. how shall I interpret the two coefficients I get from this regression, $\beta_1 \text{ and } \beta_2$?

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2
Perhaps you want to give this question a shot yourself and people can chime in with additional insight? I also recommend avoiding using the letter B as a variable name because it usually represents a coefficient, for consistency stick with x1, x2, x3, etc. Finally, you should be default include both terms in the interaction independently as well, which would give you 3 coefficients. – Michael Bishop Oct 28 '12 at 21:11
I fixed up your formatting, in part to match @MichaelBishop 's excellent suggestions. In addition, you shouldn't turn a continuous variable into a dichotomous one except under very unusual circumstances and you should include an intercept except under very unusual circumstances. – Peter Flom Oct 28 '12 at 21:20
Thanks guys for your suggestions. I've added the intercept terms. Yes, indeed I did it with the intercept term. But I couldn't figure the interpretation... could you please help me? Thank you! – Luna Oct 28 '12 at 21:49

## 1 Answer

$\beta_1$ describes how $y$ changes for a one-unit change in $x_1$ if $x_2 = 0$

The sum of $\beta_1$ and $\beta_2$ describes how $y$ changes for a one-unit change in $x_1$ if $x_2 = 1$

$\beta_2$ describes the difference in the change in $y$ per one-unit change in $x_1$ between $x_2 = 0$ and $x_2 = 1$

I think your notation is still not standard. Also, according to the principle of marginality you should include all main effects of the interactions you include, so here this means that a main effect for $x_2$ should be included (to estimate the part of the effect of $x_2$ that is independent of that of $x_1$). I think your model should look something like

$E(Y|X) = \beta_0 + \beta_1X_1 + \beta_2X_2 + \beta_3X_1X_2$

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