# regression with interactions interpretation

Suppose I have the following model

$$y = \alpha + \beta_1 X +\beta_2 D + \beta_3 XD +\epsilon$$

where $$X$$ and $$y$$ are continuous variables and $$D$$ is a dummy.

If $$\beta_1$$ is equal to some number, say, 0.2 and not statistically different from zero, and $$\beta_3$$ is equal to say $$0.15$$ and statistically significant, can I infer automatically that $$\beta_1+\beta_3$$ ( the total effect of $$X$$ on $$y$$ when $$D=1$$) is statistically different from zero?

The alternative is to run:

$$y = \alpha + \beta_1 X(1-D) +\beta_2 D + \beta_3 XD +\epsilon$$

in which case I just have to look at whether $$\beta_3$$ is significant.

In your question, are the numeric values you provide estimated values of $$\beta_1$$ and $$\beta_3$$? If so, you are better off to denote these estimated values with $$b_1$$ and $$b_3$$ to make it clear they are your best guesses from the data as to what the true unknown values of $$\beta_1$$ and $$\beta_3$$ are. In other words, $$b_1 = 0.2$$ estimates the unknown parameter $$\beta_1$$ and $$b_3 = 0.15$$ estimates the unknown parameter $$\beta_3$$.

With the notation clarified, you can set up the following test of the contrast of interest:

Ho: $$\beta_0 + \beta_3 = 0$$

Ha: $$\beta_0 + \beta_3 \neq 0$$

The test statistic for this test is $$t = (b_0 + b_1 - 0)/var(b_0 + b_1 - 0)$$, where var denotes the estimated variance of b_0 + b_1 - 0. (The 0 doesn't matter in the nominator and denominator of this test statistic, but I left it in for you to see that it comes from the right hand side of your null/alternative hypothesis.)

If the null hypothesis Ho is true, this test statistic will have a $$t$$ distribution with degrees of freedom equal to the residual degrees of freedom of the model.

You should be able to implement this contrast test in the statistical software you are using. For example, in R this can be implemented using the multcomp package.

• Thank you @Isabella Ghement. Is there an advantage to doing this compared to just running the alternative model ? Nov 18, 2019 at 17:05
• The advantage comes in when you are interacting X with a categorical dummy variable with k categories, in which case your model would include k-1 dummies and k-1 interactions. Then you can simultaneously test all contrasts you are interested in. In your example, k = 2, so you are only testing one contrast. Nov 18, 2019 at 17:07