# When is an interaction in quadratic regression significant?

I run a regression with a continuous variable $$x_2$$ and a binary grouping variable $$x_1$$. We suppose that the effect of $$x_2$$ is quadratic on the dependent variable $$y$$ and our question is whether the effect of $$x_2$$ on $$y$$ depends on $$x_1$$.

The formula is $$y\sim x_{1} + x_{2} + x_{2}^{2} + x_{1} : x_{2} + x_{1} : x_{2}^{2}$$. So I get two signifince tests for interaction, i.e. $$p$$ for $$x_{1} : x_{2}$$ and $$p$$ for $$x_{1} : x_{2}^{2}$$. What of the following statements is true:

• We can say that there is a significant interaction if one of the both interaction coefficiens is significant, no matter whether it is a significant $$p$$ for $$x_{1} : x_{2}$$ or a significant $$p$$ for $$x_{1} : x_{2}^{2}$$
• We can say that there is a significant interaction if both interactions coefficients are significant
• We must first run a model without any interaction, then a model with both interaction terms, than compare both models using ANOVA. If the $$p$$ of the F test is significant we can say that there is a significant interaction.

Or is there another approach one must choose in order to test whether the effect of $$x_2$$ depends on $$x_1$$ significantly?

Edit

To make clearer point three:

model1 <- $$y\sim x_{1} + x_{2} + x_{2}^{2}$$

model2 <- $$y\sim x_{1} + x_{2} + x_{2}^{2} + x_{1} : x_{2} + x_{1} : x_{2}^{2}$$

ANOVA(model1, model2)

The p value of F in this ANOVA will show whether the effect of x2 depends on x1, i.e. whether the interaction is significant.

Don't think of indidivual coefficients when a variable is represented by more than one term in the model. Think of composite (AKA "chunk") tests which test more general hypotheses. Chunk tests use likelihood ratio, Wald, score, or $$F$$-tests and have multiple degrees of freedom. From the model specified above here are some useful chunk tests:

• test all coefficients jointly that relate simultaneously to $$x_{1}, x_{2}$$ (2 d.f., tests whether effect of $$x_2$$ depends on $$x_1$$
• test all coefficients relating to $$x_1$$ (3 d.f., tests whether $$x_1$$ is associated with $$Y$$ for any value of $$x_2$$)
• test all coefficients relating to $$x_2$$ (4 d.f., tests whether $$x_2$$ is associated with $$Y$$ for any value of $$x_1$$)

All these chunk tests are provided automatically in the R rms package's anova function:

require(rms)
f <- ols(y ~ x1 * pol(x2, 2))
anova(f)  # provides F stats since using a linear (ordinary least squares) model


Chunk tests have interpretability and importantly do not depend on the coding/centering of variables. They also provide perfect multiplicity adjustment in frequentist modeling.

• Does this correspond to point three in my question? Also see edit of my question.
– LulY
Commented Aug 13, 2023 at 14:53
• Shouldn't "test all coefficients relating to x2" be df 4? x2 has four coefficiens
– LulY
Commented Aug 13, 2023 at 14:59
• Thanks for the correction; fixed. Yes point 3. Commented Aug 13, 2023 at 15:35

The first one seems right to me.

The second one seems wrong.

The third one answers a somewhat different question: Do the interactions add a significant amount to the model?

However, I think it would be better to be more specific in the question and answer, by specifying the two different interactions.

• You can state the 3rd either way, i.e., it tests whether the effect of one variable varies with the level of the other variable. I wouldn't stress "two different interactions". $x_2$ is expressed as 2 terms in the model and these terms need to be kept together, otherwise the tests are very coding/origin-dependent. Commented Aug 13, 2023 at 15:38