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?


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.


2 Answers 2


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:

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.

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

  • $\begingroup$ 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. $\endgroup$ Commented Aug 13, 2023 at 15:38

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