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Say I have a model with one response variable ($y$, a continuous variable) and two predictor variables ($x_1$ & $x_2$, both continuous). My model includes both the additive effects of these variables and their interaction.

$$ y = \beta_0 + \beta_1x_1 + \beta_2x_2 + \delta(x_1 \cdot x_2) +\epsilon $$

The interpretation of $\delta$ in this case (i.e. the interaction coefficient) is the change in slope of $\beta_2$ given a unit change in $x_1$ (correct me if I'm wrong).

However, the data I'm working with (and modelling for the purpose of prediction rather than inference) exhibit a more complex interaction than this. The slope of $\beta_2$ seems to change as a quadratic function of $x_1$. Is there any way to incorporate this into my linear model?

The dataset I'm working with is too large to paste here, and if I could easily reproduce the problem I probably wouldn't be asking the question. I expect there is a simple solution and lack of coffee is just getting the better of me...

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  • $\begingroup$ Welcome to the site. Can you post some output from the model to help us understand what you're dealing with? $\endgroup$
    – Marquis de Carabas
    Apr 28, 2016 at 21:01
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    $\begingroup$ Why not include a term $x_3\equiv x_2^2$? $\endgroup$
    – StatsStudent
    Apr 28, 2016 at 22:42

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In principle you could try to model

$y = \beta_0 + \beta_1 x_1 + \beta_2 x_2 + (\beta_3 x_1 + \beta_4 x_1^2)x_2 + \varepsilon$

In practice I would model the main effects at least as flexible as the interaction effects, as otherwise any non-linearity in the effect of the main effect tends to have a lot of influence on the interaction effect. So I would try

$y = \beta_0 + \beta_1 x_1 + \beta_2 x_1^2 + \beta_3 x_2 + (\beta_4 x_1 + \beta_5 x_1^2)x_2 + \varepsilon$

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