7.5 Summary

Chapter 7 in review

Chapter 7 introduced interactions — the way that the effect of one predictor depends on the value of another. Interactions are essential for capturing context-dependent effects.

We covered:

Building an interaction (Section 7.1): The interaction term $\beta_3 x_i z_i$ allows the slope of $x$ to depend on $z$ . Interactions should be included when theory demands them.
Symmetry (Section 7.2): Interactions are symmetric — the effect of $x$ depends on $z$ and vice versa. The same coefficient governs both perspectives.
Continuous interactions (Section 7.3): When both predictors are continuous, the interaction creates a twisted plane. Centering and plotting predictions are essential for interpretation.
Design formulas (Section 7.4): R's * operator includes main effects and interaction; : includes only the interaction. The formula is an interface, not the model.

Key lessons

Interactions allow the effect of one variable to depend on another.
The slope of $x$ is $\beta_x + \beta_{xz} z$ — it varies with $z$ .
Interactions are symmetric: both variables moderate each other.
Center predictors and plot predictions to interpret continuous interactions.
Include interactions when theory demands them, not just when they are significant.

Rethinking

Interactions are the gateway to complexity. Until now, our models assumed that each predictor has a constant effect. Interactions break this assumption, allowing the model to capture the reality that effects depend on context. This is the gateway to more complex models — multilevel models (Chapter 12) allow interactions at the group level, and generalized linear models (Chapter 9) allow interactions with the link function. The key lesson is: if your theory says the effect depends on context, you must model that dependence. The golem will not do it for you.

Interactions in design formulas

Practice