Interaction term

An interaction term lets the effect of one predictor depend on another. Inthe slope with respect to $X$ is ${\beta }_1+{\beta }_{3}Z$.

A two-factor interaction means that changing one factor has a different effect at different levels of another. For a two-level factorial design, its sign column is $x_i{x}_j$, and four times its coefficient is the corresponding difference of differences. It can be confounded with other factorial contrasts in a fractional factorial design.

A coefficient multiplying an interaction term controls its strength relative to the chosen field normalization. Its engineering dimension determines whether it is dimensionless, relevant or irrelevant. A renormalized coupling constant constant is specified at a renormalization scale; its change with that scale is described by the renormalization-group beta function.

A hierarchical statistical model that contains an interaction term retains its associated lower-order effects. For two categorical predictors this means keeping both main effects when keeping their interaction. A nonsignificant averaged main effect may coexist with large opposite effects in different groups. Dropping its lower-order term while retaining the interaction can make the model depend on arbitrary factor coding.