2^k Factorial Linear Model
The linear model for a two-level factorial design includes main effects and interaction terms:
$$ y = \beta_0 + \beta_A x_A + \beta_B x_B + \beta_C x_C
- \beta_{AB} x_A x_B + \beta_{AC} x_A x_C + \beta_{BC} x_B x_C
- \beta_{ABC} x_A x_B x_C + \varepsilon $$
for a $2^3$ design. The same logic extends to general $2^k$ designs.
This model is the regression version of factorial-effect analysis.