# Applied Statistics Lesson of the Day – Additive Models vs. Interaction Models in 2-Factor Experimental Designs

In a recent “Machine Learning Lesson of the Day“, I discussed the difference between a supervised learning model in machine learning and a regression model in statistics.  In that lesson, I mentioned that a statistical regression model usually consists of a systematic component and a random component.  Today’s lesson strictly concerns the systematic component.

An additive model is a statistical regression model in which the systematic component is the arithmetic sum of the individual effects of the predictors.  Consider the simple case of an experiment with 2 factors.  If $Y$ is the response and $X_1$ and $X_2$ are the 2 predictors, then an additive linear model for the relationship between the response and the predictors is

$Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \varepsilon$

In other words, the effect of $X_1$ on $Y$ does not depend on the value of $X_2$, and the effect of $X_2$ on $Y$ does not depend on the value of $X_1$.

In contrast, an interaction model is a statistical regression model in which the systematic component is not the arithmetic sum of the individual effects of the predictors.  In other words, the effect of $X_1$ on $Y$ depends on the value of $X_2$, or the effect of $X_2$ on $Y$ depends on the value of $X_1$.  Thus, such a regression model would have 3 effects on the response:

1. $X_1$
2. $X_2$
3. the interaction effect of $X_1$ and $X_2$

full factorial design with 2 factors uses the 2-factor ANOVA model, which is an example of an interaction model.  It assumes a linear relationship between the response and the above 3 effects.

$Y = \beta_0 + \beta_1 X_1 + \beta_2 X_2 + \beta_3 X_1 X_2 + \varepsilon$

Note that additive models and interaction models are not confined to experimental design; I have merely used experimental design to provide examples for these 2 types of models.