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

March 7, 2014 Leave a comment

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 is the response and and are the 2 predictors, then an additive linear model for the relationship between the response and the predictors is

**In other words, the effect of on does not depend on the value of , and the effect of on does not depend on the value of .**

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 on depends on the value of , or the effect of on depends on the value of .** Thus, such a regression model would have 3 effects on the response:

- the interaction effect of and

A 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.

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.

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