Thus, for a response y and two variables x₁ and x₂ an additive model would be:

Very often the interacting variables are categorical variables rather than real numbers. For example, members of a population may be classified by religion and by occupation. If one wishes to predict a person's height based only on the person's religion and occupation, a simple additive model, i.e., a model without interaction, would add to an overall average height an adjustment for a particular religion and another for a particular occupation. A model with interaction, unlike an additive model, could add a further adjustment for the "interaction" between that religion and that occupation. This example may cause one to suspect that the word interaction is something of a misnomer.

The consequence of an interaction is that the effect of one variable depends on the value of another. This has implications in design of experiments as it is misleading to vary one factor at a time.

Real-world examples of systems that manifest interactions include:

• Interaction between adding sugar to coffee and stiring the coffee. Neither of the two individual variables has much effect on sweetness but a combination of the two does.

• Interaction between adding carbon to steel and quenching. Neither of the two individually has much effect on strength but a combination of the two has a dramatic effect.

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