# What is an Interaction?

• Interactions are when the effect of two, or more, variables is not simply additive. This page describes the interaction between two variables. It is possible to examine the interactions of three or more variables but this is beyond the scope of this page.

## Examples

• Imagine a study about the effect of energy bars and energy drinks on time to run the 1500 meters. The quantity of energy bars and energy drinks represent two variables. The dependent variable is the time taken to run 1500 meters.
1. Example 1 - An interaction occurs if running speed improves by more than just the additive effect of having either an energy bar or an energy drink. For example, imagine eating a certain amount of energy bars increases running speed by 3 seconds, and drinking energy drinks increases running speed by 2 seconds. An interaction occurs if the joint effect of energy bars and energy drinks increases running speed by more than 5 seconds, such as liquid in the drink amplifying the ability to digest the energy in the bar leading to faster times.
Chart 1a below shows an additive effect
Chart 1b below shows an Interaction.
2. Example 2 - A second example of an interaction is that alone neither variable may have an effect on running speed, such as imagining that an energy bar by itself, or an energy drink by itself, is unable to increase running speed. But, there might be an interaction effect that influences running speed when you eat the bar and drink the drink, such as the energy bar having a chemical that unleashes the power of the energy drink to increase running speed.
Chart 2a shows when neither variable has an effect, with no Interaction
Chart 2b also shows when neither variable has an effect, but now with an Interaction
3. Example 3 - A final example is when one of the variables has an effect but not the other. When a variable has an effect (such as the energy bar increasing running speed, or the energy drink increasing running speed) that is called a Main Effect.
Chart 3a shows a Main Effect for the energy bar, with no Interaction
Chart 3b shows the same Main Effect for the energy bar, but now with an Interaction
Chart 4a shows a Main Effect for the energy drink, with no Interaction
Chart 4b shows the same Main Effect for the energy drink, but now with an Interaction

## Statistical formula behind interactions

• For those more technically minded, here is the algebra. An interaction effect reflects the effect of the interaction controlling for the two predictors themselves.
1. In the following examples:
energy bar = X1,
energy drink = X2
the interaction = X1*X2,
Y = running speed
2. Here is the formula for: Running speed = intercept + b1energy drink + b2energy bar + b3(bar * drink) + ei
Yi = b0 + b1X1i + b2X2i + b3(X1i X2i) + ei
3. This formula can be rewritten as
Yi = (b0 + b2X2i) + (b1+ b3X2i) X1i + ei
where (b1+ b3X2i) represents the effect of X1 on Y at specific levels of X2
and b3 indicates how much the slope of X1 changes as X2 goes up or down one unit.
4. It is then possible to factor out X2
Yi = (b0 + b1X1i) + (b2+ b3X1i) X2i + ei
where (b2+ b3X1i) represents the effect of X2 on Y at specific levels of X1
and b3 indicates how much the slope of X2 changes as X2 goes up or down one unit.