What is an Interaction?
From PsychWiki - A Collaborative Psychology Wiki
- 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.
- 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 1 below shows an additive effect, and Chart 2 shows an Interaction.
- 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. Both Chart 3 and Chart 4 below show when neither variable has an effect, with Chart 3 also showing no Interaction, and Chart 4 showing the Interaction.
Graphical representations of interactions
- ►When an energy bar increases running speed, that is called a "Main Effect" (ME) for the energy bar; and when the energy drink also increases running speed, that is a ME for the energy drink. Chart 1 below shows the graphical result when there are two ME but no Interactions.
- ►
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.
- In the following examples:
- energy bar = X1,
- energy drink = X2
- the interaction = X1*X2,
- Y = running speed
- Here is the formula for: Running speed = intercept + b1energy drink + b2energy bar + b3(bar * drink) + e_{i}
- Y_{i} = b_{0} + b_{1}X1_{i} + b_{2}X2_{i} + b_{3}(X1_{i} X2_{i}) + e_{i}
- This formula can be rewritten as
- Y_{i} = (b_{0} + b_{2}X_{2i}) + (b_{1}+ b_{3}X_{2i}) X_{1i} + e_{i}
- where (b_{1}+ b_{3}X_{2i}) represents the effect of X_{1} on Y at specific levels of X_{2}
- and b_{3} indicates how much the slope of X_{1} changes as X_{2} goes up or down one unit.
- It is then possible to factor out X_{2}
- Y_{i} = (b_{0} + b_{1}X_{1i}) + (b_{2}+ b_{3}X_{1i}) X_{2i} + e_{i}
- where (b_{2}+ b_{3}X_{1i}) represents the effect of X_{2} on Y at specific levels of X_{1}
- and b_{3} indicates how much the slope of X_{2} changes as X_{2} goes up or down one unit.
- In the following examples: