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• 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.

Interactions and Main Effects

• 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

<nowiki>*</nowiki>(FYI - Chart 1b shows an ordinal interaction. A disordinal interaction occurs when the two lines cross).

Graphical representations of Interactions

<table width=100% cellpadding=20> <td width=50% > This column shows the four possibilities WITHOUT an interaction:

1. Two Main Effects
2. No Main Effects
3. Main Effect for Variable A
4. Main Effect for Variable B

</td> <td width=50% valign=top> This column shows the four possibilities WITH an interaction:

1. Two Main Effects
2. No Main Effects
3. Main Effect for Variable A
4. Main Effect for Variable B

</td> </table> <table> <td> There are other graphical representations of Interactions and Main Effects such as effects being reduced (e.g., imagine all the graphs below being flipped down) or effects going in the opposite direction (e.g., imagine all the graphs being flipped to the left), but for simplicity sake they are not displayed because you can discern those graphs after identifying the major types described below. </td> </table> <table> <td> </td> <td> </td> </table> <table> <td> </td> <td> </td> </table> <table> <td> </td> <td> </td> </table> <table> <td> </td> <td> </td> </table>

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) + e_i

     Y_i = b₀ + b₁X1_i + b₂X2_i + b₃(X1_i X2_i) + e_i

  3. This formula can be rewritten as

     Y_i = (b₀ + b₂X_2i) + (b₁+ b₃X_2i) X_1i + e_i

     where (b₁+ b₃X_2i) 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

     Y_i = (b₀ + b₁X_1i) + (b₂+ b₃X_1i) X_2i + e_i

     where (b₂+ b₃X_1i) 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.

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