What is an interaction effect?

 

Two independent variables interact if the effect of one of the variables differs depending on the level of the other variable. The means from the hypothetical experiment described in the section on factorial designs are reproduced below.
Notice that the effect of drug dosage differs depending on whether the task is simple or complex. For the simple task, the higher the dosage, the shorter the time to complete the task. For the complex task, the higher the dosage, the longer the time to complete the task. Thus, there is an interaction between dosage and task complexity. It is usually much easier to interpret an interaction from a graph than from a table. A graph of the means for the interaction between task complexity and drug dosage is shown on the next page. The dependent variable (response time) is shown on the Y axis. The levels of drug dosage are shown on the X axis. The two levels of task complexity are graphed separately.


A look at this graph shows that the effect of dosage differs as a function of task complexity. It also shows that the effect of task complexity differs as a function of drug dosage: The larger the drug dosage, the greater the difference between the simple task and the complex task. An interaction does not necessarily imply that the direction of an effect is different at different levels of a variable. There is interaction as long as the magnitude of an effect is greater at one level of a variable than at another. In the example, the complex task always takes longer than the simple task. There is an interaction because the magnitude of the difference between the simple and complex tasks is different at different levels of the variable drug dosage.

Two variables interact if a particular combination of variables leads to results that would not be anticipated on the basis of the main effects of those variables. For instance, it is known that both drinking alcohol and smoking increase the chance of throat cancer. However, people who both drink and smoke have a much higher chance of getting cancer than would be predicted if one knew only how much more likely smokers are than nonsmokers to get throat cancer and how much more likely drinkers are than nondrinkers to get throat cancer. The combination of smoking and drinking is particularly dangerous: these drugs interact. This definition of interaction in terms of a particular combination of variables is consistent with the previously-given definition that there is an interaction if the effect of one variable differs depending on the level of another variable. In the tobacco and alcohol example, the effect of smoking on the probability of getting cancer is greater for people who drink than for people who do not drink: the effect of smoking differs depending on whether drinkers or nondrinkers are being considered. Similarly, the effect of drinking differs depending on whether smokers or nonsmokers are being considered.

Interactions can be described in a variety of ways. Examples of graphs of interactions and possible verbal descriptions of each follow.

a. The difference between the treatment and control conditions was greater for subjects performing Task 1 than for subjects performing Task 2.
b. There was a greater difference between Task 1 and Task 2 for subjects in the treatment condition than there was for subjects in the control condition.
c. Tasks 1 and 2 were performed about equally well in the control condition but Task 1 was performed considerably better than Task 2 in the treatment condition.


a. On the well-learned task, increased magnitude of reward was associated with better performance whereas on the novel task, increased magnitude of reward was associated with reduced performance.
b. Under low magnitude of reward, the novel task was performed better than the well-learned task. Under high magnitude of reward, the well-learned task was performed better than the novel task.
c. The difference between the novel task and the well-learned task changed from positive for low magnitude of reward to slightly negative for medium magnitude of reward to very negative for high magnitude of reward.

 

a. Overall, Condition B2 led to better performance than did either B1 or B3. This effect was much more pronounced for subjects performing Task 1 than for subjects performing Task 2.
b. The difference between Tasks 1 and 2 was greatest for subjects in Condition B2.
c. The combination of Task 1 and Condition B2 led to especially high performance.
 

 

In Example 4 there is no interaction. The effect of task is the same at all three levels of B and the effect of B is the same for both tasks. Notice that the two lines are parallel. When there is no interaction, the lines will always be parallel.
 

Higher Order Interactions
So far, all the interactions that have been described are called "two-way" interactions. They are two-way interactions because they involve the interaction of two variables. A three-way interaction is an interaction among three variables.
There is a three-way interaction whenever a two-way interaction differs depending on the level of a third variable. Consider the two figures on the left side of this page. The upper figure shows the interaction between task and condition (B) for well-rested subjects; the lower figure shows the same interaction for sleep-deprived subjects. The forms of these interactions are different. For the well-rested subjects, the difference between Tasks 1 and 2 is largest under condition B2 whereas for the sleep-deprived subjects the difference between Tasks 1 and 2 is smallest under condition B2. The two-way interactions are therefore different for the two levels of the variable "sleep deprivation." This means that there is a three-way interaction among the variables sleep deprivation, task, and condition.
 

Four-way interactions occur when three-way interactions differ as a function of the level of a fourth variable. Four-way and higher interactions are usually very difficult to interpret and are rarely meaningful.

Note that an interaction effect limits the generalizability of a main effect! Read more about this in the section about simple effects.

Reference:
http://davidmlane.com/hyperstat/