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.