**Understanding and Interpreting Interactions**

**Contents**

Plotting Data

Main Effects, Simple Effects, and Interactions

The Meaning of an Interaction

Which Way to Draw the Graph

Identifying Effects

Main Effects in the Presence of Interactions

Interactions with Three Independent Variables

Interactions with Four Independent Variables

Crossover Interactions

Before we discuss interactions, you should review the basic
principles of interpreting and plotting data graphs. A visual representation of
results is easier for most people to understand than a table of numbers, and it
is especially helpful when thinking about interactions.

You may use
bar graphs or line graphs, whichever is easier to understand. However, if you
have more than one independent variable with several levels of each, or if the
independent variable is numerical (e.g., number of practice trials), a line graph
is usually better.

In every case, the vertical axis (the ordinate,
or Y axis) represents the dependent variable, and the horizontal axis (the abscissa
, or X axis) represent an independent (or classification) variable.

If
you have two independent variables (including classification variables), one will
be plotted on the horizontal axis. The other will be represented by plotting multiple
lines on the graph. A third independent or classification variable can be plotted
by using additional lines, or by drawing multiple graphs side by side.

Independent
and classification variables will be referred to as **factors**. When deciding
which factor to plot on the horizontal axis, you should almost always choose the
variable with most levels. If they have the same number of levels, choose a variable
that has numerical values, if there is one. If there's no other basis for a choice,
use a repeated measures variable for the horizontal axis.

Here's an example. Reaction times were measured (in milliseconds) in a discrimination task. Three levels of task difficulty were used. Participants were tested one hour after waking from a good night's sleep, and after 24 hours without sleep. In this case the horizontal axis is a repeated measures factor, and is the factor with more levels.

By the way, Excel and other spreadsheet programs can be used to produce these graphs. In Excel, you can enter the data as a table, then use the "Chart Wizard" to generate the graph. It is best to edit the colors that Excel uses to produce a black and white figure.

**Reaction times following one hour and 24 hours without sleep**

There are some other points to note about this graph. Both axes are labeled, and the graph itself is given a title. A legend identifies the two sleep conditions. The table was set up so that the higher line (24 hours) corresponds to the upper label in the legend. This is always helpful to the reader.

In the reaction time experiment, suppose that Age was a third independent variable (actually, a classification variable). The performance of college students and senior citizens was compared. How would you present the data for the both groups of subjects? |

** Main
Effects, Simple Effects, and Interactions**

Look at the graph above.
A number of conclusions can be drawn from that graph. Assume that the error variance
is small, so that differences in reaction time that are apparent on the graph
are also statistically significant.

It appears that, as the task becomes
more difficult, reaction times get longer for both conditions (one hour and 24
hours since sleep). However, the effect is greater following the 24 hours sleep
deprivation.

If we were to calculate the average reaction time across
all three tasks, it would be about 550 ms for the one hour group, and 680 ms for
the 24 hour group. On the average, then, reaction times were longer following
24 hours sleep deprivation.

If we calculate the average reaction time for
each of the three tasks, it would be about 520 ms for the easy task, 590 ms for
the intermediate task, and 710 ms for the difficult task. On the average, reaction
times were longer for the more difficult tasks.

The difference between
the one hour group and the 24 hour group is a **Sleep Deprivation Main Effect**.
The difference in reaction time as a function of task difficulty is a **Task
Difficulty Main Effect**. A main effect is the overall effect for one factor

The fact that the task difficulty effect is much greater for the 24 hour
group than it is for the one hour group is the **Sleep Deprivation by Task Difficulty
Interaction**.

Looking only at the 24 Hour group, the reaction times
get longer as the task becomes more difficult. This is a **Task Difficulty Simple
Effect for the 24 hour group**. There is a smaller but still apparent **Task
Difficulty Simple Effect for the one hour group**.

Looking only at
the Difficult task, reaction times are longer for the 24 hour group. This is a
**Sleep Deprivation Simple Effect for the difficult task**. There is a much
smaller, possibly non-significant **Sleep Deprivation Simple Effect for the intermediate
task**. For the easy task, there appears to be no Sleep Deprivation Simple Effect.

If you followed all of this, it should now be clear that **we observe
an interaction between two factors whenever the simple effects of one change as
the levels of the other factor are changed**.

Another, probably easier
way to recognize an interaction is to notice that, in a graph of the results,
**the lines connecting the points are not parallel**.

Be sure to
notice that an interaction involves two factors (i.e., two independent or classification
variables) and a dependent variable. In this example we say that Task Difficulty
interacts with Sleep Deprivation in their effects on Reaction Time. We *never*
talk about an interaction between an independent variable and a dependent variable.

Here's an example using a table rather than a graph. Infants aged 4 months and 12 months were tested in an "intuitive addition" task. The time spent looking at a stimulus indicates whether or not the infant expects, say, two toys plus three toys to add up to five toys. Infants were tested under two conditions, where the total number of toys displayed was actually five (correct), or was three (incorrect). The results show the average looking time for each condition. If infants understand addition, it's expected they will look longer if the total is "incorrect" (i.e., three). Assume that differences less than 0.5 sec are not significant. Describe the main effects, the simple effects, and the interaction effects, in these data. | ||

Age 4 months | "Correct" Total Displayed | 4.9 sec |

"Incorrect" Total Displayed |
4.8 sec | |

Age 12 months | "Correct" Total Displayed |
3.5 sec |

"Incorrect" TotalDisplayed | 6.3 sec |

When two factors interact, it means that changes in the dependent variable
cannot be explained by independent effects of the two factors. Rather, the explanation
must be more complicated. The effect of one factor depends on what has happened
to the other factor. In the example above, we cannot explain changes in reaction
time by stating that responses are slower when the task is more difficult, and
responses are slower when one is suffering from sleep deprivation. Both of these
statements are true, but we must point out also that the effect of sleep deprivation
is much greater on difficult tasks than on simple tasks.

As psychologists, interactions complicate life for us, but they also make life
more interesting. They suggest that there are multiple psychological processes
going on. Indeed, if a theory predicts that two or more processes are involved
in explaining some behavior, then testing for an interaction is the best way
to test the theory.

As noted above, you must choose which factor to plot on the horizontal axis. Often one way of plotting the graph is much clearer than the other. However, for purposes of recognizing interactions it does not matter how the graph is plotted. The following graph shows the same data as before, with hours since sleep on the horizontal axis. Notice that all of the conclusions reached above are still true.

**Reaction times following one hour and 24 hours without sleep **

You should become adept at looking at a graph and determining
whether or not there are interactions or main effects. For two factors, there
are eight possible results. The following graphs illustrate those possibilities.
In each case the factors are referred to as Group and Condition, with Condition
plotted on the horizontal axis. In these graphs assume that differences less than
1 full point are merely random error.

Study these graphs, and be sure
you understand why the results are as described. Be able to recognize the patterns
that are illustrated here.

**Both main effects and an interaction**: The lines are not parallel. Group
1 scores higher then Group 2 (notice how the line for Group 1 is above the line
for Group 2) and Condition B scores higher then Condition A (on the average,
the scores for B are higher than the scores for A).

**Both main effects and no interaction**: Group 1 scores higher then Group
2 and Condition B scores higher then Condition A. The lines are parallel.

**Group main effect, an interaction, and no Condition effect**: There are
non-parallel lines. On the average, Group 1 scores higher then Group 2, while
averages for Conditions A and B are about the same.

**Group main effect , no Condition effect, and no interaction**: Note that
averages for Conditions A and B are about the same, while Group 1 clearly scores
higher than Group 2.

**Condition main effect and an interaction, no Group effect**: The averages
for Groups 1 and 2 are about the same, while on the average scores for Condition
1 are higher than scores for Condition 2.

**Condition main effect, no Group effect and no interaction**: Condition
B scores are higher than Condition A. The groups are about the same.

**An interaction and no main effects**: The averages for Conditions A and
B, and for Groups 1 and 2, are about the same.

**No main effects, no interaction**: All averages are about the same.

**Main Effects in the Presence of Interactions**

Whenever two factors interact, if there is a main effect for
either of those factors, it may not be very meaningful. Remember, an interaction
means that the simple effects of a factor are not the same at all levels of the
other factor. The main effect is essentially the average of all the simple effects.
If the differences in simple effects are large (they may even be in a different
direction in some cases), it may be very misleading to report the main effect.

The graph above illustrates a case where both main effects exist. On the average, Group 1 scores higher than Group 2, and scores for Condition B are higher than scores for Condition A. The second of these statements is rather misleading, though. If we look only at Group 2, there is no difference between the two conditions. To report the Group main effect is perhaps less misleading. For both conditions, Group 1 does better than Group 2, although the difference is smaller for Condition A.

For this reason we generally withhold comment on any main effects until
we have examined the interaction.

**Interactions
with Three Independent Variables**

If we introduce a third factor
(a third independent or classification variable), more complex interaction effects
are possible. Let the three factors be A, B, and C. Then the following effects
are all possible:

A, B, and C main effects

AB, AC, and BC interactions

An ABC triple interaction

The main
effects and two-way (or double) interactions can be recognized in the same way
they are for a two factor design. The main effect for A, for example, exists if
the average value of the dependent variable is different for different levels
of A. The two-way interaction AB exists if the simple effects of A are different
at different levels of B, or vice versa. A plot of the data will show non-parallel
lines.

With a third factor, C, we can ask if the AB two-way interaction
changes as we change the level of C. If it does, then we have a three-way (or
triple) interaction. Again, a graph is the easiest way to display such an interaction.

The first example above examined the effects of sleep deprivation on reaction
times in tasks that varied in difficulty. Suppose we were to repeat the experiment,
but using subjects who have had lots of practice on the task. The data might look
like the following. The graph on the left, for unpracticed subjects, is the same
as the first graph above. The second shows results for practiced subjects.

What is clear from these data is that, with practiced subjects, the interaction between Sleep Deprivation and Task Difficulty disappears. That is, the two way interaction for practiced subjects is different from the two-way interaction for non-practiced subjects. There is a three-way interaction of Practice, Sleep Deprivation, and Task Difficulty.

A three-way interaction, then, is a change in the two way interaction as
the level of the third factor changes. This can be seen in a graph of the results
as a change in the pattern of the graphs for two-way interactions. Note that in
the example above, there is a two-way interaction in one case and no two-way interaction
in the other case. This is just one kind of three-way interaction, though not
the only kind.

In the graphs above, one factor (Difficulty) is plotted
on the horizontal axis. A second factor (Sleep Deprivation) is represented by
different lines on the same graph. The third (Practice) is represented by creating
different graphs. The choice of which factor to represent which way is determined
by the clarity of the results. However, a three-way interaction will be apparent
no matter how the data are drawn. When you examine the graphs for each level of
the third factor you will see changes in the two-way interaction.

Below are three more pairs of graphs that illustrate three-way interactions (in one case, there is no three-way interaction). In each example there are two graphs. Assume the graph on the left is for males, the graph of gthe right for females. There are two groups of males and two groups of females, and subjects in all four groups have been tested under two conditions, A and B.

Three-way interaction - two different two-way interactions:

Three-way interaction - a two-way interaction on the left, and no two-way interaction on the right:

No three-way interaction - the two-way interactions are the same:

Note that in the third example, the average score on the left is different from the average score on the right. That is, there's a main effect. However, the shape of the two-way interaction does not change

Look at the last graph above. List all of the effects (main effects and interactions), and indicate which ones appear to be significant. |

**Interactions
with Four Independent Variables**

If we have four factors, the following
effects are possible:

A, B, C, and D main effects

AB, AC, AD, BC, BD, and CD two-way interactions

ABC, ABD, ACD, and BCD three-way interactions

An ABCD four-way interaction

You need not
worry about such complex designs for this course. However, you may be able to
deduce what a four-way interaction would mean. They arise whenever the shape of
a three-way interaction changes as the level of the fourth factor changes.

Consider the three pairs of graphs shown above. Suppose these represent data from a single experiment, with each pair representing a level of a fourth factor. Then we have a four-way interaction: there are different three-way interactions in the first two pairs, and none in the third pair. That is, the three-way interaction changes, so there is a four-way interaction.

A study by Halford et al (2005)
suggests that a four-way interaction is at the limit of even an experienced researcher's
processing ability. They found that the ability to interpret correctly a five-way
interaction was no better than chance.

Compare the two two-way interactions in the two graphs below. In the graph on the left the two lines both have negative slope, and they do not intersect. In the graph on the right the two lines again both have negative slope, but they do intersect. The second is referred to as a "crossover" interaction.

The graph below uses the same data as the second graph above, but plotted with Group on the horizontal axis. This time there is no crossover. But notice that the lines are opposite in slope. One is positive, the other negative. When the lines have opposite slope, this too is called a cross-over interaction, because if plotted the other way the lines would cross.

Why is this important? Because in psychology we can practically never guarantee
that the measurement of our dependent variable is an interval scale. Now, normally
this does not matter. We can proceed as if we had an interval scale, using all
the statistics like means and correlations that require us to make the assumption
of equal intervals. In the case of interactions, though, this problem can come
back to bite us.

If we do not have an interval scale, then in principle
we can change the way the dependent variable is measured in any way we wish, as
long as the ordering of the data remains unchanged. For example, instead of using
raw scores, we could use the square of the raw scores. For any two scores, if
one is larger than the other, then its square is also large than the square of
the other.

Now, suppose in the left graph above we change all the scores by squaring them. The result is shown on the next graph. Do you see what has happened? The two-way interaction has almost disappeared.

Therein lies the problem with non-crossover interactions. If it is not
a crossover interaction, then it is possible to make the interaction disappear
by using a perfectly permissible transformation in the dependent variable. In
other words, we must be very careful when interpreting a non-crossover interaction,
because it might be nothing more than an accident of the way the dependent variable
has been measured.

If we have a crossover interaction the problem is
does not arise. It is impossible to make the interaction disappear using permissible
transformations if it is a crossover interaction. Thus, a crossover interaction
tells us that the two factors really do interact, no matter how we scale the dependent
variable.

One point to bear in mind. The next graph represents a special case where, for one level of one factor, there is no simple effect of the other factor (for Group 1 there is no difference between the conditions). This counts as a crossover interaction. In this case too it is impossible to make the interaction disappear - there will always be a simple effect for one level of the factor and no simple effect for the other level.

Examine the examples of interactions in the early parts of this document. Do any of them represent non-crossover interactions? |

It is not uncommon for researchers to find non-crossover interactions, and then to make a strong theoretical statement based on that interaction. The point being made here: beware of making strong conclusions about an interaction unless it shows a crossover effect.

Dr. Mesmer hypothesized that students who are good at math solve problems in a different way from students who are poor at math. He proposed that if students are required to remember a list of words while they work on a problem, the performance of poor students will be affected far more than the performance of good students. To test the hypothesis, Dr. Mesmer tested subjects who were good at math, and subjects who were poor at math. Subjects solved problems while remembering different numbers of words - zero, four, or eight. He found the interaction he was expecting - performance declined more for the poorer students (see below). Dr. Mesmer's colleague, Dr. Klesmer, repeated the experiment, using a different math test. Dr. Klesmer found no interaction (see below). Examine the graphs, and explain why the two investigators reached different conclusions. | |

Dr. Mesmer's
data | |

Dr. Klesmer's data |

**References**

Halford,
G. S., Baker, R., McCredden, J. E., & Bain, J. D. (2005) How many varables
can humans process? *Psychological Science*, 16, 70-76.