- Define main effect, simple effect, interaction, and marginal mean
- State the relationship between simple effects and interaction
- Compute the source of variation and df for each effect in a factorial
design
- Plot the means for an interaction
- Define three-way interaction
Basic Concepts and Terms

In the Bias
Against Associates of the Obese case study, the researcher
was interested in whether the weight of a companion of a job
applicant would affect judgments of a male applicant's qualifications
for a job. Two independent
variables were investigated: (1) whether the companion was
obese or of typical weight and (2) whether the companion was
a girl friend or just an acquaintance. One approach would have
been to conduct two separate studies, one with each independent
variable. However, it is more efficient to conduct one study
that includes both independent variables. Moreover, there is
a much bigger advantage than efficiency for including two variables
in the same study: it allows a test of the interaction between
the variables. There is an interaction when the effect of one
variable differs depending on the level of a second variable. For example, it is possible that
the effect of having an obese companion would differ depending
on the relationship to the companion. Perhaps there is more
prejudice against a person with an obese companion if the companion
is a girl friend than if she is just an acquaintance. If so,
there would be an interaction between the obesity factor and
the relationship factor.

There are three effects of interest in this experiment:

- Weight: Are applicants judged differently depending on the
weight of their companion?

- Relationship: Are applicants judged differently depending
on their relationship with their companion?

- Weight x Relationship Interaction: Does the effect of weight
differ depending on the relationship with the companion?

The first two effects (Weight and Relationship) are both main
effects. A main effect of an independent variable is the
effect of the variable averaging over the levels of the other
variable(s). It is convenient to talk about main effects in terms
of marginal means. A marginal mean for
a level of a variable is the mean of the means of all levels of
the other variable. For example, the marginal mean for the level "Obese" is
the average of "Girl-Friend-Obese"
and "Acquaintance-Obese." Table 1 shows that this marginal
mean is equal to the average of 5.65 and 6.15 which is 5.90. Similarly,
the marginal mean for Typical is the average of 6.19 and 6.59
which is 6.39. The main effect of Weight is based on a comparison
of the these two marginal means. Similarly, the marginal means
for Girl Friend and Acquaintance are 5.92 and 6.37.

In contrast to a main effect which is the effect
of a variable averaged across levels of another variable, the simple
effect of a variable is the effect of the variable at a
single level of another variable. The simple effect of "Weight" at
the level of "Girl Friend" is the difference between
the "Girl-Friend Typical"
condition and the "Girl-Friend Obese" conditions. The
difference is 6.19-5.65 = 0.54. Similarly, the simple effect of "Weight" at
the level of "Acquaintance" is the difference between
the "Acquaintance Typical" condition and the "Acquaintance
Obese" conditions. The difference is 6.59-6.15 = 0.44.

Recall that there is an interaction when the effect
of one variable differs depending on the level of another variable.
This is equivalent to saying that there
is an interaction when the simple effects differ. In this
example, the simple effects are 0.54 and 0.44. As shown below,
these simple effects are not significantly different.

Tests of Significance

The important questions are not whether there
are main effects and interactions in the sample data. Instead,
what is important is whether the sample data allow you to conclude
about the population. This is where Analysis of Variance comes
it. ANOVA tests main effects and interactions for significance.
An ANOVA Summary Table for these data is shown in Table 2.

Consider first the effect of "Weight." The degrees
of freedom (df) for "Weight" is 1. The degrees
of freedom for a main effect is always equal to the number of
levels of the variable minus one. Since there are two levels
of the "Weight"
variable (typical and obese) the df is 2 -1 = 1. We skip the calculation
of the sum of squares (SSQ) not because it is difficult, but because
it is so much easier to rely on computer programs to compute it.
The mean square (MS) is the sum of squares divided by the df.
The F ratio is computed by dividing the MS for the effect by the
MS for error (MSE). For the effect of "Weight," F =
10.4673/1.6844 = 6.214. The last column, p, is is the probability
of getting an F of 6.214 or larger given that there is no effect
of weight in the population. The p value is 0.0136 which is quite
low and therefore the null hypothesis of no main
effect of "Weight" is rejected. The conclusion is that
the weight of the companion lowers judgments of qualifications.

The effect "Relation" is interpreted the
same way. The conclusion is that being accompanied by a girl friend
leads to lower ratings than being accompanied by an acquaintance.

The df for an interaction is the product of the
df of variables in the interaction. For the Weight x Relation
interaction (W x R), the df = 1 since both Weight and Relation
have one df: 1 x 1 = 1. The p value for the interaction is 0.8043
which is the probability of getting an interaction as big or bigger
than the one obtained in the experiment if there were no interaction
in the population. Therefore, these data provide no evidence for
an interaction. Always keep in mind that the lack of evidence
for an effect does not justify the conclusion that there is no
effect. In other words, you do not accept the null hypothesis
just because you do not reject it.

For "Error," the degrees of freedom is
equal to the total number of observations minus the total number
of groups. The sample sizes for this experiment are shown in Table
3. The total number of observations is 40 + 42 + 40 + 54 = 176.
Since there are four groups, df = 176 - 4 = 172.

The final row is "Total." The degrees
of freedom total is equal to the sum of all degrees of freedom.
It is also equal to the number of observations minus 1, or 176
-1 = 175. When there are equal sample sizes, the sum of squares
total will equal the sum all other sums of squares. However, when
there are unequal sample sizes, as there are here, this will not
generally be true. The reasons for this are complex and are discussed
in the section Unequal n .

Plotting Means

Although the plot shown in Figure 1 illustrates
the main effects and the interaction (or lack of interaction)
clearly, it is called an interaction plot.
It is important to carefully consider the components of this plot.
First, the dependent variable is on the Y axis. Second, one of
the independent variables is on the X axis. In this case, it is
the variable is "Weight." Finally, a separate line is
drawn for each level of the other independent variable. It is
better to label the lines right on the graph as shown here than
with a legend.

Three-Factor Designs

Three-factor designs are analyzed in much the
same way as two-factor designs. Table 4 shows the analysis of
a study described by Franklin and
Cooley investigating three factors on the strength of industrial
fans: (1) Hole Shape (Hex or Round), (2) Assembly Type (Stake
or Spun), and (3) Barrel Shape (knurled or smooth). The dependent
variable, breaking torque, was measured in foot-pounds. There
were eight observations in each of the eight combinations of the
three factors.

As you can see in Table 4, there are three main
effect, three two-way interactions, and one three-way interaction.
The degrees of freedom for the main effects are, as in a two-factor
design, equal to the number of levels of the factor minus one.
Since all the factors here have two levels, all the main effects
have one degree of freedom. The interaction degrees of freedom
are always equal to the product of the degrees of freedom of the
component parts. This holds for the three-factor interaction as
well as the two-factor interactions. The error degrees of freedom
is equal to the number of observations (64) minus the number of
groups (8) and is 56.

A three-way interaction means
that the two-way interactions differ as a function of the level
of the third variable. The usual way to portray a three-way
interaction is to plot the two-way interactions separately.
Figure 5 shows the Barrel (knurled or smooth) x Assembly (Staked
or Spun) separately for the two levels of Hole Shape (Hex or
Round). For the Hex Shape, there is very little interaction
with the lines being close to parallel with a very slight tendency
for the effect of Barrel to be bigger for Staked than for Spun.
The two-way interaction for the Round Shape is different: The
effect of Barrel is bigger for Spun than for Staked. The finding
of a significant three-way interaction indicates that this difference
in two-way interactions is not due to chance.