Comparing more than two means: two factor ANOVA with independent samples; the important role of interaction - Improving the User Experience through Practical Data Analytics - page 193

Table 8.5 2 × 2 Table with Extreme Interaction and No Other Effect!!

Low High

Low 3.4 3.9

High 3.9 3.4

From Table 8.4 , we just demonstrated that the effect on the rating of satisfaction of the
design by changing the level of length depends on the level of shading (+0.4 or +0.6, for
a difference of +0.2). If we go the other way, we get the same +0.2. What is the effect of
changing the shading from black and white to multicolored? If we have length = shorter,
it's +0.8 (going from 3.5 to 4.3); if we have length = longer, it's +1 (going from 3.9 to 4.9).
It is no coincidence that the difference between +0.8 and +1 is also 0.2. And,
indeed, the difference between 4.9 and 4.7 (the latter value indicative of no interac-
tion) is also +0.2. The bottom line here is that we should always conduct an experi-
ment that tells us if we have a signiicant interaction effect between two factors when
there are multiple factors 1 . However, as we noted in Chapter 7, we need to have
replication or else need to assume that there is no interaction. Happily, we do have
replication in this example; each [age, gender] combination occurs more than once .

SIDEBAR: AN EXTREME CASE OF (NEGATIVE) INTERACTION
In an extreme case, you could have data such as in the ratings in Table 8.5 , in which we use two
generic factors, and for each we have levels that are called “low” and “high.”
The overall effect of the row factor is zero (each row averages the same: 3.65) and the overall
effect of the column factor is zero (each column averages the same: 3.65). But there is something
else going on: we have gigantic (negative) interaction!! The lower right cell has a mean of 3.4, not
the sum of the horizontal and vertical increases of 0.5, which would result in 4.4.

8.4 WORKING THE EXAMPLE IN SPSS
We illustrate two-factor ANOVA using SPSS.

SIDEBAR: EXPERIMENTAL DESIGN TO OVERCOME THE LIMITS OF EXCEL
To perform ANOVA with Excel, the data have to be comprised in a way that is extremely restrictive.
You need to have the same number of people of each age group and need the same number of males and
females in each age group! That is not even close to what we have in our example; we have a different
number of people in different age groups (sample sizes by age group, respectively, are 28, 26, 26, 23, 23),
and we have nowhere close to the same male/female split for each age group (males are, respectively, by

Continued

1 There are possible “three-way” and higher order interactions, but they are usually assumed to be zero
or negligible. Discussion of these interactions is beyond the scope of the text. For further discussion
of these higher-order interactions, we recommend the Berger and Maurer text referenced at the end of
Chapter 6.