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 191

differ by how to practically deine interaction in the most intuitively understand-
able way.

8.3.1 INTERACTION—DEFINITION 1
Consider the following two-by-two table, Table 8.3 , where the two rows stand
for the length of a Web page (Shorter, Longer), the two columns stand for the
shading (black and white, multicolored), and the values in the cells are the mean
satisfaction rating (using the 1-5 Likert scale) of the design by a large number of
respondents, with independent samples—each person sees only one of the four
(combinations) designs.
Let us assume for the moment that there is no error in the means in the
cells. This is similar to saying that each cell mean is the mean of “a million”
responders, so that the sample means (the ones in the cells) are only itsy-bitsy,
teeny-weeny, virtually invisible amounts different from the respective true
μ values.
We can see in Table 8.3 that if we start in the upper left-hand corner, we gain 0.4
in rating (3.9 - 3.5 = 0.4) if we hold the shading at Black and White and go from shorter
to longer page. If we hold length constant at shorter, and move to the right, we see
that we gain 0.8 (4.3 - 3.5 = 0.8) going from black and white to multicolored. What if
we move from upper left to lower right (where there is now a question mark)—(i.e.,
shorter, black and white, to longer, multicolored)?
Do we obtain a mean rating of 4.7? This would be taking the original 3.5 and
adding the 0.4 for longer length and adding the 0.8 for multicolored shading, getting
3.5 + 0.4 + 0.8 = 4.7.
This may sound odd, but in this case we'd say that we have no interaction. Adding
both boosts to the design—longer and multicolored—has an effect on ratings that is
simply additive . Strictly speaking, there is no interaction effect. This leads to our irst
deinition of interaction:

Interaction = degree of difference from the sum of the separate effects.

If the value in the lower right-hand cell (longer, multicolored) is 4.7, there is
no interaction. If the value were, for example, 4.5, we would say we have negative
interaction (even though 4.5 would be the largest value in the table). If the value were
4.85, we would say that we have positive interaction .

Table 8.3 2 × 2 Table of Ratings of Length and Shading of Pages

Black and White Multicolored

Shorter 3.5 4.3

Longer 3.9 ?