Page 209 - Intermediate Statistics for Dummies

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Part III: Comparing Many Means with ANOVA
Suppose factor A has i levels and factor B has j levels, with a sample of size m
collected on each combination of A and B. The degrees of freedom for factor
A, factor B, and the interaction term AB are (i – 1), (j – 1), and (i – 1) (j – 1)
respectively. This formula is just an extension of the degrees of freedom for
the one-way model for factors A and B. The degrees of freedom for SSTO is
j
j
m – 1, and the degrees of freedom for SSE is i
i
* *
Understanding Interaction Effects
The interaction effect is the heart of the two-way ANOVA model. Knowing
that the two factors may act together in a different way than they would sepa-
rately is important and must be taken into account. In this section, you see
the many ways in which the interaction term AB and the main effects of fac-
tors A and B affect the response variable in a two-way ANOVA model.
What is interaction anyway? * * (m – 1). *
Interaction is when two factors meet, or interact with each other, on the
response in a way that’s different from how each factor affects the response
separately. For example, before you can test to see whether dosage of medi-
cine (factor A) or number of times taken (factor B) are important in explaining
changes in blood pressure, you have to look at how they operate together to
affect blood pressure. That is, you have to examine the interaction term.
Suppose you’re taking one type of medicine for cholesterol and one medicine
for a heart problem. Suppose researchers only looked at the effects of each
drug alone, saying each one was good for managing the problem for which it
was designed, with little to no side effects. Now you come along and mix the
two drugs in your system. As far as the individual study results are con-
cerned, all bets are off. With only those separate studies to go on, they will
have no idea how the drugs will interact with each other, and you can be in a
great deal of trouble very quickly. Fortunately, drug companies and medical
researchers do a great deal of work studying drug interactions, and your
pharmacist knows which drugs interact as well. You can bet a statistician
was involved in this work from day one!
Baking is another good example of how interaction works. Slurp down one
raw egg, drink a cup of milk, and eat a cup of sugar, a cup of flour, and a stick
of margarine. Then eat a cup of chocolate chips. Each one of these items has
a certain taste, texture, and affect on your taste buds that, in most cases,
won’t be all that great. But mix them all together in a bowl and voilà! You
have a batch of chocolate chip cookie dough, thanks to the magic effects of
interaction.