Statistical Methods for Psychology

where

From this model it can be shown that with fixed variables the expected mean squares are

those given in Table 13.3. It is apparent that the error term is the proper denominator for

each Fratio, since the E(MS) for any effect contains only one term other than s^2 e.

Xijk=any observation

m=the grand mean

ai=the effect of Factor Ai=mAi2m

bj=the effect of Factor Bj=mBj2m

abij=the interaction effect of Factor Ai and Factor Bj

=m2mAi2mBj1mij; (^) iaabij= a
j
abij= 0
eijk=the unit of error associated with observation Xijk
=N(0, s^2 e)
Section 13.3 Interactions 421
Table 13.3 Expected mean squares for two-way analysis of variance (fixed)
Source E(MS)
A
B
AB
Error s^2 e
s^2 e 1 nu^2 ab
s^2 e 1 nau^2 b
s^2 e 1 nbu^2 a
where
Consider for a moment the test of the effect of Factor A:
If is true, then and , and thus nb , will be 0. In this case, Fwill have
an expectation of approximately 1 and will be distributed as the standard (central) Fdistri-
bution. If is false, however, will not be 0 and Fwill have an expectation greater than
1 and will not follow the central Fdistribution. The same logic applies to tests on the ef-
fects of Band AB. We will return to structural models and expected mean squares in Section
13.8 when we discuss alternative designs that we might use. There we will see that the ex-
pected mean squares can become much more complicated, but the decision on the error
term for a particular effect will reflect what we have seen here.

13.3 Interactions

One of the major benefits of factorial designs is that they allow us to examine the interac-

tion of variables. Indeed, in many cases, the interaction term may well be of greater inter-

est than are the main effects (the effects of factors taken individually). Consider, for

example, the study by Eysenck. The means are plotted in Figure 13.1 for each age group

separately. Here you can see clearly what I referred to in the interpretation of the results

when I said that the differences due to Condition were greater for younger participants than

H 0 u^2 a

H 0 mA 1 =mA 2 =m ua^2 u^2 a

E(MSA)

E(MSerror)

=

s^2 e 1 nbu^2 a

s^2 e

u^2 a=

©a^2 j

a 21

=

©(mi2m)^2

a 21