Statistical Methods for Psychology

The same kind of argument holds for our test on the interaction, because

and the result will be significant only if the interaction component is significant.^1

But now look at the test on A, the fixed effect. If we form our usual Fratio

we no longer have a legitimate test on A. The ratio could be large if either the interaction is

significant or the effect of Ais significant, and we can’t tell which is causing a result. This

creates a problem, and the only way we can form a legitimate Ffor Ais to divide MSAby

MSAB, giving us

I know from experience that people are going to tell me that I made an error here be-

cause I have altered the test on the fixed effect rather than on the random effect, which is

the effect that is causing all of the problems. I wish I were wrong, but I’m not. Having a

random effect alters the test for the fixed effect. For a very nice explanation of why this

happens I strongly recommend looking at Maxwell and Delaney (2004, p. 475).

For our example we can create our Ftests as

The results of this analysis are presented in Table 13.8.

FL 3 C=

MSL 3 C

MSerror

=

47.575

8.026

=5.93

FLetter=

MSLetter

MSerror

=

378.735

8.026

=47.19

FCase=

MSCase

MSC 3 L

=

240.25

47.575

=5.05

E(F)=

MSA

MSAB

=Ea

s^2 e 1 ns^2 ab 1 nbs^2 a

s^2 e 1 ns^2 ab

b

E(F)=Ea

s^2 e 1 ns^2 ab 1 nbs^2 a

s^2 e

b

E(F)=Ea

MSAB

MSerror

b=

s^2 e 1 ns^2 ab

s^2 e

434 Chapter 13 Factorial Analysis of Variance

(^1) If an interaction is the product of both a fixed and a random factor, the interaction is treated as random.
(^2) These results differ from those produced by some software packages, which treat the mixed model as a random
model when it comes to the denominator for F. But they are consistent with the expected mean squares given
above and with the results obtained by other texts. You can reproduce these results in SPSS by using the following
syntax:
Manova dv by Case(1,2) Letter(1,5)
/design 5 Case vs 1
Case by Letter 5 1 vs within
Letter vs within.
Table 13.8 Analysis of variance with one fixed and
one random variable^2
Source df SS MS F
Case 1 240.25 240.250 5.05
Letter 4 1514.94 378.735 47.19
C 3 L 4 190.30 47.575 5.93
Error 90 722.30 8.026
Total 99 2667.79
p,.05