Statistical Methods for Psychology

estimated by the numerator and denominator in an Fstatistic. Rather than providing a deri-

vation of expected mean squares, as I have in the past (See Howell, 2007 for that develop-

ment), I will simply present a table showing the expected mean squares for fixed, random,

and mixed models. Here a random model is one in which both factors are random, and is

not often found in the behavioral sciences. A mixed model is one with both a random and a

fixed factor, as we are dealing with here, and they are much more common. (I present the

expected mean squares of completely random models only to be complete.) Notice that for

fixed factors the “variance” for that term is shown as rather than as. The reason for

this is that the term is formed by dividing the sum of squared deviations by the degrees of

freedom. For example,

But since we are treating the levels of the factor that we actually used as the entire popula-

tion of that factor in which we are interested, it is not actually a variance because, as the

parameter, it would have to be divided by the number of levels of A, not the dffor A. This

is not going to make any difference in what you do, but the distinction needs to be made

for accuracy. The variance terms for the random factors are represented as. Thus the

variance of Letter means is and the error variance, which is the variance due to subjects,

which is always considered a random term, is.

If you look at the column for a completely fixed model you will see that the expected

mean squares for the main effects and interaction contain a component due to error and a

single component reflecting differences among the means for the main effect or interaction.

The error term, on the other hand, contains only an error component. So if you form a ratio

of the mean squares for A, B, or ABdivided by MSerrorthe only reason that the expected

value of Fwill depart much from 1 will be if there is an effect for the term in question. (We

saw something like this when we first developed the Fstatistic in Section 11.4.) This means

that for all factors in fixed models MSerroris the appropriate error term.

Look now at the column representing the mixed model, which is the one that applies to

our current example. Leaving aside the test on our fixed effect (A) for a moment, we will

focus on the other two effects. If we form the ratio

that ratio will be significantly different from 1 only if the component for the Beffect ( )

is nonzero. Thus MSerroris an appropriate denominator for the Ftest on B. In this case we

can divide MSLetterby MSerrorand have a legitimate test.

nbs^2 b

E(F)=Ea

MSB

MSerror

b=

s^2 e 1 nbs^2 b

s^2 e

s^2 e

s^2 b

s^2

u^2 a=

aa

2

j

a 21

u^2 s^2

Section 13.8 Expected Mean Squares and Alternative Designs 433

Table 13.7 Expected mean squares for fixed, random, and mixed models

Fixed Random Mixed
Afixed Arandom Afixed
Source Bfixed Brandom Brandom

A
B
AB

Error

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

s^2 e 1 nas^2 b

s^2 e 1 ns^2 ab

s^2 e

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

s^2 e 1 ns^2 ab 1 nas^2 b

s^2 e 1 ns^2 ab

s^2 e

s^2 e 1 nbu^2 a

s^2 e 1 nau^2 b

s^2 e 1 nu^2 ab

s^2 e